# Self-Similar Solutions to the Curvature Flow and its Inverse on the
2-dimensional Light Cone

Fábio Nunes da Silva111 Universidade de Brasília, Department of Mathematics,
70910-900, Brasília-DF, and Universidade Federal do Oeste da Bahia, Brazil,
<EMAIL_ADDRESS>Supported by the Universidade Federal do Oeste da
Bahia during his graduate program at the Universidade de Brasília. Keti
Tenenblat222 Universidade de Brasília, Department of Mathematics, 70910-900,
Brasília-DF, Brazil<EMAIL_ADDRESS>Partially supported by CNPq Proc.
312462/2014-0, Ministry of Science and Technology, Brazil and CAPES/Brazil-
Finance Code 001.

###### Abstract

We show that the solutions to the curvature flow (CF) for curves on the
2-dimensional light cone are in correspondence with the solutions to the
inverse curvature flow (ICF). We prove that the ellipses and the hyperboles
are the only curves that evolve under homotheties. The ellipses are the only
closed ones and they are ancient solutions. We show that a spacelike curve on
the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its
curvature (resp. inverse of its curvature) differs by a constant $c$ from
being the inner product between its tangent vector field and a fixed vector
$v$ of the 3-dimensional Minkowski space. The curve is a soliton solution when
$c=0$. We prove that, for each vector $v$ there exist a 2-parameter family of
self-similar solutions to the CF and to the ICF, on the light cone. Moreover,
at each end of such a curve the curvature is either unbounded or it tends to
$0$ or to the constant $c$. Explicitly given soliton solutions are included
and some self-similar solutions on the light cone, are visualized.
_Keywords_ : Curvature Flow, Inverse Curvature FLow, Light Cone, self-similar
solutions, soliton solutions, ancient solutions.
_Mathematics Subject Classification 2010_ : 53C44, 53C50, 37E35.

##  1 Introduction

We consider the 3-dimensional Minkowski space as
$\mathbb{R}_{1}^{3}=(\mathbb{R}^{3},\langle,\rangle)$, where $\mathbb{R}^{3}$
is the 3-dimensional vector space and $\langle,\rangle$ is the Minkowski
metric defined by $\langle u,v\rangle=-u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}.$ We
define the light cone as the lightlike surface
$Q^{2}:=\\{p=(p_{1},p_{2},p_{3})\in\mathbb{R}_{1}^{3}\setminus\\{0\\}:\langle
p,p\rangle=0\\}.$

Let $\displaystyle X:I\subset\mathbb{R}\rightarrow
Q^{2}\subset\mathbb{R}_{1}^{3}$ be a spacelike curve parametrized by arc
length $s$. The curve $X$ is characterized by a trihedrom
$\\{X(s),T(s),Y(s)\\}$, $s\in I$, where $X^{\prime}(s)=T(s)$ is the unit
tangent vector field and $Y(s)$ is the unique lightlike vector field
orthogonal to $T(s)$, such that $\langle X(s),Y(s)\rangle=1$. The curvature
$k(s)$ of $X(s)$, at $s\in I$, is defined by

$T^{\prime}(s)=k(s)X(s)-Y(s).$ (1.1)

The curvature measures how much the curve $X$ bends from a parable, whose
curvature vanishes. When the curvature is a positive (resp. negative) constant
function, then $X(s)$ is a hyperbole (resp. ellipse). Morover we say that
$Y(s)$ is the associated curve to $X(s)$. If $k(s)\neq 0$, $s\in I$, then
$Y(s)$ is a spacelike curve and its curvature is given by
$\tilde{k}(s)=1/k(s)$.

The light cone has two connected components, namely $Q^{2}=Q^{2}_{+}\cup
Q^{2}_{-}$, where $Q_{+}^{2}=\\{p\in Q^{2}:p_{1}>0\\}$ and $Q_{-}^{2}=\\{p\in
Q^{2}:p_{1}<0\\}$. Let $X(s)$ be a spacelike curve on $Q_{+}^{2}$, then it
follows from $\langle X(s),Y(s)\rangle=1$ that $Y(s)$ is a curve on
$Q_{-}^{2}$ , hence $-Y(s)\in Q_{+}^{2}$ for each $s$. Without loss of
generality, we will consider only $Q_{+}^{2}$, since the same results hold for
$Q_{-}^{2}$.

A 1-parameter family of spacelike curves $\displaystyle\hat{X}:I\times
J\rightarrow Q_{+}^{2}\subset\mathbb{R}_{1}^{3}$ is a solution to the
curvature flow (CF) (resp. inverse curvature flow (ICF)), with initial
condition $X(u)$, $u\in I$, if

$\left\\{\begin{array}[]{ll}\displaystyle{<\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot),\hat{Y}^{t}(\cdot)>}=\hat{k}^{t}(\cdot)\\\ \
\hat{X}^{0}(\cdot)=X(\cdot),\end{array}\right.\qquad\mbox{(}resp.)\left\\{\begin{array}[]{ll}\displaystyle{<\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot),\hat{Y}^{t}(\cdot)>}=-\frac{1}{\hat{k}^{t}(\cdot)}\\\ \
\hat{X}^{0}(\cdot)=X(\cdot),\end{array}\right.$ (1.2)

where $\hat{k}^{t}(\cdot)=\hat{k}(\cdot,t)$ is the curvature of
$\hat{X}^{t}(\cdot)=\hat{X}(\cdot,t)$ and
$\hat{Y}^{t}(\cdot)=\hat{Y}(\cdot,t)$ is the lightlike vector field associated
to $\hat{X}^{t}(\cdot)$ for each $t\in J$. When $X(u)$ is a parable i.e.
$k\equiv 0$, then the family $\hat{X}^{t}(u)=X(u)$, for all $t$, is a trivial
solution to the CF.

The definition above was motivated by the curve shortening flow for curves on
a 2-dimensional manifold $M^{2}$, where one considers the inner product
$<\partial/\partial t\;\hat{X}^{t},\hat{N}^{t}>=\hat{k}^{t}$ where $N^{t}$ is
the unit vector field normal to the curve. This is a gradient type flow for
the length functional. The curve shortening flow for curves on the
2-dimensional Euclidean space was studied by several authors in [1], [7] \-
[11] and [13]. In particular, the solutions that evolve by isometries and/or
homotheties were investigated. These are the so called self-similar solutions
of the curve shortening flow and they are called solitons, when the curve
evolves by isometries. The importance of such flows is due to the fact, that
after partial results obtained by several authors, Angenent [3] proved that
for closed curves, under general conditions, the curve shortening flow turns
into a self-similar solution and eventually collapses into a point.
Halldorsson [13] gave a complete description of the self-similar solutions for
curves on the plane. An increased interest also appeared in investigating the
flows for curves on the plane, when one replaces the curvature $k$ by a
function of the curvature such as $1/k^{\alpha}$ in [18] and [19], or
$k^{\alpha}/\alpha$ studied in [2]. Considering the function $-1/k$, Drugan et
al. [6] investigated the curves on the plane that evolve by translations and
Andrews [2] showed that the only simple closed curves on the plane that evolve
by homotheties are circles. There are a few results in [9], [12] and [16] for
curves on an ambient space wich is not the Euclidean plane. Moreover, Dos Reis
and Tenenblat [5] characterized and described all the soliton solutions of the
curve shortening flow on the sphere and Nunes da Silva and Tenenblat [17]
described all the soliton solutions of this flow, on the 2-dimensional
hyperbolic space.

Halldorsson [14] in 2015, considered curves on the Minkowski plane and
classified all the self similar solutions of the curvature flow. He also
proved that this flow is not necessarily length decreasing. In this paper, we
study curvature flows for curves on the light cone
$Q^{2}_{+}\subset\mathbb{R}^{3}_{1}$.

Considering curves whose curvature does not vanish, we show that studying the
solutions to the curvature flow (CF) on the light cone is equivalent to
studying the solutions to the inverse curvature flow (ICF). We prove, in
Theorems 1.5 (resp. 1.6), that a spacelike curve on the light cone is a self-
similar solution to the CF (resp. (ICF)) if, only if, its curvature function
$k$ (resp. $1/k$, for $k\neq 0$) differs by a constant $c$ from being the
inner product between its tangent vector field and a fixed vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$.

We investigate the self-similar solutions to the flows i.e. the curves that
evolve by isometries and/or homotheties of $Q^{2}_{+}$. When the constant
$c=0$, then the curve evolves only by isometries and it is called a soliton
solution. We will show that curves whose curvature is a non zero constant
(ellipses and hyperboles) are the only curves that evolve by homotheties. The
ellipses are the only closed ones and they are ancient solutions (see [4]),
i.e. they evolve for time $t\in(-\infty,A)$.

We prove that the self-similar flows are characterized in terms of a system of
ordinary differential equations, which admits initial condition on three
disjoint sets. We prove existence results, which show that for each vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$, there exists a 2-parameter family of
self-similar solutions to the CF (and consequently to the ICF) on $Q^{2}_{+}$.
There are three classes of such solutions which are associated to the type of
the vector $v$. Moreover, considering non trivial solutions to the CF we prove
that the curvature may vanish at most on two points and therefore the
corresponding solutions to the ICF may have at most three connected
components. We study the behaviour of the curvature function at each end of
the maximal interval of definition of the curve.

In what follows we state our main results. We start establishing the
correspondence between the CF and the ICF on $Q_{+}^{2}$.

######  Remark 1.1.

Let $X:I\rightarrow Q_{+}^{2}$, $u\in I$, be a spacelike curve and let $Y(u)$
be the curve associated to $X$. Let $\hat{X}^{t}(u)=\hat{X}(u,t)$,
$\hat{X}:I\times J\rightarrow Q_{+}^{2}$, $(u,t)\in I\times J$, $0\in J$ be a
1-parameter family of curves with non vanishing curvature for all $t$ and let
$\hat{Y}^{t}(\cdot)=\hat{Y}(\cdot,t)$ be the associated curve to
$\hat{X}^{t}(\cdot)$. For each $t$, $\hat{Y}^{t}$ is a curve on $Q^{2}_{-}$.
Moreover, $<\hat{X}(u,t),\hat{Y}(u,t)>=1$. Taking the derivative with respect
to $t$ and considering the curves on $Q^{2}_{+}$, we conclude that
$\hat{X}^{t}(\cdot)$ is a solution to the CF, with initial condition $X(u)$
if, and only if, $-\hat{Y}(\cdot,-t)$ is a solution to the ICF, with initial
condition $-Y(u)$.

We investigate solutions to the CF and the ICF that evolve by homotheties
and/or isometries of $Q_{+}^{2}$. We remark that an isometry of $Q_{+}^{2}$ is
an element of the Lie group $O_{1}(3)$, acting on $\mathbb{R}^{3}_{1}$, that
preserves $Q_{+}^{2}$. Let $\displaystyle\hat{X}:I\times J\rightarrow
Q_{+}^{2}$ be a solution to the CF (resp. ICF) on $Q_{+}^{2}$, with initial
condition $\displaystyle X:I\rightarrow Q_{+}^{2}$. The curve $X$ is a self-
similar solution to the CF (resp. ICF) if $\hat{X}^{t}(s)=f(t)M(t)X(s)$, where
$f(t)>0$ is a smooth map with $f(0)=1$ and $M(t)$, $t\in J$ is a family of
isometries of $Q_{+}^{2}$, such that $M(0)=Id$ is the identity map. If
$f(t)\equiv 1$ for all $t\in J$, then $X$ is a soliton solution to the CF
(resp. ICF). Our next two theorems classify the solutions to the CF and ICF
that evolve by homotheties on $Q^{2}_{+}$.

######  Theorem 1.2.

Let $X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized by arc length
$s$ with curvature $k(s)\not\equiv 0$ and let
$\hat{X}(s,t)=f(t)X(s),\,(s,t)\in I\times J$, $f(t)>0,\;f(0)=1$ be an
evolution of $X$ by homotheties. The family $\hat{X}(s,t)$ is a solution to
the CF if, and only if, the curvature $k$ of $X$ is constant,
$f(t)=\sqrt{2kt+1}$ and $\hat{k}(x,t)=k/(2kt+1)$. In particular

  1. 1.

If $k<0$ then $X$ is an ellipse of $Q_{+}^{2}$, with curvature $k$ and it is
an ancient solution, with $J=(-\infty,-\frac{1}{2k})$. At $t=-\frac{1}{2k}$,
$\hat{X}^{t}$ collapses into the origin of $R^{3}_{1}$.

  2. 2.

If $k>0$ then $X$ is a hyperbole of $Q_{+}^{2}$, with curvature $k$, and
$J=(-\frac{1}{2k},+\infty)$.

One can see, from Theorem 1.2, that the curvature flow on $Q_{+}^{2}$ is not
always a curve shortening flow, as it occurs on the Minkowski plane (see
[14]). In fact, if $\hat{X}(s,t)=f(t)X(s)$, $f(t)>0,\,f(0)=1$, is a solution
to the CF on $Q_{+}^{2}$, then the arc length of $X^{t}(s)$ is given by
$h(t)=f(t)(s_{1}-s_{0})$, for $s_{0}<s_{1}$, and
$h^{\prime}(t)={k}/({2kt+1})\,(s_{1}-s_{0})$. Therefore, when $X(s)$ is an
ellipse (resp. hyperbole) the arc length of $X^{t}(s)$ decreases (resp.
increases) along the flow.

######  Theorem 1.3.

Let $X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized by arc length
$s$ whose curvature $k(s)$ does not vanish and let
$\hat{X}(s,t)=f(t)X(s),\,(s,t)\in I\times J$, $f(t)>0,\;f(0)=1$ be an
evolution of $X$ by homotheties. The family $\hat{X}(s,t)$ satisfy the ICF if,
and only if, the curvature $k$ of $X$ is constant,
$f(t)={1}/{\sqrt{{2}t/k+1}}$ and $\hat{k}(s,t)=2t+k$. In particular

  1. 1.

If $k<0$, then $X$ is an ellipse of $Q_{+}^{2}$, with curvature $k$. It is an
ancient solution with $J=\left(-\infty,-\frac{k}{2}\right)$, evolving from the
origin of $\mathbb{R}^{3}_{1}$.

  2. 2.

If $k>0$, then $X$ is a hyperbole of $Q_{+}^{2}$, with curvature $k$, and
$J=\left(-\frac{k}{2},+\infty\right)$.

######  Corollary 1.4.

The ellipses are the only closed curves on the light cone that evolve by
homotheties, along the curvature flow or the inverse curvature flow. They are
ancient solutions that collapse into $0\in\mathbb{R}^{3}_{1}$ along the CF and
they evolve from $0\in\mathbb{R}^{3}_{1}$ along the ICF.

The following two results provide a characterization of the self-similar
solutions to the CF and to the ICF.

######  Theorem 1.5.

Let $\displaystyle X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized
by arc length $s\in I$. Then $X$ is a self-similar solution to the curvature
flow on $Q_{+}^{2}$ if, and only if, there exist a vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and a constant $c\in\mathbb{R}$ such
that

$c+\langle T(s),v\rangle=k(s),$ (1.3)

where $T$ is the unit tangent vector field and $k$ is the curvature of $X$. In
particular, $X$ is a soliton solution to the CF whenever $c=0$.

######  Theorem 1.6.

Let $\displaystyle X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized
by arc length $s$, such that $k(s)\neq 0$ for all $s\in I$. Then $X$ is a
self-similar solution to the inverse curvature flow on $Q_{+}^{2}$ if, and
only if, there exist a vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and
$c\in\mathbb{R}$ such that

$c+\langle T(s),v\rangle=\frac{1}{k(s)},$ (1.4)

where $T$ is the unit tangent vector field and $k$ is the curvature of $X$. In
particular, $X$ is a soliton solution to the ICF whenever $c=0$.

Observe that a parable, an ellipse or a hyperbole is a self-similar solution
to the CF satisfying also (1.3). In fact, considering the tangent vector field
$T$, since it is a planar curve, there exits a vector
$v\in{\mathbb{R}^{3}_{1}}\setminus\\{0\\}$ such that $\displaystyle\langle
T,v\rangle=0$, hence (1.3) is satisfied for the constant $c=k$. Similarly, by
considering $c=1/k$, an ellipse or a hyperbole satisfy (1.4) hence it is a
self-similar solution to the ICF. These curves will be called trivial
solutions to the CF and to the ICF.

As a consequence of the characterizations given by Theorems 1.5 and 1.6, we
can show that obtaining self-similar solutions to the CF and to the ICF
correspond to obtaining solutions to systems of ODEs (see (2.13), (2.15) and
Propositions 2.2, 2.3 and 2.5). A long sequence of lemmas, providing
properties of the solutions of these systems, will prove the following
existence theorem.

######  Theorem 1.7.

For any $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and $c\in\mathbb{R}$, there
is a 2-parameter family of curves $X$, which are non-trivial self-similar
solutions (soliton solutions when $c=0$) to the CF, on the 2-dimensional light
cone $Q^{2}_{+}$. There are three classes of such solutions corresponding to
each type of the vector $v$. Moreover, the curvature function of $X$ has at
most two zeros and at each end, the curvature function is either unbounded or
it tends to one of the following constants $\\{c,0\\}$. Each curve $-Y$
associated to $X$ is a self-similar solution to the ICF and it has at most
three connected components.

The proofs of the theorems stated above are given in Section 2, where we also
provide explicit soliton solutions. In Section 3, some self-similar solutions
to the CF and to the ICF, on the light cone, are visualized.

##  2 Proof of the main results

We start by providing some properties of a self-similar evolution of a curve
on $Q^{2}_{+}$. In order to do so, we need the expression of the curvature
function $k$ of a curve $X$ on the light cone, parametrized by an arbitrary
parameter. The curvature is given by (see [15])

$k=\frac{(\langle X^{\prime},X^{\prime\prime}\rangle)^{2}-\langle
X^{\prime},X^{\prime}\rangle\langle
X^{\prime\prime},X^{\prime\prime}\rangle}{2(\langle
X^{\prime}X^{\prime}\rangle)^{3}}.$ (2.1)

######  Proposition 2.1.

Let $X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized by arc length
$s$, with curvature $k(s)$ and associated curve $Y(s)$. Let
$\hat{X}(s,t)=f(t)M(t)X(s),\,(s,t)\in I\times J$ be a self-similar evolution
of $X$. Then, for each $t\in J$,

$\hat{T}^{t}(s)=M(t)T(s),\hskip
14.22636pt\hat{Y}^{t}(s)=\frac{1}{f(t)}M(t)Y(s)\hskip 14.22636pt,\hskip
14.22636pt\hat{k}^{t}(s)=\frac{k(s)}{f^{2}(t)},$ (2.2)

where $\hat{T}^{t}(s)$ is the unit vector field tangent to $\hat{X}^{t}(s)$,
$\hat{Y}^{t}(s)$ is the associated curve to $\hat{X}^{t}(s)$ and
$\hat{k}^{t}(s)$ is the curvature of $\hat{X}^{t}(s)$.

###### Proof.

It follows from the expression of $\hat{X}$ that $\hat{T}^{t}(s)=M(t)T(s)$.
Moreover, considering $\hat{Y}(s)=M(t)Y(s)/f(t)$, then
$\langle\hat{Y}^{t}(s),\hat{Y}^{t}(s)\rangle=0$,
$\langle\hat{Y}^{t}(s),\hat{T}^{t}(s)\rangle=0$ and
$\langle\hat{Y}^{t}(s),\hat{X}^{t}(s)\rangle=1$. Hence $\hat{Y}^{t}(s)$ is the
curve associated to $\displaystyle\hat{X}^{t}(s)$. We conclude from (2.1) that

$\hat{k}^{t}(s)=-\frac{\langle
M(t)T^{\prime}(s),M(t)T^{\prime}(s)\rangle}{2f^{2}(t)}=\frac{k(s)}{f^{2}(t)},$

where in the last equality we used $\left\langle
M(t)T^{\prime}(s),M(t)T^{\prime}(s)\right\rangle=-2k(s)$, which follows from
(1.1). ∎

Let $X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized by arc length
$s$ such that $k(s)\neq 0$ for all $s$. It follows from Remark 1.1 and
Proposition 2.1 that $\hat{X}(s,t)=f(t)M(t)X(s)$ satisfy the CF in $Q_{+}^{2}$
if, and only if, $\displaystyle-\hat{Y}(s,-t)=-\frac{1}{f(-t)}M(-t)Y(s)$
satisfy the ICF in $Q_{+}^{2}$ i.e. $X$ is a self-similar solution to the CF
if, and only if, $-Y(s)$ is a self-similar solution to the ICF. In this
context, it is sufficient to study the behaviour of the self-similar solution
to the CF.

###### Proof of Theorem 1.2.

Consider $\hat{X}^{t}(s)=f(t)X(s)$, $(s,t)\in I\times J$, where $f(t)>0$ and
$f(0)=1$. Then Proposition 2.1 implies that $\hat{Y}^{t}(s)=Y(s)/f(t)$ and
$\hat{k}^{t}(s)=k(s)/f^{2}(t)$. If $\hat{X}$ is a solution to the CF then

$\frac{k(s)}{f^{2}(t)}=\left\langle\frac{\partial}{\partial
t}\hat{X}^{t}(s),\hat{Y}^{t}(s)\right\rangle=\frac{f^{\prime}(t)}{f(t)}.$

Therefore, $f^{\prime}(t)f(t)=k(s)$ for all $(s,t)$ and hence $k$ is constant
and $f(t)=\sqrt{2kt+1}$, since $f(0)=1$. Hence, the curve is an ellipse (resp.
hyperbole) if $k<0$ (resp. $k>0$) and the domain $J$ is determined by the
function $f$ and the sign of $k$. Moreover, since $f(-k/2)=0$, it follows that
$X^{t}$ collapses into the origin of $\mathbb{R}^{3}_{1}$.

Conversely, if $k$ is constant, by considering $f(t)=\sqrt{2kt+1}$ and
$\hat{X}^{t}(s)=f(t)X(s)$, a straightforward computation using (2.2) shows
that $\hat{X}$ is a solution to the CF.

∎

The proof of Theorem 1.3 is similar to the proof of Theorem 1.2 and it will be
omitted.

###### Proof of Theorem 1.5.

Let $\hat{X}(s,t)=f(t)M(t)X(s)$ be a self-similar evolution of $X$ that
satisfy the CF. Taking the derivative of $\hat{X}$ with respect to $t$, it
follows from (2.2) that

$\frac{k(s)}{{f^{2}(t)}}=\hat{k}(s,t)=\left\langle\frac{\partial}{\partial
t}\hat{X}(s,t),\hat{Y}(s,t)\right\rangle=\frac{f^{\prime}(t)}{f(t)}+\left\langle
M^{\prime}(t)X(s),M(t)Y(s)\right\rangle.$

Therefore, $\displaystyle k(s)=f(t)f^{\prime}(t)+{f^{2}(t)}\left\langle
M^{\prime}(t)X(s),M(t)Y(s)\right\rangle$ for each $(s,t)\in I\times J$ and
hence at $t=0$

$k(s)=f^{\prime}(0)+\langle M^{\prime}(0)X(s),Y(s)\rangle.$

$M^{\prime}(0)$ is an element of the Lie algebra $\mathfrak{o}_{1}(3)$ of the
Lie group $O_{1}(3)$. We consider a basis of $\mathfrak{o}_{1}(3)$

$\displaystyle A_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&-1&0\end{array}\right),\,\,\,A_{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&1\\\
1&-1&0\end{array}\right)\,\,\,\,\text{and}\,\,\,\,\,A_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\end{array}\right).$

Then $M^{\prime}(0)=a_{1}A_{1}+a_{2}A_{2}+a_{3}A_{3}$. Denoting
$X(s)=(x_{1}(s),x_{2}(s),x_{3}(s))$, we have

$\displaystyle
M^{\prime}(0)X(s)=\left(a_{3}x_{2}(s)+a_{2}x_{3}(s),\,a_{3}x_{1}(s)+(a_{1}+a_{2})x_{3}(s),\,a_{2}x_{1}(s)-(a_{1}+a_{2})x_{2}(s)\right).$

Since $X(s)\times Y(s)=T(s)$, it follows that

$\langle M^{\prime}(0)X(s),Y(s)\rangle=\langle X(s)\times
Y(s),(-a_{1}-a_{2},-a_{2},a_{3})\rangle=\langle T(s),v\rangle.$

Therefore, taking $v=(-a_{1}-a_{2},-a_{2},a_{3})$ and $f^{\prime}(0)=c$, we
have $k(s)=c+\langle T(s),v\rangle$.

Conversely, let $X$ be a spacelike curve on
$Q_{+}^{2}\subset\mathbb{R}_{1}^{3}$ parametrized by arc length $s$ such that
$k(s)=c+\langle T(s),v\rangle$ for a vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and $c\in\mathbb{R}$. Without loss of
generality, up to isometries of $Q_{+}^{2}$, we can consider $v$ to be a
multiple of $e_{1}=(1,0,0)$ if $v$ is a timelike vector, a multiple of
$e_{2}=(1,1,0)$ if $v$ is a lightlike vector and a multiple of $e_{3}=(0,0,1)$
if $v$ is a spacelike vector. Thus, depending on the type of the vector $v$,
the curvature $k_{i}(s)=c+\langle T(s),v_{i}\rangle$ where $v_{i}=a\,e_{i}$,
$a>0$ and $i=1,2,3$. Now, we define the evolution of $X$ in $Q_{+}^{2}$ to be
$\hat{X}_{i}(s,t)=f(t)M_{i}(t)X(s)$, where

$\displaystyle M_{1}(t)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{lll}1&0&0\\\
0&\cos(\varphi(t))&-\sin(\varphi(t))\\\
0&\sin(\varphi(t))&\cos(\varphi(t))\end{array}\right),\,\,M_{2}(t)=\left(\begin{array}[]{ccc}1+\frac{(\varphi(t))^{2}}{2}&-\frac{(\varphi(t))^{2}}{2}&-\varphi(t)\\\
\frac{(\varphi(t))^{2}}{2}&1-\frac{(\varphi(t))^{2}}{2}&-\varphi(t)\\\
-\varphi(t)&\varphi(t)&1\end{array}\right),$ $\displaystyle M_{3}(t)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{lll}\cosh(\varphi(t))&\sinh(\varphi(t))&0\\\
\sinh(\varphi(t))&\cosh(\varphi(t))&0\\\
0&0&1\end{array}\right),\,\,\,\varphi(t)=\left\\{\begin{array}[]{ll}\displaystyle\frac{a}{c}\log(f(t)),\,\,\text{if}\,\,\,c\neq
0,\\\ at,\,\,\text{if}\,\,c=0,\end{array}\right.$

and $f(t)=\sqrt{2ct+1}$. Taking the derivative with respect to $t$, it follows
from (2.2) that

$\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{Y}_{i}(s,t)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{f^{\prime}(t)}{f(t)}+\left\langle
M^{\prime}_{i}(t)X(s),M_{i}(t)Y(s)\right\rangle.$

A straightforward computation shows that $\displaystyle\left\langle
M^{\prime}_{i}(t)X(s),M_{i}(t)Y(s)\right\rangle=\varphi^{\prime}_{i}(t)\langle
X(s)\times Y(s),w_{i}\rangle$. Thus

$\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{Y}_{i}(s,t)\right\rangle=\frac{f^{\prime}(t)}{f(t)}+\varphi^{\prime}_{i}(t)\langle
T(s),w_{i}\rangle$

for each $i=1,2,3$. If $c\neq 0$, then

$\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{Y}_{i}(s,t)\right\rangle=\frac{f^{\prime}(t)}{cf(t)}\left(c+a\langle
T(s),w_{i}\rangle\right)=\frac{k_{i}(s)}{{f^{2}(t)}}).$

It follows from (2.2) that $\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{Y}_{i}(s,t)\right\rangle=\frac{k_{i}(s)}{{f^{2}(t)}})=\hat{k_{i}}(s,t)$.

When $c=0$, $k_{i}(s)=\langle T(s),v_{i}\rangle$, $f(t)\equiv 1$ and
$\varphi_{i}(t)=at$. Hence the evolution is composed only by isometries and

$\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{Y}_{i}(s,t)\right\rangle=a\langle
T(s),w_{i}\rangle=k_{i}(s)=\hat{k}_{i}(s,t),$

where the last equality follows from fact that isometries preserve curvature.

∎

The proof of Theorem 1.6 is analogue to the proof of Theorem 1.5 and it will
be omitted. It follows from Theorem 1.5 that the investigation of the self-
similar solutions to the CF in $Q_{+}^{2}$ reduces to studying the curves that
satisfy equation (1.3) for some vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and $c\in\mathbb{R}$. Up to
isometries of $Q_{+}^{2}$ and depending on the type of vector $v$, we consider
$v$ as being $v_{i}=ae_{i}$, where $a>0$, $e_{1}=(1,0,0)$, $e_{2}=(1,1,0)$ or
$e_{3}=(0,0,1)$. Our next result characterizes (1.3) in terms of a system of
differential equations.

######  Proposition 2.2.

Let $X:I\rightarrow Q^{2}_{+}$ be a spacelike curve parametrized by arc length
$s$. Consider the vectors

$e_{1}=(1,0,0),\hskip 8.5359pte_{2}=(1,1,0),\hskip 8.5359pt\text{e}\hskip
8.5359pte_{3}=(0,0,1).$ (2.6)

For each $i\in\\{1,2,3\\}$, define the functions

$\alpha_{i}(s)=\langle X(s),e_{i}\rangle,\qquad\tau_{i}(s)=\langle
T(s),e_{i}\rangle,\qquad\eta_{i}(s)=\langle Y(s),e_{i}\rangle,$ (2.7)

where $T$ is the unit tangent vector and $Y$ is the vector field associated to
$X$. For fixed $a>0$ and $c\in\mathbb{R}$,

$k_{i}(s)=c+a\tau_{i}(s)$

is satisfied for all $s\in I$, i.e. $X$ is a self-similar solution to the CF,
if and only if, the functions $\alpha_{i}(s)$, $\tau_{i}(s)$ and $\eta_{i}(s)$
satisfy the system

$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=[c+a\tau_{i}(s)]\alpha_{i}(s)-\eta_{i}(s),\\\
\eta^{\prime}_{i}(s)=-[c+a\tau_{i}(s)]\tau_{i}(s),\end{array}\right.$ (2.8)

with initial conditions $(\alpha_{i}(0),\tau_{i}(0),\eta_{i}(0))\neq(0,0,0)$
satisfying

$2\alpha_{i}(0)\eta_{i}(0)+\tau^{2}_{i}(0)=\left\\{\begin{array}[]{lll}-1,\hskip
8.5359pt\text{se}\hskip 8.5359pti=1,\\\ 0,\hskip 8.5359pt\text{se}\hskip
8.5359pti=2,\\\ 1,\hskip 8.5359pt\text{se}\hskip
8.5359pti=3.\end{array}\right.$ (2.9)

For such a solution, $2\alpha_{i}(s)\eta_{i}(s)+\tau^{2}_{i}(s)=-1$ (resp. $0$
or $1$) when $i=1$ (resp. $2$ or $3$), for all $s$.

###### Proof.

The vector fields $X$, $T$ and $Y$ satisfy the following equations (see [15])

$\left\\{\begin{array}[]{lll}X^{\prime}(s)=T(s),\\\
T^{\prime}(s)=k(s)X(s)-Y(s),\\\ Y^{\prime}(s)=-k(s)T(s),\end{array}\right.$
(2.10)

for all $s\in I$. Thus, taking the inner product with $e_{i}$, we obtain

$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=k_{i}(s)\alpha_{i}(s)-\eta_{i}(s),\\\
\eta^{\prime}_{i}(s)=-k_{i}(s)\tau_{i}(s).\end{array}\right.$ (2.11)

Assume that $k_{i}(s)=c+a\tau_{i}(s)$ for all $s\in I$, then (2.8) is
satisfied. Moreover, since $\\{X(s),T(s),Y(s)\\}$ is a basis for
$\mathbb{R}_{1}^{3}$ for each $s$, it follows from (2.7) that
$e_{i}=\eta_{i}(s)X(s)+\tau_{i}(s)T(s)+\alpha_{i}(s)Y(s)$ i.e. $\langle
e_{i},e_{i}\rangle=2\alpha_{i}(s)\eta_{i}(s)+\tau^{2}_{i}(s)$ for all $s\in
I$. In particular, at $s=0$ we have that (2.9) holds.

Conversely, suppose that the functions $\alpha_{i}(s)$, $\tau_{i}(s)$ and
$\eta_{i}(s)$ satisfying (2.8) and (2.9) for each $i\in\\{1,2,3\\}$. Since
(2.11) holds, then $\left[c+a\tau_{i}(s)-k_{i}(s)\right]\alpha_{i}(s)=0$ and
$\left[c+a\tau_{i}(s)-k_{i}(s)\right]\tau_{i}(s)=0$, for all $s\in I$. Assume
by contradiction that $c+a\tau_{i}(s)-k_{i}(s)\neq 0$ on some interval
$J\subset I$. Then $\alpha_{i}(s)=\langle X(s),e_{i}\rangle=0$,
$\tau_{i}(s)=\langle T(s),e_{i}\rangle=0$ for all $s\in J$ and hence $\langle
T^{\prime}(s),e_{i}\rangle=\langle Y(s),e_{i}\rangle=0$, which is a
contradiction since $e_{i}\neq 0$. Therefore, $k_{i}(s)=c+a\tau_{i}(s)$ for
all $s\in I$ and $i\in\\{1,2,3\\}$. ∎

Our next proposition shows that a solution of the system (2.8), with initial
conditions satisfying (2.9), provides a self-similar solution to the CF in
$Q_{+}^{2}$.

######  Proposition 2.3.

Given a solution $\alpha(s),\tau(s),\eta(s):I\rightarrow\mathbb{R}$ of the
system (2.8) on some interval $J$ for fixed $a>0$, $c\in\mathbb{R}$ and
initial conditions $(\alpha(0),\tau(0),\eta(0))\neq(0,0,0)$ satisfying
$2\alpha(0)\eta(0)+\tau^{2}(0)=-1$ (resp. $0$ and $1$), there exists smooth
spacelike curve $X:I\rightarrow Q_{+}^{2}$, parametrized by arc length $s$,
which is a self-similar solution to the CF, with curvature $k(s)=c+a\tau(s)$,
such that its tangent and associated vector fields $T$ and $Y$ satisfy

$\alpha(s)=\langle X(s),e\rangle,\hskip 14.22636pt\tau(s)=\langle
T(s),e\rangle\hskip 14.22636pt\text{\rm e}\hskip 14.22636pt\eta(s)=\langle
Y(s),e\rangle,$ (2.12)

where $e=(1,0,0)$ (resp. $e=(1,1,0)$ and $e=(0,0,1)$).

###### Proof.

Given a solution of (2.8), define $k(s)=c+a\tau(s)$. Up to isometries of
$Q_{+}^{2}$, there exists a unique curve $\displaystyle X:I\rightarrow
Q_{+}^{2}$, whose curvature is $k(s)$ i.e. $X(s)$ and its tangent and
associated vector fields $T(s)$ and $Y(s)$ satisfy (2.10). The curve $X(s)$ is
uniquely determined by the initial conditions $X(0)$, $T(0)$ and $Y(0)$, that
can be chosen such that $\eta(0)X(0)+\tau(0)T(0)+\alpha(0)Y(0)=e,$ where
$e=(1,0,0)$ (resp. $e=(1,1,0)$ and $e=(0,0,1)$). A straightforward computation
shows that (2.8) and (2.10) imply
$\frac{d}{ds}\left(\eta(s)X(s)+\tau(s)T(s)+\alpha(s)Y(s)\right)=0$ for all
$s$. Therefore, (2.12) is satisfied and Theorem 1.5 implies that $X$ is a
self-similar solution to the CF. ∎

Propositions 2.2 and 2.3, show that investigating the self-similar solutions
to the CF on $Q_{+}^{2}$ is equivalent to studying the solutions
$\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of the system

$\left\\{\begin{array}[]{lll}\alpha^{\prime}(s)=\tau(s),\\\
\tau^{\prime}(s)=[c+a\tau(s)]\alpha(s)-\eta(s),\\\
\eta^{\prime}(s)=-[c+a\tau(s)]\tau(s),\end{array}\right.$ (2.13)

for given constants $c\in\mathbb{R}$, $a>0$ and initial condition $\psi(0)\in
H\cup C\cup S$, where

$\begin{array}[]{l}H:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:2\alpha\eta+\tau^{2}=-1\,\,\text{and}\,\,\alpha<0\\},\\\
C:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}\setminus\\{0\\}:2\alpha\eta+\tau^{2}=0\,\,\text{and}\,\,\alpha\leq
0\\},\\\
S:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:2\alpha\eta+\tau^{2}=1\\}.\end{array}$
(2.14)

These are disjoint sets and if the initial condition $\psi(0)\in H$ (resp. $C$
or $S$) then the solution $\psi(s)$, defined on the maximal interval $I$, will
be contained in $H$ (resp. $C$ or $S$).

######  Remark 2.4.

Let $\displaystyle X:I\rightarrow Q_{+}^{2}$ be a spacelike curve parametrized
by arc length $s$ given by $X(s)=(x_{1}(s),x_{2}(s),x_{3}(s))$ and let
$Y(s)=(y_{1}(s),y_{2}(s),y_{3}(s))$ be the vector field associated to $X(s)$.
The function $\alpha(s)$ defined by (2.12) has the following geometric
interpretation.

  * •

If $e=(1,0,0)$ (timelike vector), then $\alpha(s)=-x_{1}(s)<0$ and
$\eta(s)=-y_{1}(s)>0$ for all $s\in I$. Moreover, $\alpha(s)$ is the Euclidean
height function with respect to the vector $(-1,0,0).$

  * •

If $e=(1,1,0)$ (lightlike vector), then $\alpha(s)=-x_{1}(s)+x_{2}(s)\leq 0$
and $\eta(s)=-y_{1}(s)+y_{2}(s)\geq 0$ for all $s\in I$. Moreover,
$\alpha(s)/\sqrt{2}$ is the Euclidean height function with respect to the
vector $(-1/\sqrt{2},1/\sqrt{2},0).$

  * •

If $e=(0,0,1)$ (spacelike vector), then $\alpha(s)=x_{3}(s)$ for all $s\in I$.
Moreover, $\alpha(s)$ is the Euclidean height function (with sign) with
respect to the vector $(0,0,1).$

We can prove results similar to Propositions 2.2 and 2.3 for the ICF. In fact,
using the same arguments, we have the following

######  Proposition 2.5.

A spacelike curve parametrized by arc length $s$, $\tilde{X}:I\rightarrow
Q_{+}^{2}$, $s\in I$, whose curvature $\tilde{k}(s)\neq 0$, is a self-similar
solution to the ICF, i.e. $c+\langle\tilde{T},v\rangle=1/\tilde{k}$ for a
vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ and $c\in\mathbb{R}$ if, and
only if, $\tilde{X}$ is determined by a solution
$\tilde{\psi}(s)=(\tilde{\alpha}(s),\tilde{\tau}(s),\tilde{\eta}(s))$ defined
on the maximal interval $I$ of the system

$\left\\{\begin{array}[]{lll}\displaystyle\tilde{\alpha}^{\prime}(s)=\tilde{\tau}(s),\\\
\displaystyle\tilde{\tau}^{\prime}(s)=\frac{\tilde{\alpha}(s)}{c+a\tilde{\tau}(s)}-\tilde{\eta}(s),\\\
\displaystyle\tilde{\eta}^{\prime}(s)=-\frac{\tilde{\tau}(s)}{c+a\tilde{\tau}(s)},\end{array}\right.$
(2.15)

for a constant $a>0$ and initial condition $\psi(0)\in H\cup C\cup S$, where
$\tilde{\alpha}(s)=\langle\tilde{X}(s),e\rangle$,
$\tilde{\tau}(s)=\langle\tilde{T}(s),e\rangle$,
$\tilde{\eta}(s)=\langle\tilde{T}(s),e\rangle$ ,
$e\in\\{(1,0,0),(1,1,0),(0,0,1)\\}$ and $v=ae$.

From now on, unless explicitly stated, we will restrict ourselves to studying
the self-similar solutions to the CF on $Q^{2}_{+}$. The proof of Theorem 1.7
will follow from a long series of lemmas on the properties of the solutions of
the system of differential equations (2.13). Our first lemma will be used
repeatedly and it provides two relations between the functions $\alpha(s)$,
$\tau(s)$ and $\eta(s)$, which depend on the initial condition.

######  Lemma 2.6.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$, $s\in I\subset\mathbb{R}$ be a
solution of (2.13), $a>0$, $c\in\mathbb{R}$ and initial condition $\psi(0)\in
H\cup C\cup S$. Then for all $s\in I$

$[c+a\tau(s)]\alpha^{2}(s)-\alpha(s)\tau^{\prime}(s)+\frac{\tau^{2}(s)}{2}=\displaystyle\left\\{\begin{array}[]{ccc}\displaystyle-\frac{1}{2}&\text{if}&\psi(0)\in
H,\\\ \displaystyle 0&\text{if}&\psi(0)\in C,\\\
\displaystyle\frac{1}{2}&\text{if}&\psi(0)\in S\end{array}\right.$ (2.16)

and

$\eta^{2}(s)+\eta(s)\tau^{\prime}(s)+\frac{[c+a\tau(s)]\tau^{2}(s)}{2}=\displaystyle{\left\\{\begin{array}[]{lll}\displaystyle-\,\frac{[c+a\tau(s)]}{2}&\text{if}&\psi(0)\in
H,\\\ 0&\text{if}&\psi(0)\in C,\\\
\displaystyle\frac{[c+a\tau(s)]}{2}&\text{if}&\psi(0)\in
S.\end{array}\right.}$ (2.17)

###### Proof.

Multiplying the second equation of (2.13) by $\alpha(s)$ (resp. $\eta(s)$) and
using the fact that $2\alpha(s)\eta(s)+\tau^{2}(s)=-1$ (resp. $0,1$) in $H$
(resp. $C,$ $S$), we obtain (2.16) (resp. (2.17)). ∎

In our next lemma, we study the solutions of (2.13) with constant function
$\tau(s)$.

######  Lemma 2.7.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non null solution of (2.13)
defined on the maximal interval $I$, $a>0$, $c\in\mathbb{R}$ and initial
condition $\psi(0)\in H\cup C\cup S$. The function $\tau(s)=b$, $\forall s\in
I$, where $b\in\mathbb{R}$ if, and only if, $b\in\\{-1,0,1\\}$ and
$I=\mathbb{R}$. Moreover,

  * i)

if $b=0$, then $\psi(s)=(\alpha_{0},0,c\alpha_{0})$ is a singular solution of
(2.13) and $\psi(s)\in H$ (resp. $C$, $S$), for all $s\in\mathbb{R}$, when
$c<0$ (resp. $c=0$, $c>0$);

  * ii)

if $b^{2}=1$, then $a^{2}=c^{2}$ and $\psi(s)=(\pm s+\alpha_{0},\pm 1,0)\in S$
for all $s\in\mathbb{R}.$

###### Proof.

It folows from (2.13) that $\tau(s)=b$, if and only if
$\alpha(s)=bs+\alpha_{0}$, $\eta(s)=(c+ab)(bs+\alpha_{0})$ and $2(c+ab)b=0$.
Since $2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$, where $\gamma\in\\{-1,0,1\\}$
then if $b=0$, then i) holds and if $b\neq 0$ then $c+ab=0$, $\eta=0$,
$b^{2}=1$ and $a^{2}=c^{2}$, which proves ii). ∎

Lemma 2.7 shows that the solutions $\psi(s)$ of (2.13), for which $\tau(s)$ is
constant, are explicitly given and they correspond to the conic sections of
$Q_{+}^{2}$, with constant curvature $k(s)=c+ab$. In this context, we define a
trivial solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (2.13) when
$\tau(s)$ is a constant function. From now on, we will study only non trivial
solutions $\psi(s)$.

Our next three lemmas analyze the critical points of the functions
$\alpha(s)$, $\tau(s)$ and $\eta(s)$.

######  Lemma 2.8.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I$, $a>0$, $c\in\mathbb{R}$ and initial
condition $\psi(0)\in H\cup C\cup S$. If $s_{0}$ is a critical point of
$\alpha(s)$ then it is a local minimum (resp. maximum) point of $\alpha(s)$
if, and only if, $c\alpha(s_{0})-\eta(s_{0})>0$ (resp.
$c\alpha(s_{0})-\eta(s_{0})<0$). Moreover:

  * i)

if $c\geq 0$ and $\psi(0)\in H\cup C$, then $s_{0}$ is a global maximum point
of $\alpha(s)$;

  * ii)

if $c<0$ and $\psi(0)\in H$, then $s_{0}$ is a local minimum (resp. maximum)
point of $\alpha(s)$ if, and only if, $2c\alpha^{2}(s_{0})+1<0$ (resp.
$2c\alpha^{2}(s_{0})+1>0$);

  * iii)

if $c<0$ and $\psi(0))\in C$, then $s_{0}$ is a local minimum (resp. maximum)
point of $\alpha(s)$ if, and only if, $\alpha(s_{0})<0$ (resp.
$\alpha(s_{0})=0$).

###### Proof.

Let $s_{0}$ be a critical point of $\alpha(s)$. It follows from (2.13) that
$\alpha^{\prime\prime}(s_{0})=\tau^{\prime}(s_{0})=c\alpha(s_{0})-\eta(s_{0}).$
If there exists $s_{0}\in I$ such that
$\tau(s_{0})=\tau^{\prime}(s_{0})=\alpha^{\prime\prime}(s_{0})=0$, then
$(\alpha(s_{0}),0,c\alpha(s_{0}))$ is a singular solution of (2.13), which
contradicts the hypothesis. Thus, $\alpha^{\prime\prime}(s_{0})\neq 0$ and the
result holds.

i) When $\psi(0)\in H\cup C$ and $c\geq 0$ we have $\alpha(s_{0})\leq 0$ and
$\eta(s_{0})\geq 0$ for all $s\in I$ and hence
$\alpha^{\prime\prime}(s_{0})=c\alpha(s_{0})-\eta(s_{0})<0$ i.e. $s_{0}$ is a
global maximum point of $\alpha(s).$

ii) If $\psi(0)\in H$ and $c<0$ then $\alpha(s_{0})<0$, $\eta(s_{0})>0$,
$2\alpha(s_{0})\eta(s_{0})=-1$ and
$2\alpha(s_{0})\alpha^{\prime\prime}(s_{0})=2c\alpha^{2}(s_{0})+1.$

iii) If $\psi(0)\in C$ and $c<0$, then $\alpha(s_{0})\leq 0$, $\eta(s_{0})\geq
0$, $2\alpha(s_{0})\eta(s_{0})=0$, $\alpha^{2}(s_{0})+\eta^{2}(s_{0})\neq 0$
and $\alpha^{\prime\prime}(s_{0})=c\alpha(s_{0})-\eta(s_{0}).$ This proves
item iii). ∎

######  Lemma 2.9.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I$, $a>0$, $c\in\mathbb{R}$ and initial
condition $\psi(0)\in H\cup C\cup S$. Let $s_{0}$ be a critical point of
$\eta(s).$ If $c=0$ then $\eta(s)$ is a decreasing function and $s_{0}$ is an
inflection point of $\eta(s)$. If $c\neq 0$ then
$[c+a\tau(s_{0})]\tau(s_{0})=0.$ In this case

  * i)

If $\tau(s_{0})=0$ then $\eta^{\prime\prime}(s_{0})\neq 0$ and $s_{0}$ is a
local minimum (resp. maximum) point of $\eta(s)$ if, and only if,
$-c^{2}\alpha(s_{0})+c\eta(s_{0})>0$ (resp.
$-c^{2}\alpha(s_{0})+c\eta(s_{0})<0$).

  * ii)

If $c+a\tau(s_{0})=0$ then $\eta^{\prime\prime}(s_{0})\neq 0$ and $s_{0}$ is a
local minimum (resp. maximum) point of $\eta(s)$ if, and only if,
$c\eta(s_{0})<0$ (resp. $c\eta(s_{0})>0$).

###### Proof.

Let $s_{0}$ be a critical point of $\eta(s)$, then
$[c+a\tau(s_{0})]\tau(s_{0})=0$. When $c=0$ it follows from (2.13) that
$\eta(s)$ is a decreasing function on $I$.

i) Suppose that $c\neq 0$ and $\tau(s_{0})=0$, then (2.13) implies that
$\eta^{\prime\prime}(s_{0})=-c^{2}\alpha(s_{0})+c\eta(s_{0})$. Moreover,
$\eta^{\prime\prime}(s_{0})\neq 0$, otherwise,
$\psi(s)=(\alpha(s_{0}),0,\eta(s_{0}))$, $s\in\mathbb{R}$ would be a singular
solution of (2.13), which contradicts the hypothesis.

ii) Suppose $c\neq 0$ and $\displaystyle c+a\tau(s_{0})=0$, then (2.13)
implies that $\eta^{\prime\prime}(s_{0})=c\tau^{\prime}(s_{0})=-c\eta(s_{0})$.
Moreover, $\eta^{\prime\prime}(s_{0})\neq 0$. In fact, otherwise
$\alpha(s_{0})\eta(s_{0})=0,$ $\displaystyle a^{2}\tau^{2}(s_{0})=c^{2}$.
Thus, $\psi(s_{0})\in S$, $c^{2}=a^{2}$, $\tau^{2}(s_{0})=1$ and
$\psi(s_{0})=(\alpha(s_{0}),\pm 1,0)$. However, it follows from Lemma 2.7 that
$\psi(s)=(\pm s+\alpha(s_{0}),\pm 1,0)$, $s\in\mathbb{R}$ are trivial
solutions of (2.13), contradicting the hypothesis. ∎

######  Lemma 2.10.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I$, $a>0$, $c\in\mathbb{R}$ and initial
condition $\psi(0)\in H\cup C\cup S$. If $s_{0}$ is a critical point of
$\tau(s)$ then $\tau^{\prime\prime}(s_{0})\neq 0$ and it is a local minimum
(resp. maximum) point of $\tau(s)$ if, and only if,
$\displaystyle\tau(s_{0})[c+a\tau(s_{0})]>0$ (resp.
$\displaystyle\tau(s_{0})[c+a\tau(s_{0})]<0$). In particular:

  * i)

if $c=0$ then $s_{0}$ is a global minimum point of $\tau(s);$

  * ii)

if $\psi(0)\in H\cup C$ then $c+a\tau(s_{0})<0;$

  * iii)

if $c>0$ and $\psi(0)\in H\cup C$ then $s_{0}$ is a global minimum point of
$\tau(s)$.

###### Proof.

Let $s_{0}$ be a critical point of $\tau(s)$, then
$[c+a\tau(s_{0})]\alpha(s_{0})=\eta(s_{0})$. Taking the second derivative of
$\tau(s)$ at $s_{0}$, it follows from(2.13), that
$\tau^{\prime\prime}(s_{0})=2[c+a\tau(s_{0})]\tau(s_{0})$. Assume by
contradiction that $\tau^{\prime\prime}(s_{0})=0$, i.e.
$2[c+a\tau(s_{0})]\tau(s_{0})=0$, then either $\tau(s_{0})=0$ or
$c+a\tau(s_{0})=0$. If $\tau(s_{0})=0$ then $c\alpha(s_{0})=\eta(s_{0})$ and
$\psi(s)=(\alpha(s_{0}),0,c\alpha(s_{0}))$, $s\in\mathbb{R}$ would be a
singular solution of (2.13), which contradicts the hypothesis. If
$c+a\tau(s_{0})=0$ we would have $\eta(s_{0})=0$,
$a^{2}\tau^{2}(s_{0})=c^{2}.$ Thus, $\psi(s_{0})\in S$, $c^{2}=a^{2}$,
$\tau^{2}(s_{0})=1$ and $\psi(s_{0})=(\alpha(s_{0}),\pm 1,0)$. However, it
follows from Lemma 2.7 that $\psi(s)=(\pm s+\alpha(s_{0}),\pm 1,0)$,
$s\in\mathbb{R}$ are trivial solutions of (2.13), contradicting the
hypothesis.

i) Suppose that $c=0$ then $\tau^{\prime\prime}(s_{0})=a\tau^{2}(s_{0})>0$
i.e. $s_{0}$ is a global minimum point of $\tau(s)$.

ii) If $\psi(0)\in H\cup C$ then $\alpha(s_{0})<0$ and $\eta(s_{0})>0$. Thus,
$[c+a\tau(s_{0})]\alpha(s_{0})=\eta(s_{0})>0$ and $c+a\tau(s_{0})<0.$

iii) If $c>0$ and $\psi(0)\in H\cup C$ then it follows from ii) that
$c+\tau(s_{0})<0$ and $\tau(s_{0})<0$. Therefore,
$\tau^{\prime\prime}(s_{0})>0$ and $s_{0}$ is a global minimum point of
$\tau(s).$ ∎

Our next lemma shows that the curvature function $k(s)=c+a\tau(s)$ of a self-
similar solution to the CF has at most two zeros. The importance of the number
of zeros of the curvature is due to the fact that it determines the number of
connected components of the associated self-similar solutions to the ICF.

######  Lemma 2.11.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$,
$c\in\mathbb{R}$ and initial condition $\psi(0)\in H\cup C\cup S$.

  * i)

If $c\geq 0$ then $\tau(s)$ has at most two zeros.

  * ii)

If $c<0$, then $\tau(s)$ has an infinite number of zeros in $I$ if, and only
if, the functions $\alpha(s)$, $\tau(s)$ and $\eta(s)$ have an infinite number
of critical point in $I$.

  * iii)

The function $c+a\tau(s)$ has at most two zeros.

###### Proof.

i) If $c\geq 0$ and $\tau({s})>0$, for $s\in J\subset I$, then
$[c+a\tau(s)]\tau(s)>0$, and it follows from Lemma 2.10 that $\tau(s)$ has no
points of local maximum in $J$. Moreover, if $s_{0}\in I$ satisfies
$\tau(s_{0})=0$ then $s_{0}$ is not a critical point of $\tau$. Hence, if
$\tau^{\prime}(s_{0})>0$ then $\tau(s)$ is increasing on $(s_{0},\omega_{+})$
and if $\tau^{\prime}(s_{0})<0$ then $\tau(s)$ is decreasing
on$(\omega_{-},s_{0})$. Therefore, the function $\tau(s)$ has at most two
zeros.

ii) It follows from (2.13) that the zeros of $\tau(s)$ are critical points of
$\alpha(s)$ and $\eta(s)$. Suppose that $c<0$ then Lemma 2.10 implies that
$\tau(s)$ may have local maximum (resp. minimum) points when $0<a\tau(s)<-c$
(resp. $\tau(s)<0$). Therefore, if $c<0$ then $\tau(s)$ has an infinite number
of zeros if, and only if, $\tau(s)$ has an infinite number of local critical
points.

iii) When $c\leq 0$ and $c+a\tau(s)>0$ for $s\in J\subset I$, then
$[c+a\tau(s)]\tau(s)>0$ and it follows from Lemma 2.10 that $\tau(s)$ has no
local maximum in $J$. Moreover, if $s_{0}\in I$ is such that
$c+a\tau(s_{0})=0$ then $s_{0}$ is not a critical point of $\tau$. Therefore,
when $c\leq 0$, the function $c+a\tau(s)$ has at most two zeros.

Suppose that $c>0$ and $\psi(0)\in H\cup C$, it follows from item iii) of
Lemma 2.10 that $\tau(s)$ has at most one critical point, thus, $c+a\tau(s)$
has at most two zeros. Finally, suppose that $c>0$ and $\psi(0)\in S$ and
assume by contradiction that there exist three consecutive zeros
$s_{1},s_{2},s_{3}\in I$ such that $s_{1}<s_{2}<s_{3}$, $c+a\tau(s_{i})=0$,
for $i=1,23$ and $c+a\tau(s)\neq 0$ for all
$s\in(s_{1},s_{2})\cup(s_{2},s_{3})$. It follows from Lemma 2.10 that
$\tau(s)<0$ for all $s\in(s_{1},s_{3})$, because $\tau(s)$ has no local
maximum points when $\tau(s)>0$. Moreover, $s_{1},s_{2}$ and $s_{3}$ are also
critical points of $\eta(s)$. If $s_{1}$ is a local maximum of $\eta(s)$, then
it follows from (2.13) and Lemma 2.9 that: $\eta(s_{1})>0$, $\tau(s)$ is
decreasing at $s=s_{1}$, $s_{2}$ is a local minimum of $\eta(s)$,
$\eta(s_{2})<0$, $\displaystyle c+a\tau(s)<0$ for all $s\in(s_{1},s_{2})$,
$s_{3}$ is a local maximum of $\eta(s)$, $\eta(s_{3})>0$ and $c+a\tau(s)>0$
for all $s\in(s_{2},s_{3})$. Thus, there exist $b\in(s_{1},s_{2})$ and
$d\in(s_{2},s_{3})$ such that $\eta(b)=\eta(d)=0$. Since
$2\alpha(s)\eta(s)+\tau^{2}(s)=1$ and $\tau(s)<0$ for all $s\in(s_{1},s_{3})$,
it follows that $\tau(b)=\tau(d)=-1$ i.e. $\displaystyle c-a<0$ and
$\displaystyle c-a>0$. This is a contradiction. With similar arguments we
obtain a contradiction if $s_{1}$ is a local minimum of $\eta(s)$. Therefore,
$c+a\tau(s)$ has at most two zeros. ∎

######  Lemma 2.12.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c<0$ and
initial condition $\psi(0)\in H\cup C\cup S$. If $\tau(s)$ has an infinite
number of zeros in the interval $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$) for some $\overline{s}\in I$ fixed, then the
functions $\alpha(s),$ $\tau(s)$ and $\eta(s)$ are bounded on the intervals
$(\omega_{-},\overline{s})$ (resp. $(\overline{s},\omega_{+})$).

###### Proof.

Note that, the function $g(s)=c\alpha(s)+\eta(s)$, $s\in I$ is decreasing on
$I$, since $g^{\prime}(s)=-a\tau^{2}(s)$ for all $s\in I.$ When $c<0$ it
follows from item ii) of Lemma 2.11 that the functions $\alpha(s)$, $\tau(s)$
and $\eta(s)$ have an infinite number of critical points if $\tau(s)$ has an
infinite number of zeros. Thus, suppose that
$(s_{k})_{k\in\mathbb{N}}\subset(\omega_{-},\overline{s})$ such that
$\tau(s_{k})=0$, $k\in\mathbb{N}$, $\tau(s)\neq 0$ for all
$s\in(s_{k+1},s_{k})$ and $\lim_{k\to+\infty}s_{k}=\omega_{-}$. Without loss
generality, we can consider: $(s_{2k})_{k\in\mathbb{N}}$ the local maximum
points of $\alpha(s)$; $(s_{2k+1})_{\in\mathbb{N}}$ the local minimum points
of $\alpha(s)$; $\tilde{s}_{2k}\in(s_{2k+1},s_{2k})$ the local maximum points
of $\tau(s)$ and $\tilde{s}_{2k+1}\in(s_{2k+2},s_{2k+1})$ the local minimum
points of $\tau(s).$ Hence $\tau(s)>0$ on $(s_{2k+1},s_{2k})$ and $\tau(s)<0$
on $(s_{2k+2},s_{2k+1})$. It follows from Lemma 2.10 that the local maximum
points of $\tau(s)$ satisfy $\tau(\tilde{s}_{2k})>0$ and
$c+a\tau(\tilde{s}_{2k})<0$ for all $k\in\mathbb{N}$, and $a\tau(s)+c<0$ for
all $s\in(s_{2k+1},s_{2k}).$ In the similar way, suppose that
$\displaystyle(s_{j})_{j\in\mathbb{N}}\subset(\overline{s},\omega_{+})$ such
that $\tau(s_{j})=0$, $j\in\mathbb{N}$, $\tau(s)\neq 0$ for all
$s\in(s_{j},s_{j+1})$ and $\lim_{j\to+\infty}s_{j}=\omega_{+}$. Without loss
generality, we can consider: $\displaystyle(s_{2j})_{j\in\mathbb{N}}$ the
local maximum points of $\alpha(s)$;
$\displaystyle(s_{2j+1})_{j\in\mathbb{N}}$ the local minimum points of
$\alpha(s)$; $\tilde{s}_{2j}\in(s_{2j+1},s_{2j+2})$ the local maximum points
of $\tau(s)$ and $\tilde{s}_{2j+1}\in(s_{2j+2},s_{2j+3})$ the local minimum
points of $\tau(s).$ Hence $\tau(s)>0$ on $(s_{2j+1},s_{2j+2})$ and
$\tau(s)<0$ on $(s_{2j+2},s_{2j+3})$. Thus, it follows from Lemma 2.10 that
the local maximum points of $\tau(s)$ satisfy $c+a\tau(\tilde{s}_{2j})<0$,
$k\in\mathbb{N}$ and $c+a\tau(s)<0$ for all $s\in(s_{2j+1},s_{2j+2}).$ We will
divide the proof in two cases: $\psi(0)\in H\cup C$ and $\psi(0)\in S.$

When $\psi(0)\in H\cup C$ then $\alpha(s)\leq 0$ and $\eta(s)\geq 0$ for all
$s\in I$ i.e. the decreasing function $g(s)=c\alpha(s)+\eta(s)$ is non
negative and bounded on the interval $(\overline{s},\omega_{+})$. Moreover,
$0\leq c\alpha(s)\leq g(s)\leq g(\overline{s})$, $0\leq\eta(s)\leq g(s)\leq
g(\overline{s})$ for all $s>\overline{s},$ i.e. the functions $\alpha(s)$ and
$\eta(s)$ are bounded on $(\overline{s},\omega_{+})$ and it follows from
$2\alpha(s)\eta(s)+\tau^{2}(s)=\delta$, where $\delta\in\\{-1,0\\}$ that
$\tau(s)$ is also bounded.

We claim that $\alpha(s)$ is bounded on the interval
$(\omega_{-},\overline{s}).$ In fact, assume by contradiction that $\alpha(s)$
is unbounded i.e. the sequence of the local minimum value of $\alpha(s)$,
$(\alpha(s_{2k+1}))_{k\in\mathbb{N}}\subset(\omega_{-},\overline{s})$ is
unbounded. When $\psi(0)\in H,$ then it follows from item ii) of Lemma 2.8
that $\sqrt{-2c}\alpha(s_{2k+1})<-1$ and $\sqrt{-2c}\alpha(s_{2k})>-1$ for all
$k\in\mathbb{N}$ i.e. there exists $t_{2k}\in(s_{2k+1},s_{2k})$ such that
$\sqrt{-2c}\alpha(t_{2k})=-1$ for all $k\in\mathbb{N}$. When $\psi(0)\in C,$
it follows from item iii) of Lemma 2.8 that $\alpha(s_{2k+1})<0$ and
$\displaystyle\alpha(s_{2k})=0$ for all $k\in\mathbb{N}$, moreover,
$\displaystyle\alpha(s_{2k+3})<\alpha(s_{2k+1})$ for all $k\in\mathbb{N}$,
because $g(s)=c\alpha(s)+\eta(s)$ is a decreasing function and
$\eta(s_{2k+1})=\eta(s_{2k+3})=0$, $k\in\mathbb{N}.$ Thus, there exist
$k_{0}\in\mathbb{N}$ and $t_{2k}\in(s_{2k+1},s_{2k})$ satisfying
$\displaystyle\sqrt{-2c}\alpha(t_{2k})=-1$ for all $k>k_{0}$. Since
$c+a\tau(s)<0$ and $\tau(s)>0$ on $(s_{2k+1},s_{2k})$, and
$2\alpha(s)\eta(s)+\tau^{2}(s)=\delta$, where $\delta\in\\{-1,0\\}$, then
$\displaystyle(\eta(t_{2k}))_{k>k_{0}}$ is also bounded. Hence,
$\displaystyle(g(t_{2k}))_{k>k_{0}}$ is bounded and monotone. Therefore,
$g(s)$ is bounded and $0\leq c\alpha(s)\leq g(s)$ for all
$s\in(\omega_{-},\overline{s})$. But this contradicts the assumption that
$\alpha(s)$ is unbounded. Thus, $\alpha(s)$ is bounded on the interval
$(\omega_{-},\overline{s}).$

It follows from (2.17) that

$\eta^{2}(\tilde{s}_{2k})=\displaystyle{\left\\{\displaystyle{\begin{array}[]{lll}\displaystyle-\frac{[c+a\tau(\tilde{s}_{2k})][1+\tau^{2}(\tilde{s}_{2k})]}{2}&\text{if}&\psi(0)\in
H,\\\ \\\
\displaystyle-\frac{[c+a\tau(\tilde{s}_{2k})]\tau^{2}(\tilde{s}_{2k})}{2}&\text{if}&\psi(0)\in
C.\end{array}}\right.}$

Since $\tau(\tilde{s}_{2k})>0$ and $c+a\tau(\tilde{s}_{2k})<0$, then
$(\eta(\tilde{s}_{2k}))_{k\in\mathbb{N}}$ is bounded. Therefore,
$g(\tilde{s}_{2k})=c\alpha(\tilde{s}_{2k})+\eta(\tilde{s}_{2k})$,
$k\in\mathbb{N}$ is bounded and monotone. Thus, $g(s)$ is bounded, $0\leq
c\alpha(s)\leq g(s)$, $0\leq\eta(s)\leq g(s)$ for all
$s\in(\omega_{-},\overline{s})$. Therefore, the functions $\alpha(s)$,
$\eta(s)$ and $\tau(s)$ are bounded on the interval
$(\omega_{-},\overline{s}).$

Finally, we will study the case $\psi(0)\in S$. Note that, when $s_{0}$
satisfy $\tau(s_{0})=0$, then $s_{0}$ is a critical point of $\alpha(s)$ and
of $\eta(s)$, and $2\alpha(s_{0})\eta(s_{0})=1$. Thus, it follows from Lemmas
2.8 and 2.9 that, if $\alpha(s_{0})>0$ (resp. $\alpha(s_{0})<0$) then $s_{0}$
is a local maximum (resp. minimum) point of $\alpha(s)$ and of $\eta(s)$

Claim: If $s_{0},\overline{s}_{0}\in I$, $s_{0}<\overline{s}_{0}$ are local
maximum or local minimum points of $\alpha(s)$ (resp. $\eta(s)$), then
$\alpha(s_{0})\leq\alpha(\overline{s}_{0})$ (resp.
$\eta(s_{0})\geq\eta(\overline{s}_{0})$). In fact, since
$\tau(s_{0})=\tau(\overline{s}_{0})=0$, $\displaystyle
2\alpha(s_{0})\eta(s_{0})=1$, $\displaystyle
2\alpha(\overline{s}_{0})\eta(\overline{s}_{0})=1$ and
$g(s)=c\alpha(s)+\eta(s)$ is a decreasing function, we have $g(s_{0})\geq
g(\overline{s}_{0})$ and

$\frac{2c\alpha^{2}(s_{0})+1}{2\alpha(s_{0})}\geq\frac{2c\alpha^{2}(\overline{s}_{0})+1}{2\alpha(\overline{s}_{0})}.$

It follows from Lemma 2.8 that the local maximum (resp. minimum) values of
$\alpha$ are positive (resp. negative). Thus,
$\alpha(s_{0})\alpha(\overline{s}_{0})>0$ and $\displaystyle
2c\alpha(s_{0})\alpha(\overline{s}_{0})\left[\alpha(s_{0})-\alpha(\overline{s}_{0})\right]\geq\alpha(s_{0})-\alpha(\overline{s}_{0}).$
If $\alpha(s_{0})>\alpha(\overline{s}_{0})$ we would have
$2c\alpha(s_{0})\alpha(\overline{s}_{0})\geq 1$, this is a contradiction,
since $c<0$ and $\alpha(s_{0})\alpha(\overline{s}_{0})>0$ i.e.
$\alpha(s_{0})\leq\alpha(\overline{s}_{0})$. In a similar way we can prove the
other inequality of the Claim.

Let $\displaystyle(s_{k})_{k\in\mathbb{N}}\subset(\omega_{-},\overline{s})$ be
such that $\tau(s_{k})=0$ for all $k\in\mathbb{N}$. It follows from our Claim
that the local maximum values of $\alpha(s)$ and the local minimum values of
$\eta(s)$ are bounded i.e. $0<\alpha(s_{2k})\leq\alpha(s_{2})$ and
$0>\eta(s_{2k+1})\geq\eta(s_{1})$ for all $k\in\mathbb{N}$. Moreover, since
$\alpha(s_{2k+3})\leq\alpha(s_{2k+1})<0$ then there exists
$t_{2k}\in[s_{2k+1},s_{2k}]$ such that
$\displaystyle\alpha(t_{2k})=\alpha(s_{1})$ for all $k\in\mathbb{N}$. Since
$\tau(s)$ is also bounded on $[s_{2k+1},s_{2k}]$ it follows that
$\tau(t_{2k})$ is bounded for all $k\in\mathbb{N}$. Hence, it follows from
$2\alpha(s)\eta(s)+\tau^{2}(s)=1$ that $\eta(t_{2k})$ is also bounded for all
$k\in\mathbb{N}$. Therefore, the sequence $(g(t_{2k}))_{k\in N}$ is monotone
and bounded, and consequently the function $g(s)$ is bounded on
$(\omega_{-},\overline{s})$ i.e. there exists $M\in\mathbb{R}^{+}$ such that
$|g(s)|\leq M$ for all $s<\overline{s}$. Thus,

$\left\\{\begin{array}[]{l}\displaystyle|c\alpha(s_{2k+1})|\leq|g(s_{2k+1})|+|\eta(s_{2k+1})|\leq|\eta(s_{1})|+M,\\\
\displaystyle|\eta(s_{2k})|\leq|g(s_{2k})|+|c\alpha(s_{2k})|\leq
M+|c|\alpha(s_{2})\end{array}\right.$

for all $k\in\mathbb{N}$ i.e. the local minimum values of $\alpha(s)$ and the
local maximum values of $\eta(s)$ are also bounded. Therefore, $\alpha(s)$ and
$\eta(s)$ are bounded on $(\omega_{-},\overline{s})$, and it follows from
$2\alpha(s)\eta(s)+\tau^{2}(s)=1$ that $\tau(s)$ is also bounded. Using
similar arguments we can prove that $\alpha(s)$, $\tau(s)$ and $\eta(s)$ are
bounded on $(\overline{s},\omega_{+})$. ∎

######  Lemma 2.13.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$,
$c\in\mathbb{R}$ and initial condition $\psi(0)\in H\cup C\cup S$.

  * i)

If $c\geq 0$, then there exist $s_{1},s_{2}\in I$, $s_{1}\leq s_{2}$, such
that the functions $\alpha(s),$ $\tau(s)$ and $\eta(s)$ are monotone on the
intervals $(\omega_{-},s_{1})$ and $(s_{2},\omega_{+}).$

  * ii)

If $c<0$ and at least one the functions $\alpha(s)$, $\tau(s)$ and $\eta(s)$
is unbounded on $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$) with $\overline{s}\in I$ fixed, then there exists
$s_{1}\in(\omega_{-},\overline{s})$ (resp.
$s_{2}\in(\overline{s},\omega_{+})$) such that the functions $\alpha(s)$,
$\tau(s)$ and $\eta(s)$ are monotone on $(\omega_{-},s_{1})$ (resp.
$(s_{2},\omega_{+})$).

###### Proof.

i) If $c\geq 0$, it follows from Lemma 2.11 that $\tau(s)$ and $c+a\tau(s)$
have at most two zeros. Therefore, it follows from (2.13) that $\alpha(s)$ has
at most two critical points and $\eta(s)$ has at most four critical points. We
claim that $\tau(s)$ has at most three critical points. In fact, assume by
contradiction that $s_{1},s_{2},s_{3}$ and $s_{4}$ are four consecutive
critical points of $\tau(s)$ and $\tau^{\prime}(s)\neq 0$ for all
$s\in(s_{1},s_{2})\cup(s_{2},s_{3})\cup(s_{3},s_{4})$. If $\tau(s)>0$, for
$s\in J\subset I$, since $[c+\tau(s)]\tau(s)>0$, it follows from Lemma 2.10
that $\tau(s)$ has no local maximum in $J$. Hence, $\tau(s)<0$ for
$s\in[s_{1},s_{4}]$ and Lemma 2.10 implies that $c+a\tau(s_{i})>0$ when
$s_{i}$ is a local maximum point of $\tau(s)$ and $c+a\tau(s_{i})<0$ when
$s_{i}$ is a local minimum point of $\tau(s)$, where $i\in\\{1,2,3,4\\}$.
However, this implies that $c+a\tau(s)$ has three zeros, which contradicts the
fact that $c+a\tau(s)$ have at most two zeros. Hence, $\tau(s)$ has at most
three critical points. Therefore, there exist $s_{1},s_{2}\in I$, $s_{1}\leq
s_{2}$ such that $\alpha(s),$ $\tau(s)$ and $\eta(s)$ are monotone on the
intervals $(\omega_{-},s_{1})$ and $(s_{2},\omega_{+}).$

ii) If $c<0$, it follows from Lemma 2.12 that, when one of the functions
$\alpha(s)$, $\tau(s)$ and $\eta(s)$ is unbounded on
$(\omega_{-},\overline{s})$ (resp. $(\overline{s},\omega_{+})$) then $\tau(s)$
has at most a finite number of zeros on $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$). Hence, $\alpha$ and $\eta$ have a finite number
of critical points.

Therefore, there exists $s_{1}\in(\omega_{-},\overline{s})$ (resp.
$s_{2}\in(\overline{s},\omega_{+})$) such that the functions $\alpha(s)$,
$\tau(s)$ and $\eta(s)$ are monotone on $(\omega_{-},s_{1})$ (resp.
$(s_{2},\omega_{+})$). ∎

######  Lemma 2.14.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$,
$c\in\mathbb{R}$ and initial condition $\psi(0)\in H\cup C\cup S$. Let
$\overline{s}\in I$.

If $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$), then the functions $\tau(s)$ and $\eta(s)$ are
bounded on $(\omega_{-},\overline{s})$ and $\omega_{-}=-\infty$ (resp.
$(\overline{s},\omega_{+})$ and $\omega_{+}=+\infty$).

If $\eta(s)$ is bounded on $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$), then $\tau(s)$ is bounded on
$(\omega_{-},\overline{s})$ and $\omega_{-}=-\infty$ (resp.
$(\overline{s},\omega_{+})$ and $\omega_{+}=+\infty$).

###### Proof.

Note that, if $\alpha(s)\neq 0$, $\eta(s)\neq 0$ and $[c+a\tau(s)]\tau(s)\neq
0$, since $\psi(s)\in H\cup C\cup S$, we have
$2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$, where $\gamma\in\\{-1,0,1\\}$ and it
follows from (2.13) that

$\frac{\displaystyle{\frac{d}{ds}}\left(\tau^{2}(s)\right)}{\displaystyle{\frac{d}{ds}}\left(2\eta(s)\right)}=-\alpha(s)+\frac{\gamma}{2\alpha(s)[c+a\tau(s)]}-\frac{\tau(s)}{2\alpha(s)\left[\displaystyle{\frac{c}{\tau(s)}}+a\right]}.$
(2.18)

i) We will only consider the interval $(\omega_{-},\overline{s})$, since the
proof for the interval $(\overline{s},\omega_{+})$ follows from similar
arguments. Suppose that $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$.
Assume by contradiction that $\tau(s)$ is unbounded on
$(\omega_{-},\overline{s}).$ It follows from Lemma 2.13 that there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that $\alpha(s)$, $\tau(s)$ and
$\eta(s)$ are monotone on $(\omega_{-},s_{1})$. Thus,

$\displaystyle\lim_{s\to\omega_{-}}\tau(s)=\pm\infty,\hskip
8.5359pt\displaystyle\lim_{s\to\omega_{-}}\tau(s)[c+a\tau(s)]=+\infty\hskip
8.5359pt\text{and}\hskip 8.5359pt\eta^{\prime}(s)=-[c+a\tau(s)]\tau(s)<0$

for all $s<s_{1}$. Hence, it follows from
$2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$ that $\eta(s)$ is also unbounded,
$\lim_{s\to\omega_{-}}\eta(s)=+\infty$ and
$\lim_{s\to\omega_{-}}\eta(s)\alpha(s)=-\infty$. Moreover, $s_{1}$ can be
chosen such that $\alpha(s)<0$ for all $s<s_{1}.$

Suppose that $\lim_{s\to\omega_{-}}\tau(s)=-\infty$, then there exists $s_{1}$
such that $\tau^{\prime}(s)>0$, $\gamma-\tau^{2}(s)<0$ and $c+a\tau(s)<0$ for
all $s<s_{1}$, where $\gamma\in\\{-1,0,1\\}.$ Thus,
$\alpha(s)\tau^{\prime}(s)<0$ and (2.16) implies that

$\alpha(s)\tau^{\prime}(s)=[c+a\tau(s)]\alpha^{2}(s)-\frac{\gamma}{2}+\frac{\tau^{2}(s)}{2}<0\,\,\,\,\,\,\text{i.e.}\,\,\,\,\,\,\alpha^{2}(s)>\frac{\gamma-\tau^{2}(s)}{c+a\tau(s)}>0$

for all $s<\overline{s}_{1}$, this is a contradiction, we assumed that
$\lim_{s\to\omega_{-}}\tau(s)=-\infty$ and $\alpha(s)$ is bounded.

Suppose that $\lim_{s\to\omega_{-}}\tau(s)=+\infty$. Then there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that $\alpha(s)$ is increasing and
negative on $(\omega_{-},s_{1})$. Since that $\alpha(s)$ is bounded, we have
$\lim_{s\to\omega_{-}}\alpha(s)=L$, where $L\in\mathbb{R}^{-}$ and
$\lim_{s\to\omega_{-}}\alpha(s)[c+a\tau(s)]=-\infty$. Thus, it follows from
(2.18) and L’Hospital rule that

$\displaystyle\lim_{s\to\omega_{-}}\frac{\displaystyle{\frac{d}{ds}}\left(\tau^{2}(s)\right)}{\displaystyle{\frac{d}{ds}}\left(2\eta(s)\right)}=+\infty\,\,\,\,\text{and}\,\,\,\,\displaystyle\lim_{s\to\omega_{-}}\frac{\tau^{2}(s)}{2\eta(s)}=\displaystyle\lim_{s\to\omega_{-}}\frac{\displaystyle{\frac{d}{ds}}\left(\tau^{2}(s)\right)}{\displaystyle{\frac{d}{ds}}\left(2\eta(s)\right)}=+\infty.$

Since $2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$, then
$\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{-}}[\gamma/(2\eta(s))-\tau^{2}/(2\eta(s))]=-\infty$.
This contradicts that $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$.
Therefore, $\tau(s)$ is bounded on $(\omega_{-},\overline{s}).$

Let $M>0$ be such that $|\tau(s)|<M$, $s\in(\omega_{-},\overline{s})$. It
follows from (2.13) that

$|\eta(\overline{s})-\eta(s)|=\left|-\int^{\overline{s}}_{s}[c+\tau(u)]\tau(u)du\right|<\left(|c|M+aM^{2}\right)(\overline{s}-s)$

for each $s<\overline{s}$. Since $I$ is a maximal interval we conclude that
the solution $\psi(s)$ of (2.13) leaves any compact subset of
$(\omega_{-},\overline{s})\times\mathbb{R}^{3}$, then $\omega_{-}=-\infty$.

We will now proof that $\eta(s)$ is bounded. Assume by contradiction that
$\eta(s)$ is unbounded. It follows from Lemma 2.13 that there exists $s_{1}$
such that the functions $\alpha(s)$, $\tau(s)$ are $\eta(s)$ are monotone on
$(-\infty,s_{1})$ and $\lim_{s\to-\infty}\eta(s)=\pm\infty$. Moreover, the
limits $\lim_{s\to-\infty}\alpha(s)$ and $\lim_{s\to-\infty}\tau(s)$ exist.
Hence, the improper integral

$\int^{\overline{s}}_{-\infty}\tau^{\prime}(u)du=\tau(\overline{s})-\lim_{s\to-\infty}\tau(s)$

is convergent, which is a contradiction, since
$\lim_{s\to-\infty}[c+a\tau(s)]\alpha(s)-\eta(s)=\mp\infty$. Therefore,
$\eta(s)$ is also bounded on $(-\infty,\overline{s})$.

ii) We will give the proof only for the interval $(\omega_{-},\overline{s})$,
since the proof for the interval $(\overline{s},\omega_{+})$ follows by
similar arguments. Let $\eta(s)$ be bounded on $(\omega_{-},\overline{s})$.
Assume by contradiction that $\tau(s)$ is unbounded on
$(\omega_{-},\overline{s})$. It follows from Lemma 2.13 that there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that $\alpha(s)$, $\tau(s)$ and
$\eta(s)$ are monotone on $(\omega_{-},s_{1})$ and
$\lim_{s\to\omega_{-}}\tau(s)=\pm\infty$. Since
$2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$, we have that $\alpha(s)$ is also
unbounded, $\lim_{s\to\omega_{-}}\alpha(s)=\pm\infty$ and
$\lim_{s\to\omega_{-}}\tau^{\prime}(s)=\lim_{s\to\omega_{-}}[c+a\tau(s)]\alpha(s)-\eta(s)=\pm\infty.$
Using L’Hospital rule we obtain

$\displaystyle\lim_{s\to\omega_{-}}\frac{\tau^{2}(s)}{2\alpha(s)}=\lim_{s\to\omega_{-}}\frac{2\tau(s)\tau^{\prime}(s)}{2\tau(s)}=\lim_{s\to\omega_{-}}\tau^{\prime}(s)=\pm\infty.$

Hence,
$\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}[\gamma/{(2\alpha(s))}-\tau^{2}{(2\alpha(s))}]=\mp\infty$,
which contradicts the hypothesis thet $\eta(s)$ is bounded on
$(\omega_{-},\overline{s})$. Therefore, $\tau(s)$ is bounded on
$(\omega_{-},\overline{s}).$ Let $M>0$ be such that $|\tau(s)|<M$ for all
$s\in(\omega_{-},\overline{s})$. From (2.13) we obtain

$|\alpha(\overline{s})-\alpha(s)|=\left|\int^{\overline{s}}_{s}\tau(u)du\right|<M(\overline{s}-s)$

for each $s<\overline{s}$. Since $I$ is a maximal interval, the solution
$\psi(s)$ of (2.13) leaves any compact subset of
$(\omega_{-},\overline{s})\times\mathbb{R}^{3}$. Hence $\omega_{-}=-\infty$. ∎

######  Lemma 2.15.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$,
$c\in\mathbb{R}$ and initial condition $\psi(0)\in H\cup C\cup S$. If
$\alpha(s)$ is bounded on the interval $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$), $\overline{s}\in I$ fixed, then
$\omega_{-}=-\infty$, $\lim_{s\to-\infty}\tau(s)=0$ and
$\lim_{s\to-\infty}\tau^{\prime}(s)=0$ (resp. $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}\tau(s)=0$ and $\lim_{s\to+\infty}\tau^{\prime}(s)=0.$)

###### Proof.

We will only consider the interval $(\omega_{-},\overline{s})$, since the
proof for the interval $(\overline{s},\omega_{+})$ follows with similar
arguments. We define the function $g(s)=c\alpha(s)+\eta(s)$, $s\in I$. It
follows from (2.13) that $g^{\prime}(s)=-a\tau^{2}(s)$ and $g(s)$ is
decreasing for all $s\in I$.

Suppose that $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$,
$\overline{s}\in I$ fixed, then it follows from item i) of Lemma 2.14 that
$\eta(s)$ and $\tau(s)$ are bounded on $(\omega_{-},\overline{s})$ and
$\omega_{-}=-\infty$. Thus, the function $g(s)$ is decreasing and bounded on
$(-\infty,\overline{s})$ and $\lim_{s\to-\infty}g(s)$ exists. Note that
$g^{\prime}(s)$ is uniformly continuous on $(-\infty,\overline{s})$, since
$g^{\prime\prime}(s)=2a\tau(s)[c+a\tau(s)]\alpha(s)-\eta(s)$ is bounded on
$(-\infty,\overline{s})$. Hence, it follows from Barbalat’s Lemma that
$\lim_{s\to-\infty}g^{\prime}(s)=0$. Since $\displaystyle
g^{\prime}(s)=-a\tau^{2}(s)$, we conclude that $\lim_{s\to-\infty}\tau(s)=0$.
Moreover, it follows from (2.13) that $\tau^{\prime\prime}(s)$ is also bounded
on $(-\infty,\overline{s})$ i.e. $\tau^{\prime}(s)$ is uniformly continuous on
$(-\infty,\overline{s})$. Since $\lim_{s\to-\infty}\tau(s)=0$ then it follows
from Barbalat’s Lemma that $\lim_{s\to-\infty}\tau^{\prime}(s)=0.$ ∎

######  Lemma 2.16.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$,
$c\in\mathbb{R}$ and initial condition $\psi(0)\in H\cup C\cup S$. Let
$\overline{s}\in I$ be fixed. If $\alpha(s)$ is unbounded on
$(\omega_{-},\overline{s})$ , then either
$\lim_{s\to\omega_{-}}|\eta(s)|=\lim_{s\to\omega_{-}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}c+a\tau(s)=0$. If $\alpha(s)$ is
unbounded on $(\overline{s},\omega_{+})$, then either
$\lim_{s\to\omega_{+}}|\eta(s)|=\lim_{s\to\omega_{+}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to+\infty}\eta(s)=\lim_{s\to+\infty}c+a\tau(s)=0$.

###### Proof.

We will only consider the interval $(\omega_{-},\overline{s})$, since the
proof for the interval $(\overline{s},\omega_{+})$ follows with similar
arguments. Since $\alpha(s)$ is unbounded on $(\omega_{-},\overline{s})$ it
follows from Lemma 2.13 that there exists $s_{1}\in(\omega_{-},\overline{s})$
such that $\alpha(s)$, $\tau(s)$ and $\eta(s)$ are monotone on
$(\omega_{-},s_{1})$. Thus, the monotone function $\eta(s)$ is either bounded
or unbounded on $(\omega_{-},s_{1})$.

If $\eta(s)$ is bounded, it follows item ii) of Lemma 2.14 that $\tau(s)$ is
bounded on $(\omega_{-},s_{1})$, $\omega_{-}=-\infty$ and the limits
$\lim_{s\to-\infty}\tau(s)$ and $\lim_{s\to-\infty}\eta(s)$ exist. Hence,
$\lim_{s\to-\infty}\eta^{\prime}(s)=-\lim_{s\to-\infty}\tau(s)[c+a\tau(s)]$
also exist and therefore $\lim_{s\to-\infty}\tau(s)[c+a\tau(s)]=0.$ We will
prove that $\lim_{s\to-\infty}\tau(s)\neq 0$ when $c\neq 0$. Assume by
contradiction that $\lim_{s\to-\infty}\tau(s)=0$. Hence,
$\lim_{s\to-\infty}\tau^{\prime}(s)=\lim_{s\to-\infty}[c+\tau(s)]\alpha(s)-\eta(s)=\pm\infty$
and the improper integral

$\int_{-\infty}^{s_{1}}\tau^{\prime}(u)du=\tau(s_{1})-\lim_{s\to-\infty}\tau(s)=\tau(s_{1}),$

which is a contradiction. Therefore, $\lim_{s\to-\infty}[c+a\tau(s)]=0$. Note
that $2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma,$ where $\gamma\in\\{-1,0,1\\}$. By
hypothesis $\alpha(s)$ is unbounded, hence $\lim_{s\to-\infty}\eta(s)=0$.

If $\eta(s)$ is unbounded on $(\omega_{-},s_{1})$ then
$2\alpha(s)\eta(s)+\tau^{2}(s)=\gamma$ implies that $\tau(s)$ is unbounded on
$(\omega_{-},s_{1})$ and thus
$\lim_{s\to\omega_{-}}|\eta(s)|=\lim_{s\to\omega_{-}}|c+a\tau(s)|=+\infty$. ∎

######  Lemma 2.17.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c\geq 0$
and initial condition $\psi(0)\in H$. Then there exists a global maximum point
$s_{0}$ of $\alpha(s)$.

###### Proof.

Assume by contradiction that $\alpha(s)$ has no critical points i.e.
$\tau(s)>0$ or $\tau(s)<0$ for all $s\in I.$ If $\tau(s)>0$, then $\alpha(s)$
is negative and strictly increasing on $I$ and $\eta(s)$ is positive and
strictly decreasing on $I$. Thus, the limits $\lim_{s\to\omega_{+}}\alpha(s)$,
$\lim_{s\to\omega_{+}}\eta(s)$ exist and consequently
$\lim_{s\to\omega_{+}}\tau(s)$ exists, since
$-2\alpha(s)\eta(s)=\tau^{2}(s)+1$. Hence, there exists $p\in H$ such that
$\lim_{s\to\omega_{+}}\psi(s)=p$ and thus, $\omega_{+}=+\infty$ and
$\psi(s)=p$ is a singular solution of (2.13). This contradicts Lemma 2.7 which
shows that the system (2.13) has no singular solutions on the set $H$, when
$c\geq 0$. If $\tau(s)<0$, we obtain a similar contradiction. Therefore, there
exists $s_{0}\in I$ such that $s_{0}$ is a critical point of $\alpha(s)$ and
it follows from item i) of Lemma 2.8 that $s_{0}$ is the global maximum point.
∎

Our next lemma shows that for non trivial solutions $\psi$ of (2.13), when
$c\geq 0$ (resp. $c\leq 0$) and $\psi\in H$ (resp. $\psi\in S$), then $\alpha$
is unbounded on both ends of the maximal interval $I$.

######  Lemma 2.18.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, with $a>0$.

i) If $c\geq 0$ and $\psi(0)\in H$, then
$\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=-\infty$.

ii) If $c\leq 0$ and $\psi(0)\in S$, then
$\lim_{s\to\omega_{-}}|\alpha(s)|=\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$.

###### Proof.

i) Lemma 2.17 implies that $\alpha(s)$ has a unique critical point $s_{0}\in
I$. It follows from Lemma 2.10 that $\tau(s)$ has at most one critical point
$s_{1}$ and $\tau(s_{1})<0$. Thus, the functions $\alpha(s)$, $\tau(s)$ and
$\eta(s)$ are monotone on $(\omega_{-},s_{0})$ and $(s_{1},\omega_{+})$.
Assume by contradiction that $\alpha(s)$ is bounded on $(\omega_{-},s_{0})$
(resp. $(s_{1},\omega_{+})$). It follows from item i) of Lemma 2.14 that
$\tau(s)$ and $\eta(s)$ are also bounded on $(\omega_{-},s_{0})$ and
$\omega_{-}=-\infty$ (resp. $(s_{1},\omega_{+})$ and $\omega_{+}=+\infty$).
Hence, there exist $p\in H$ (resp. $\overline{p}\in H$) such that
$\lim_{s\to-\infty}\psi(s)=p$, (resp.
$\lim_{s\to+\infty}\psi(s)=\overline{p}$) and $\psi(s)=p$ (resp.
$\psi(s)=\overline{p}$) is a singular solution of (2.13) in $H$. This
contradicts Lemma 2.7 that asserts that (2.13) has no singular solutions in
the set $H$, when $c\geq 0$. Since $\alpha(s)<0$ we conclude that
$\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=-\infty.$

ii) Let us consider $\overline{s}\in I$ fixed. We will only consider the
interval $(\omega_{-},\overline{s})$, since the proof for the interval
$(\overline{s},\omega_{+})$ follows from similar arguments. Assume by
contradiction that $\alpha(s)$ is bounded on $(\omega_{-},s_{0})$. It follows
from item i) of Lemma 2.14 that $\tau(s)$ and $\eta(s)$ are also bounded on
$(\omega_{-},s_{0})$ and $\omega_{-}=-\infty$. Thus, Lemma 2.15 implies that
$\lim_{s\to-\infty}\tau(s)=0$ and $\lim_{s\to-\infty}\tau^{\prime}(s)=0$. From
(2.16) and (2.17) we have that $\lim_{s\to-\infty}\alpha(s)$ and
$\lim_{s\to-\infty}\eta(s)$ exist. Hence, there exists $p\in S$ such that
$\lim_{s\to-\infty}\psi(s)=p$ and $\psi(s)=p$, $s\in\mathbb{R}$ is a singular
solution of (2.13) in $S$, which contradicts Lemma 2.7 that asserts that
(2.13) has no singular solutions in $S$, when $c\leq 0$. Therefore,
$\alpha(s)$ is unbounded on $(\omega_{-},\overline{s})$ and it follows from
item ii) of Lemma 2.13 that $\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty.$ ∎

In order to obtain additional properties on the solutions of (2.13), we will
separate the study in three cases: $c=0$, $c<0$ and $c>0$.

When $c=0$ the solutions of (2.13) provide the soliton solutions to the CF in
$Q^{2}_{+}$. In this case, (2.13) has no singular solutions in the set $H\cup
S$. We also recall that, if $\psi(s)\in H$, $s\in I$, then $\alpha(s)<0$ and
$\eta(s)>0$, and if $\psi(s)\in C$, $s\in I$, then $\alpha(s)\leq 0$ and
$\eta(s)\geq 0$. Note that, when $c=0$, $\psi(s)=(\alpha_{0},0,0)$, $s\in I$,
$\alpha_{0}\neq 0$ are singular solutions of (2.13) in $C$. These solutions
correspond to the parables on the light cone i.e. they are trivial solutions
to the CF in $Q^{2}_{+}$. In the next lemma we will study the behaviour of
$\tau(s)$, when $c=0$ and $\psi(s)\in H\cup C$ is a non trivial solution of
(2.13).

######  Lemma 2.19.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c=0$ and
initial condition $\psi(0)\in H\cup C$. Then $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}\tau(s)=0$ and $\lim_{s\to\omega_{-}}\tau(s)=+\infty$.

###### Proof.

It follows from item i) of Lemma 2.13 that there exist $s_{1},s_{2}\in I$ such
that $\alpha(s)$, $\tau(s)$ and $\eta(s)$ are monotone on $(\omega_{-},s_{1})$
and $(s_{2},\omega_{+})$. Since $\eta(s)$ is a non negative function when
$\psi(0)\in H\cup C$, the third equation of (2.13) and $c=0$ imply that
$\eta(s)$ is decreasing and bounded on $(s_{2},\omega_{+})$. Thus, it follows
from item ii) of Lemma 2.14 that $\tau(s)$ is bounded on $(s_{2},\omega_{+})$
and $\omega_{+}=+\infty$. Hence, the limits $\lim_{s\to+\infty}\eta(s),$
$\lim_{s\to+\infty}\tau(s)$ exist. Therefore,
$\lim_{s\to+\infty}\eta^{\prime}(s)=-\lim_{s\to+\infty}a\tau^{2}$ exists and
consequently $\lim_{s\to+\infty}\tau(s)=0$.

We will now prove that $\tau(s)$ is unbounded on $(\omega_{-},s_{1})$. Lemma
2.18 asserts that $\alpha$ is unbounded on $(\omega_{-},s_{1})$ when
$\psi(0)\in H$ and $\lim_{s\to\omega_{-}}\alpha(s)=-\infty$. We claim that
$\alpha(s)$ is also unbounded on $(\omega_{-},s_{1})$ when $\psi(0)\in C$. In
fact, assume by contradiction that $\alpha(s)$ is bounded on
$(\omega_{-},s_{1}).$ Then, it follows from item i) of Lemma 2.14 that
$\tau(s)$ and $\eta(s)$ are also bounded on $(\omega_{-},s_{1})$ and
$\omega_{-}=-\infty$. Moreover, Lemma 2.15 implies that
$\lim_{s\to-\infty}\tau(s)=0$ and $\lim_{s\to-\infty}\tau^{\prime}(s)=0$.
Thus,
$\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}\tau^{\prime}(s)-\alpha(s)\tau(s)=0.$
Since $\eta(s)$ is non negative and decreasing we conclude that $\eta(s)$
vanishes on $(\omega_{-},s_{1})$, which contradicts the assumption that we are
considering only non trivial solutions. Hence, $\alpha(s)$ is unbounded when
$\psi(0)\in C$ and $\lim_{s\to\omega_{-}}\alpha(s)=-\infty$ i.e. $\alpha(s)$
is increasing on $(\omega_{-},s_{1})$. Therefore, if $\psi(0)\in H\cup C$ and
$\psi(s)$ is a non trivial solution then
$\lim_{s\to\omega_{-}}\tau(s)=+\infty$. ∎

We will now study the solutions of (2.13) with initial condition in $S$. In
this case, (2.13) has no singular solutions. Moreover, the functions
$\alpha(s)$ and $\eta(s)$ may change sign.

######  Lemma 2.20.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c=0$ and
initial condition $\psi(0)\in S$. Then,

i) either $\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}\tau(s)=+\infty$
or $\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}\tau(s)=0$ and

ii) either
$\lim_{s\to\omega_{+}}-\eta(s)=\lim_{s\to\omega_{+}}\tau(s)=+\infty$ or
$\lim_{s\to+\infty}\eta(s)=\lim_{s\to+\infty}\tau(s)=0$).

###### Proof.

Lemma 2.18 asserts that $\alpha(s)$ is unbounded on
$(\omega_{-},\overline{s})$ and $(\overline{s},\omega_{+})$, for
$\overline{s}\in I$. Thus, it follows from Lemma 2.16 that either
$\lim_{s\to\omega_{-}}|\eta(s)|=\lim_{s\to\omega_{-}}|\tau(s)|=+\infty$ or
$\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}\tau(s)=0$. Moreover, either
$\lim_{s\to\omega_{+}}|\eta(s)|=\lim_{s\to\omega_{+}}|\tau(s)|=+\infty$ or
$\lim_{s\to+\infty}\eta(s)=\lim_{s\to+\infty}\tau(s)=0$). The third equation
of (2.13) implies that $\eta(s)$ is a decreasing function on $I$. Thus, when
$\tau(s)$ is unbounded on $(\omega_{-},\overline{s})$ (resp.
$(\overline{s},\omega_{+})$) then $\eta(s)$ is unbounded on
$(\omega_{-},\overline{s})$ (resp. $(\overline{s},\omega_{+})$) and
$\lim_{s\to\omega_{-}}\eta(s)=+\infty$ (resp.
$\lim_{s\to\omega_{+}}\eta(s)=-\infty$). Thus,
$2\alpha(s)\eta(s)+\tau^{2}(s)=1$ implies that
$\lim_{s\to\omega_{-}}\alpha(s)=-\infty$ (resp.
$\lim_{s\to\omega_{+}}\alpha(s)=+\infty$) and therefore,
$\lim_{s\to\omega_{-}}\tau(s)=+\infty$ (resp.
$\lim_{s\to\omega_{+}}\tau(s)=+\infty$). ∎

Let $X(s)$ be a soliton solution to the CF on $Q^{2}_{+}$ corresponding to a
solution of (2.13). Since its curvature is given by $k(s)=a\tau(s)$, as an
immediate consequence of Lemmas 2.19 and 2.20, we get

######  Corollary 2.21.

Assume that the curve $X:I\rightarrow Q^{2}_{+}$, $s\in I$ is a soliton
solution to the CF, with curvature $k(s)$, that corresponds to a non trivial
solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (2.13) defined on the
maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c=0$, with initial
condition $\psi(0)\in H\cup C\cup S$. Then

i) If $\psi(0)\in H\cup C$ (resp. $\psi(0)\in C$) then $\omega_{+}=+\infty$.
Moreover $\lim_{s\to\omega_{-}}k(s)=+\infty$ and $\lim_{s\to+\infty}k(s)=0.$

ii) If $\psi(0)\in S$, then at each end of the curve $k(s)$ tends to $+\infty$
or converges to zero.

We will now study the solutions of (2.13) with $\psi(0)\in H\cup C\cup S$ and
$c<0$. Note that $\psi(s)=(\alpha_{0},0,c\alpha_{0})\in H$, $s\in I$ is a
singular solution and (2.13) has no singular solutions in $C$. However
$\psi(s)=(0,0,0)$ is a singular solution at the boundary of $C$. In the next
lemmas we will prove that $(\alpha_{0},0,c\alpha_{0})$ and $(0,0,0)$ are
global attractors for the solutions in $H$ and $C$ respectively.

######  Lemma 2.22.

Consider $a>0$ and $c<0$. Let $\Phi:H\rightarrow TH\subset\mathbb{R}^{3}$ be
the vector field given by
$\Phi(\alpha,\tau,\eta)=\left(\tau,c\alpha+a\tau\alpha-\eta,-c\tau-a\tau^{2}\right)$.
Then the singular point $p$ of $\Phi$ and the eigenvalues $\lambda$ of
$d\Phi_{p}$ are given by

$p=\left(-\frac{1}{\sqrt{-2c}},0,-\frac{c}{\sqrt{-2c}}\right),\qquad\qquad\lambda=\frac{-a\pm\sqrt{a^{2}-16c^{2}}}{2\sqrt{-2c}}.$
(2.19)

The eigenvalues are real numbers if $4|c|\leq a$ and they are complex numbers
if $4|c|>a$.

###### Proof.

The singular point of $\Phi$ satisfies $\tau=0$, $c\alpha=\eta$. Since
$(\alpha,\tau,\eta)\in H$ then $-1=2\alpha\eta+\tau^{2}=2\alpha\eta$. Hence,
$2c\alpha^{2}=-1$, $\alpha=-{1}/{\sqrt{-2c}}$. The singular point $p$ of
$\Phi$ in $H$ and the eigenvalues $\lambda$ of $d\Phi_{p}$ are given by (2.19)
and the eigenvectors are $w=(w_{1},w_{2},-cw_{1})\in
T_{p}H,w_{1},w_{2}\in\mathbb{R}$. ∎

######  Lemma 2.23.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c<0$ and
initial condition $\psi(0)\in H\cup C$. Then $\omega_{+}=+\infty$. If
$\psi(0)\in H$ then $\lim_{s\to+\infty}\psi(s)=p$, where $p$ is given by
(2.19). If $\psi(0)\in C$ then $\lim_{s\to+\infty}\psi(s)=(0,0,0)$.

###### Proof.

Define $g(s)=c\alpha(s)+\eta(s)$, for $s\in I$. Since $\alpha(s)\leq 0$,
$\eta(s)\geq 0$ and $c<0$, then $g(s)$ is a positive function and
$g^{\prime}(s)=-a\tau^{2}(s)<0$, i.e., $g(s)$ is a decreasing function and
$0<g(s)\leq g(\overline{s})=M$, for all $s>\overline{s}\in I$ and
$M\in\mathbb{R}$. Therefore, $0\leq c\alpha(s),\,\eta(s)<c\alpha(s)+\eta(s)<M$
, $s>\overline{s}$. Hence, $\alpha(s)$ and $\eta(s)$ are bounded on
$(\overline{s},\omega_{+})$, and Lemma 2.15 implies that $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}\tau(s)=0$ and $\lim_{s\to+\infty}\tau^{\prime}(s)=0$. It
follows from (2.16) and (2.17) that if $\psi(0)\in H$ then
$\lim_{s\to+\infty}2c\alpha^{2}(s)=-1$ and
$\lim_{s\to+\infty}2\eta^{2}(s)=-c,$ and if $\psi(0)\in C$ then
$\lim_{s\to+\infty}\alpha^{2}(s)=\lim_{s\to+\infty}\eta^{2}(s)=0$. ∎

######  Lemma 2.24.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c<0$ and
initial condition $\psi(0)\in H\cup C\cup S$.

i) If $\psi(0)\in H\cup C$, then $\lim_{s\to\omega_{-}}\alpha(s)=-\infty$ and
either $\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}c+a\tau(s)=+\infty$
or $\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}c+a\tau(s)=0$.

ii) If $\psi(0)\in S$, then on the boundary of the interval $I$ we have:
Either
$\lim_{s\to\omega_{-}}|\eta(s)|=\lim_{s\to\omega_{-}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to-\infty}\eta(s)=\lim_{s\to-\infty}c+a\tau(s)=0$.
Either
$\lim_{s\to\omega_{+}}|\eta(s)|=\lim_{s\to\omega_{+}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to+\infty}\eta(s)=\lim_{s\to+\infty}c+a\tau(s)=0$.

###### Proof.

i) Assume by contradiction that $\alpha(s)$ is bounded on
$(\omega_{-},\overline{s})$, $\overline{s}\in I$ fixed. It follows from Lemma
2.15 that $\tau(s)$ and $\eta(s)$ are bounded on $(\omega_{-},\overline{s})$,
$\omega_{-}=-\infty$, $\lim_{s\to-\infty}\tau(s)=0$ and
$\lim_{s\to-\infty}\tau^{\prime}(s)=0$. Thus, using Lemma 2.6 we have: if
$\psi(0)\in H$ then $\lim_{s\to+\infty}2c\alpha^{2}(s)=-1$ and
$\lim_{s\to+\infty}2\eta^{2}(s)=-c,$ and if $\psi(0)\in C$ then
$\lim_{s\to+\infty}\alpha^{2}(s)=\lim_{s\to+\infty}\eta^{2}(s)=0.$ Hence,
$\left(-{1}/{\sqrt{-2c}},0,-{c}/{\sqrt{-2c}}\right)$ and $(0,0,0)$ are non
attractor singular solutions of (2.13) which contradicts Lemmas 2.22 and 2.23.
Therefore, $\alpha(s)$ is unbounded on $(\omega_{-},\overline{s})$. Thus, it
follows from item ii) of Lemma 2.10 that there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that the functions $\alpha(s)$ and
$\tau(s)$ are monotone on $(\omega_{-},s_{1})$. Hence,
$\lim_{s\to\omega_{-}}\alpha(s)=-\infty$ and there exists
$\hat{s}\in(\omega_{-},s_{1})$ such that $\tau(s)>0$ for all $s<\hat{s}$.
Therefore, Lemma 2.16 implies that either
$\lim_{s\to\omega_{-}}c+a\tau(s)=+\infty$ or
$\lim_{s\to-\infty}c+a\tau(s)=\lim_{s\to-\infty}\eta(s)=0$. Moreover, if
$\lim_{s\to\omega_{-}}c+a\tau(s)=+\infty$ then we can choose $s_{1}$ such that
$\eta^{\prime}(s)=-[c+a\tau(s)]\tau(s)<0$ for all $s<s_{1}$ and thus,
$\lim_{s\to\omega_{-}}\eta(s)=+\infty$.

ii) When $c<0$ and $\psi\in S$, then $\alpha(s)$ and $\eta(s)$ may change
sign. Moreover, it follows from Lemma 2.7 that there are no singular solutions
in $S$. Lemma 2.18 implies that $\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$ and
$\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$. Thus, the result follows from
Lemma 2.16. ∎

As an immediate consequence of Lemmas 2.23 and 2.24, we get the following.

######  Corollary 2.25.

Assume that a curve $X:I\rightarrow Q^{2}_{+}$, $s\in I$ is a self-similar
solution to the CF with curvature $k(s)$ that corresponds to a non trivial
solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (2.13), defined on the
maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c<0$ with initial
condition $\psi(0)\in H\cup C\cup S$. Then

i) If $\psi(0)\in H\cup C$, then $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}k(s)=c$ and either $\lim_{s\to-\infty}k(s)=0$ or
$\lim_{s\to\omega_{-}}k(s)=+\infty.$

ii) If $\psi(0)\in S$, then at each end of $X$ the curvature either converges
to zero or it is unbounded.

We will now study the solutions of (2.13) when $c>0$. In this case, there are
no singular solutions in $H\cup C$. In the next lemma we study the functions
$\eta(s)$ and $\tau(s)$, when $c>0$ and $\psi(0)\in H$.

######  Lemma 2.26.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c>0$ and
initial condition $\psi(0)\in H$. Then $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}c+a\tau(s)=\lim_{s\to+\infty}\eta(s)=0$. Moreover,
$\lim_{s\to\omega_{-}}\tau(s)=\lim_{s\to\omega_{-}}\eta(s)=+\infty$.

###### Proof.

Lemma 2.17 implies that there exists a global maximum point $s_{0}$ of
$\alpha(s)$. Thus, $\alpha(s)$ is a decreasing (resp. increasing) function on
$(s_{0},\omega_{+})$ i.e. $\tau(s)=\alpha^{\prime}(s)<0$ for all $s>s_{0}$
(resp. $(\omega_{-},s_{0})$ i.e. $\tau(s)>0$ for all $s<s_{0}$ ). Moreover,
from Lemma 2.10 we have that $\tau(s)$ does not have critical point on
$(\omega_{-},s_{0})$ and it has at most one critical point on
$[s_{0},\omega_{+})$.

Claim. There exists a unique $s_{1}>s_{0}$ such that $c+a\tau(s_{1})=0$ and
$c+a\tau(s)<0$ for all $s>s_{1}$. In fact, assume by contradiction that there
exists a solution $\psi(s)\in S$, $s\in I$, of (2.13) such that $c+a\tau(s)>0$
and $\tau(s)<0$ for all $s>s_{0}$. Then the third equation of (2.13) implies
that the positive function $\eta(s)$ is increasing on $(s_{0},\omega_{+})$. It
follows from Lemma 2.18 that $\alpha(s)$ is unbounded on $(s_{0},\omega_{+})$.
Since $\alpha(s)$ is an increasing function on $(s_{0},\omega_{+})$ and
$2\alpha(s)\eta(s)+\tau^{2}(s)=-1$, then
$\lim_{s\to\omega_{+}}2\alpha(s)\eta(s)=-\infty$ and
$\lim_{s\to\omega_{+}}\tau^{2}(s)=+\infty$, which contradicts the fact that
$c+a\tau(s)>0$ with $\tau(s)<0$ and $c>0$ for all $s>s_{0}$. Therefore, there
exists $s_{1}\in I$ such that $c+a\tau(s_{1})=0$. Suppose that there exist
$s_{1},s_{2}\in(s_{0},\omega_{+})$ such that $c+a\tau(s_{1})=c+a\tau(s_{2})=0$
and $c+a\tau(s)\neq 0$ for all $s\in(s_{1},s_{2})$. Thus, $s_{1}$ and $s_{2}$
are consecutive critical points and local maximum points of $\eta(s)$, this is
a contradiction. Therefore, $s_{1}$ is unique.

Our Claim implies that $[c+a\tau(s)]\tau(s)>0$ for all $s>s_{1}$. Thus, (2.13)
implies that the positive function $\eta(s)$ is decreasing on
$(s_{1},\omega_{+})$ i.e. $\eta(s)$ is bounded on $(s_{1},\omega_{+})$. Hence,
it follows from Lemma 2.16 that
$\lim_{s\to+\infty}c+a\tau(s)=\lim_{s\to+\infty}\eta(s)=0$.

Finally, since $\tau(s)>0$ and the functions $\alpha(s)$, $\tau(s)$ and
$\eta(s)$ are monotone on $(\omega_{-},s_{0})$, then it follows from (2.13)
that $\eta(s)$ is a decreasing positive function on $(\omega_{-},s_{0})$.
Hence, $\lim_{s\to\omega_{-}}\eta(s)\alpha(s)=-\infty$. Therefore,
$\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}\tau(s)=+\infty.$ ∎

######  Lemma 2.27.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c>0$ and
initial condition $\psi(0)\in C$. Then either
$\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}c+a\tau(s)=+\infty$ or
$\lim_{s\to-\infty}c+a\tau(s)=c$, and $\lim_{s\to+\infty}c+a\tau(s)=\beta$
where $\beta\in\\{0,c\\}$. Moreover, if $\alpha(s)$ is bounded on
$(\omega_{-},\overline{s})$ (resp. $(\overline{s},\omega_{+})$),
$\overline{s}\in I$ then
$\lim_{s\to-\infty}\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=0$ (resp.
$\lim_{s\to+\infty}\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=0$).

###### Proof.

Item i) of Lemma 2.13 implies that there exist $s_{1},s_{2}\in I$ such that
$\alpha(s)$, $\tau(s)$ and $\eta(s)$ are monotone functions on
$(\omega_{-},s_{1})$ and $(s_{2},\omega_{+})$. If $\alpha(s)$ is bounded on
$(\omega_{-},s_{1})$, then it follows from Lemma 2.15 that
$\omega_{-}=-\infty$, $\lim_{s\to-\infty}\tau(s)=0$ and
$\lim_{s\to-\infty}\tau^{\prime}(s)=0$ i.e. $\lim_{s\to-\infty}c+a\tau(s)=c$.
Lemma 2.6 implies that
$\lim_{s\to-\infty}\alpha^{2}(s)=\lim_{s\to-\infty}\eta^{2}(s)=0$. If
$\alpha(s)$ is unbounded on $(\omega_{-},s_{1})$, then
$\lim_{s\to\omega_{-}}\alpha(s)=-\infty$ and it is an increasing function on
$(\omega_{-},s_{1})$ i.e. $\tau(s)>0$ for all $s<s_{1}$. Thus,
$[c+a\tau(s)]\tau(s)>0$ and $\eta(s)$ is a decreasing positive function for
all $s<s_{1}$. Since $2\alpha(s)\eta(s)+\tau^{2}(s)=1$ it follows that
$\tau(s)$ is unbounded on $(\omega_{-},s_{1})$ and
$\lim_{s\to\omega_{-}}c+a\tau(s)=+\infty$.

If $\alpha(s)$ is bounded on $(s_{2},\omega_{+})$, similarly to the previous
case we can prove that then $\omega_{+}=+\infty$
$,\lim_{s\to+\infty}\tau(s)=0$ and
$\lim_{s\to-\infty}\alpha^{2}(s)=\lim_{s\to-\infty}\eta^{2}(s)=0$. Now,
suppose that $\alpha(s)$ is unbounded on $(s_{2},\omega_{+})$, then
$\lim_{s\to\omega_{+}}\alpha(s)=-\infty$, $\tau(s)<0$ for all $s>s_{2}$.
Assume by contradiction that $\tau(s)$ is unbounded on $(s_{2},\omega_{+})$.
Thus, there exists $s_{3}>s_{2}$ such that $[c+\tau(s)]\tau(s)>0$ for all
$s>s_{3}$ and $\eta(s)$ is a positive decreasing function on
$(s_{3},\omega_{+})$ i.e. $\eta(s)$ is bounded on $(s_{3},\omega_{+})$ which
contradicts the third equation of (2.13), because
$-\int_{s_{3}}^{s}[c+a\tau(s)]\tau(s)ds=\eta(s)-\eta(s_{3})$. Hence, $\tau(s)$
is bounded on $(s_{2},\omega_{+})$. Since
$\lim_{s\to\omega_{+}}\alpha(s)=-\infty$ then equation
$2\alpha(s)\eta(s)+\tau^{2}(s)=0$ implies that
$\lim_{s\to\omega_{+}}\eta(s)=0$ and $\eta(s)$ is bounded on
$(s_{2},\omega_{+})$. Therefore, Lemma 2.16 implies that
$\lim_{s\to+\infty}[c+a\tau(s)]=0.$ ∎

Finally, we can study the case $\psi(0)\in S$. Note that, when $a=c$ then
$\psi(s)=(-s,-1,0)$ is a trivial solution of $(\ref{s2})$ and it represents a
parable in $Q^{2}_{+}$. i.e. a trivial solution to the CF.

######  Lemma 2.28.

Consider $\Phi:S\rightarrow TS\subset\mathbb{R}^{3}$, given by
$\Phi(\alpha,\tau,\eta)=\left(\tau,c\alpha+a\tau\alpha-\eta,-c\tau-a\tau^{2}\right)$,
where $a,c>0$ and $S$ is given by (2.14). Then
$p=\left(-{1}/{\sqrt{2c}},0,-{c}/{\sqrt{2c}}\right)$ and $-p$ are singular
points of $\Phi$, and the eigenvalues of $d\Phi_{p}$ and $d\Phi_{-p}$ are

$\displaystyle\lambda_{p}=\frac{a\pm\sqrt{a^{2}+16c^{2}}}{2\sqrt{2c}}\hskip
28.45274pt\text{and}\hskip
28.45274pt\lambda_{-p}=\frac{-a\pm\sqrt{a^{2}+16c^{2}}}{2\sqrt{2c}}.$ (2.20)

###### Proof.

It follows from the fact that the singular points of $\Phi$ satisfy $\tau=0$,
$c\alpha=\eta$ and $1=2\alpha\eta+\tau^{2}$. ∎

The singular points of $\Phi$, $\pm p$, are saddle points. Each singular
solution of (2.13) with $c>0$ and $\psi(0)\in S$ corresponds to a hyperbole
obtained by intersecting the light cone with one of the planes
$x_{3}\sqrt{2c}=\pm 1$. Moreover, there exist $q,\overline{q}\in
S\setminus\\{p\\}$ such that $\lim_{s\to+\infty}\psi(s,q)=\pm p$ and
$\lim_{s\to-\infty}\psi(s,\overline{q})=\pm p$. Define the stable and unstable
sets

$\begin{array}[]{l}W^{s}(\pm p)=\\{q\in S:\lim_{s\to+\infty}\psi(s,q)=\pm
p\\},\\\ W^{u}(\pm p)=\\{q\in S:\lim_{s\to-\infty}\psi(s,q)=\pm
p\\}.\end{array}$ (2.21)

Our next lemma provides the behaviour of $\alpha(s)$ and $\tau(s)$ at the
boundary of the maximal interval of a non trivial solution of (2.13).

######  Lemma 2.29.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (2.13)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c>0$ and
initial condition $\psi(0)\in S$. Consider $W^{u}(\pm p)$ and $W^{s}(\pm p)$
as in (2.21).

i) If $\psi(0)\in S\setminus W^{u}(\pm p)$ (resp. $S\setminus W^{s}(\pm p)$)
then $\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$ (resp.
$\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$).

ii)Either $\lim_{s\to\omega_{-}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to-\infty}c+a\tau(s)=\beta$ and either
$\lim_{s\to\omega_{+}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to+\infty}c+a\tau(s)=\beta$, where the constant $\beta\in\\{0,c\\}$.

###### Proof.

i) The limits are obtained by using the same arguments of the proof of Lemma
2.18. In order to prove ii) we note that Item i) of Lemma 2.13 implies that
there exist $s_{1},s_{2}\in I$ such that $\alpha(s)$, $\tau(s)$ and $\eta(s)$
are monotone on $(\omega_{-},s_{1})$ and $(s_{2},\omega_{+})$. If $\psi(0)\in
W^{u}(\pm p)$ (resp. $\psi(0)\in W^{s}(\pm p)$) then by definition
$\omega_{-}=-\infty$, $\lim_{s\to-\infty}\tau(s)=0$ and
$\lim_{s\to-\infty}c+a\tau(s)=c$ (resp. $\omega_{+}=+\infty$,
$\lim_{s\to+\infty}\tau(s)=0$ and $\lim_{s\to+\infty}c+a\tau(s)=c$). If
$\psi(0)\in S\setminus W^{u}(\pm p)$ (resp. $S\setminus W^{s}(\pm p)$) then it
follows from i) that $\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$ (resp.
$\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$). Thus, Lemma 2.16 implies that if
$\psi(0)\in S\setminus W^{u}(\pm p)$ (resp. $\psi(0)\in S\setminus W^{s}(\pm
p)$) then either $\lim_{s\to\omega_{-}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to-\infty}c+a\tau(s)=0$ (resp. either
$\lim_{s\to\omega_{+}}|c+a\tau(s)|=+\infty$ or
$\lim_{s\to+\infty}c+a\tau(s)=0$). ∎

As a consequence of the previous lemma, we can determine the behaviour of the
curvature function at each end of a self-similar solution to the CF on
$Q^{2}_{+}$, when $c>0$.

######  Corollary 2.30.

Let $X:I\rightarrow Q^{2}_{+}$, $s\in I$ be a curve which is a self-similar
solution to the CF, with curvature $k(s)$, that corresponds to a non trivial
solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (2.13) defined on the
maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$, $c>0$ and initial
condition $\psi(0)\in H\cup C\cup S$. Then

i) If $\psi(0)\in H$, then $\omega_{+}=+\infty$,
$\lim_{s\to\omega_{-}}k(s)=+\infty$ and $\lim_{s\to+\infty}k(s)=0$.

ii) If $\psi(0)\in C\cup S$ then at each end of $X$ either $k(s)$ is unbounded
or it tends to $0$ or $c$.

###### Proof.

Since $k(s)=c+a\tau(s)$, $a>0$ $c>0$, i) and ii) follow from Lemmas 2.26, 2.27
and 2.29. ∎

We can now prove our main result

###### Proof of Theorem 1.7.

Consider any vector $v\in\mathbb{R}^{3}\setminus\\{0\\}$. Without loss of
generality we can assume that $v=ae$, where $a>0$ and

$\displaystyle e=\left\\{\begin{array}[]{l}(1,0,0)\hskip
8.5359pt\text{if}\hskip 8.5359ptv\hskip 8.5359pt\text{is a timelike
vector},\\\ (1,1,0)\hskip 8.5359pt\text{if}\hskip 8.5359ptv\hskip
8.5359pt\text{is a lightlike vector},\\\ (0,0,1)\hskip 8.5359pt\text{if}\hskip
8.5359ptv\hskip 8.5359pt\text{is a spacelike vector.}\end{array}\right.$

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of the system (2.13),
with $c\in\mathbb{R}$, defined on a maximal interval $I$ and initial condition
$\psi(0)\in\mathbb{R}^{3}$ such that

$\displaystyle
2\alpha(0)\eta(0)+\tau^{2}(0)=\left\\{\begin{array}[]{l}-1\hskip
8.5359pt\text{if}\hskip 8.5359ptv\hskip 8.5359pt\text{is a timelike
vector},\\\ 0\hskip 8.5359pt\text{if}\hskip 8.5359ptv\hskip 8.5359pt\text{is a
lightlike vector},\\\ 1\hskip 8.5359pt\text{if}\hskip 8.5359ptv\hskip
8.5359pt\text{is a spacelike vector,}\end{array}\right.$

It follows from Proposition 2.3 that, for each initial condition, there is a
non trivial self-similar solution to the CF, $X(s)$ in $Q^{2}_{+}$, with
curvature $k(s)=c+a\tau(s)$. Thus, the initial conditions of $\alpha,\tau$ and
$\eta$, which are given by two constants, determine the self-similar solution
to the CF. Therefore, for each fixed vector
$v\in\mathbb{R}_{1}^{3}\setminus\\{0\\}$ and each constant $c\in\mathbb{R}$,
there is a 2-parameter family of non trivial self-similar solutions to the CF
in ${Q}^{2}_{+}$. These curves are solitons when $c=0$. Moreover, there are
three classes of such solutions corresponding to the three types of vectors
$v$.

It follows from item iii) of Lemma 2.11 that the curvature function $k(s)$ has
at most two zeros. Moreover, Corollaries 2.21, 2.25 and 2.30 show that, at
each end, the curvature function is unbounded or it tends to one of the two
constants $\\{c,0\\}$.

Let $Y(s)$ be the associated curve to $X(s)$. When $k(s)\neq 0$ it follows
from Remark 1.1 and Proposition 2.1 that $-Y(s)$ is a self-similar solution to
the ICF on $Q_{+}^{2}$. Moreover, if $s_{1},s_{2}\in I$ are the zeros (resp.
$s_{1}$ is the zero) of $k(s)$ on $I$, then $Y_{1}=-Y|_{(\omega_{-},s_{1})}$,
$Y_{2}=-Y|_{((s_{1},s_{2})}$ and $Y_{3}=-Y|_{(s_{2},\omega_{+})}$ (resp.
$Y_{1}=-Y|_{(\omega_{-},s_{1})}$ and $Y_{2}=-Y|_{((s_{1},\omega_{+})}$) are
self-similar solutions to the ICF on $Q_{+}^{2}$. Therefore, given
$v\in\mathbb{R}_{1}^{3}\setminus\\{0\\}$ and $c\in\mathbb{R}$, there exist a
2-parameter family of self-similar solutions to the ICF in $Q_{+}^{2}$. Any
such curve may have at most three connected components. Since
$\tilde{k}(s)=k^{-1}(s)$ is the curvature of $-Y(s)$ then at each end of
$-Y(s)$ the curvature $\tilde{k}(s)$ is unbounded or it tends to one of the
following constants $\\{1/c,0\\}$. ∎

We conclude this section by providing explicit soliton solutions to the CF and
to the ICF. They are obtained by considering $c=0$ and $v$ a light like
vector.

######  Proposition 2.31.

Let $\tilde{X}:I\subset\mathbb{R}\rightarrow Q_{+}^{2}$ defined by
$\tilde{X}(s)=(\tilde{x}_{1}(s),\tilde{x}_{2}(s),\tilde{x}_{3}(s))$, $s\in I$
be a soliton solution to the ICF in $Q^{2}_{+}$ that corresponds the solution
$\tilde{\psi}(s)=(\tilde{\alpha}(s),\tilde{\tau}(s),\tilde{\eta}(s))$ of
(2.13) with $\tilde{\psi}(0)\in C$. Then $-\tilde{Y}(s)$ is a soliton solution
to the CF in $Q_{+}^{2}$, where
$\tilde{Y}(s)=(\tilde{y}_{1}(s),\tilde{y}_{2}(s),\tilde{y}_{3}(s))$ is the
associated curve of $\tilde{X}(s)$, and for $s$ such that
$s<a\tilde{\eta}_{0}$, $\tilde{n}_{0}=\tilde{\eta}(0)>0$, we have

$\tilde{X}(s)=\left(-\frac{\tilde{x}^{2}_{3}(s)+\tilde{\alpha}^{2}(s)}{2\tilde{\alpha}(s)},\frac{-\tilde{x}^{2}_{3}(s)+\tilde{\alpha}^{2}(s)}{2\tilde{\alpha}(s)},\tilde{x}_{3}(s)\right),\quad\tilde{\alpha}(s)=-\frac{\tilde{\tau}^{2}(s)}{2\tilde{\eta}(s)},\quad\tilde{\eta}(s)=-\frac{s}{a}+\tilde{\eta}_{0},$
$\tilde{Y}(s)=\left(-\frac{\tilde{y}^{2}_{3}(s)+\tilde{\eta}^{2}(s)}{2\tilde{\eta}(s)},\frac{-\tilde{y}^{2}_{3}(s)+\tilde{\eta}^{2}(s)}{2\tilde{\eta}(s)},\tilde{y}_{3}(s)\right),\quad\tilde{\tau}(s)=\frac{2a\tilde{\eta}^{2}(s)}{3}+\sqrt{\tilde{\eta}(s)}\left(\frac{3\tilde{\tau}_{0}-2a\tilde{\eta}^{2}_{0}}{3\sqrt{\tilde{\eta}_{0}}}\right),$
$\tilde{y}_{3}(s)=\frac{\tilde{\eta}(s)\tilde{x}_{3}(s)-\tilde{\tau}(s)}{\tilde{\alpha}(s)},\quad\tilde{x}_{3}(s)=\left(\tilde{\eta}^{3/2}(s)+D\right)^{2}\left(-\int{(\tilde{\eta}^{3/2}(s)+D)^{-2}}ds+C\right)$

and $\displaystyle
D=\frac{3}{2a}\left(\frac{3\tilde{\tau}_{0}-2a\tilde{\eta}^{2}_{0}}{3\sqrt{\tilde{\eta}_{0}}}\right).$
Moreover, if $D>0$

$\int{(\tilde{\eta}^{3/2}(s)+D)^{-2}}ds=\frac{1}{9{D}^{\frac{4}{3}}}\log\frac{\tilde{\eta}(s)-D^{\frac{1}{3}}\left(\sqrt{\tilde{\eta}(s)}-D^{\frac{1}{3}}\right)}{\left(\sqrt{\tilde{\eta}(s)}+D^{\frac{1}{3}}\right)^{2}}+$
$\hskip
96.73918pt+\frac{2}{3\sqrt{3}D^{\frac{4}{3}}}\left[\arctan\left(\frac{2\sqrt{\tilde{\eta}(s)}-D^{\frac{1}{3}}}{\sqrt{3}D^{\frac{1}{3}}}\right)+\frac{\sqrt{3}\,\tilde{\eta}(s)}{D^{-\frac{1}{3}}\tilde{\eta}(s)^{\frac{3}{2}}+D^{\frac{2}{3}}}\right].$
(2.24)

###### Proof.

Suppose that $\tilde{X}(s)$ is a soliton solution to the ICF, i.e. $c=0$. It
follows from system (2.15), with $c=0$, that
$\tilde{\eta}(s)=-{s}/{a}+\tilde{\eta}_{0}$. Let
$r(s)=\tilde{\eta}(s)=-{s}/{a}+\tilde{\eta}_{0}>0$,
$r_{0}=\tilde{\eta}_{0}>0$. Then

$\frac{dr}{ds}=-\frac{1}{a},\hskip
8.5359pt\displaystyle\frac{d\tilde{\alpha}}{ds}=-\frac{1}{a}\frac{d\tilde{\alpha}}{dr},\hskip
8.5359pt\displaystyle\frac{d\tilde{\tau}}{ds}=-\frac{1}{a}\frac{d\tilde{\tau}}{dr},\hskip
8.5359pt\displaystyle\frac{d\tilde{\eta}}{ds}=-\frac{1}{a}\frac{d\tilde{\eta}}{dr}\quad\text{and}\quad\left\\{\begin{array}[]{ll}\tilde{\alpha}^{\prime}(r)=-a\tilde{\tau}(r),\\\
\displaystyle\tau^{\prime}(r)=-\frac{\tilde{\alpha}(r)}{\tilde{\tau}(r)}+ar.\end{array}\right.$

Using that $2r\tilde{\alpha}(r)+\tilde{\tau}^{2}(r)=0$ we obtain
$\displaystyle-\frac{\tilde{\alpha}(r)}{\tilde{\tau}(r)}=\frac{\tilde{\tau}(r)}{2r}$
and $\displaystyle\tilde{\tau}^{\prime}(r)-\frac{\tilde{\tau}(r)}{2r}=ar$.
Taking $\tilde{\tau}(0)=\tau_{0}$ we have

$\tilde{\tau}(r)=\frac{2ar^{2}}{3}+r^{\frac{1}{2}}C_{1}=\frac{2ar^{2}}{3}+r^{\frac{1}{2}}\left(\frac{3\tilde{\tau}_{0}-2a\tilde{\eta}^{2}_{0}}{3\sqrt{\tilde{\eta}_{0}}}\right).$
(2.25)

Since $\tilde{\alpha}(s)=-\tilde{x}_{1}(s)+\tilde{x}_{2}(s)$,
$\tilde{\eta}(s)=-\tilde{y}_{1}(s)+\tilde{y}_{2}(s)$ and $\tilde{X}(s)$ and
$\tilde{Y}(s)$ are curves in $Q^{2}_{+}$ then
$-\tilde{x}^{2}_{1}(s)+[\tilde{\alpha}(s)+\tilde{x}_{1}(s)]^{2}+\tilde{x}^{2}_{3}(s)=0$,
$-\tilde{y}^{2}_{1}(s)+[\tilde{\eta}(s)+\tilde{y}_{1}(s)]^{2}+\tilde{y}^{2}_{3}(s)=0$.
Thus,

$\displaystyle
2\tilde{\eta}(s)\tilde{y}_{1}(s)=-\tilde{y}^{2}_{3}(s)-\tilde{\eta}^{2}(s),\,\,2\tilde{\eta}(s)\tilde{y}_{2}(s)=\tilde{\eta}^{2}(s)-\tilde{y}^{2}_{3}(s),\,\,\displaystyle
2\tilde{\alpha}(s)\tilde{x}_{1}(s)=-\tilde{x}^{2}_{3}(s)-\tilde{\alpha}^{2}(s)\,\,$

and $\displaystyle
2\tilde{\alpha}(s)\tilde{x}_{2}(s)=\tilde{\alpha}^{2}(s)-\tilde{x}^{2}_{3}(s).$
Now, we will determine the functions $\tilde{x}_{3}(s)$ and
$\tilde{y}_{3}(s)$. We know that
$\tilde{T}(s)=\tilde{X}(s)\times\tilde{Y}(s)$, where $\tilde{T}(s)$ is the
unit tangent vector field and $\tilde{Y}(s)$ is the associated curve to
$\tilde{X}$. Hence,

$\displaystyle\tilde{\tau}(s)=\langle\tilde{T}(s),(1,1,0)\rangle=-\tilde{x}^{\prime}_{1}(s)+\tilde{x}^{\prime}_{2}(s)=[-\tilde{x}_{1}(s)+\tilde{x}_{2}(s)]\tilde{y}_{3}(s)-[-\tilde{y}_{1}(s)+\tilde{y}_{2}(s)]\tilde{x}_{3}(s).$

Thus,
$\tilde{\tau}(s)=\tilde{\alpha}(s)\tilde{y}_{3}(s)-\tilde{\eta}(s)\tilde{x}_{3}(s)$
and
$\displaystyle\tilde{\alpha}(s)\tilde{y}_{3}(s)=\tilde{\eta}(s)\tilde{x}_{3}(s)+\tilde{\tau}(s)$.
The equation
$\tilde{\eta}(s)\tilde{X}(s)+\tilde{\tau}(s)\tilde{T}(s)+\tilde{\alpha}(s)\tilde{Y}(s)=(1,1,0)$
implies that
$\tilde{\eta}(s)\tilde{x}_{3}(s)+\tilde{\tau}(s)\tilde{x}^{\prime}_{3}(s)+\tilde{\alpha}(s)\tilde{y}_{3}(s)=0$.
Hence,
$2\tilde{\eta}(s)\tilde{x}_{3}(s)+\tilde{\tau}(s)\tilde{x}^{\prime}_{3}(s)+\tilde{\tau}(s)=0$
and

$\tilde{x}^{\prime}_{3}(s)+2\frac{\tilde{\eta(s)}}{\tilde{\tau}(s)}\tilde{x}_{3}(s)=-1.$

Taking $r(s)=\tilde{\eta}(s)$ i.e.
$\displaystyle\frac{d\tilde{x}_{3}}{ds}=-\frac{1}{a}\frac{d\tilde{x}_{3}}{dr}$
and using (2.25), we obtain
${\tilde{x}^{\prime}_{3}(r)-3\frac{r^{\frac{1}{2}}}{r^{\frac{3}{2}}+\frac{3}{2a}C_{1}}}\tilde{x}_{3}(r)=a.$
Thus,

$\tilde{x}_{3}(r)=\frac{1}{\mu(r)}\left(a\int{\mu(r)}dr+C\right),\quad\text{where}\quad\mu(r)=\left(r^{3/2}+\frac{3}{2a}C_{1}\right)^{-2}.$

Taking $\displaystyle 2aD=3C_{1}$ we have
$\tilde{x}_{3}(r)=\left(r^{3/2}+D\right)^{2}\left(a\int{(r^{3/2}+D)^{-2}}dr+C\right).$
If $D>0$ we obtain (2.24). ∎

##  3 Visualizing some Self-Similar Solutions to the CF and ICF

In this section, we visualize some examples of self-similar solutions to the
CF and ICF on $Q_{+}^{2}$. We consider $Q_{+}^{2}$ parametrized by
$\chi(\rho,\varphi)=(\rho,\rho\,\cos\varphi,\rho\,\sin\varphi),\quad\rho>0$.
If a curve $X(s)=\chi(\rho(s),\varphi(s))$ is parametrized by arc length $s\in
I$, then $[\rho(s)\varphi^{\prime}(s)]^{2}=1.$ Taking
$\rho(s)\varphi^{\prime}(s)=1$, the unit tangent vector field is given by
$T=\left(\rho^{\prime},\rho^{\prime}\cos\varphi)-\sin\varphi,\rho^{\prime}\sin\varphi+\cos\varphi\right)$.
Moreover, the functions $\rho(s)$, $\varphi(s)$ and the curvature function
$k(s)$ satisfy the following system of ordinary differential equations

$\left\\{\begin{array}[]{ll}\displaystyle\rho^{\prime\prime}=\rho
k+\frac{1+[\rho^{\prime}]^{2}}{2\rho},\\\
\varphi^{\prime}\rho=1.\end{array}\right.$ (3.1)

The curve $Y(s)$ associated to $X(s)$ can be written as

$\displaystyle
Y=\frac{1}{2\rho}\left(-{1+[\rho^{\prime}]^{2}},{2\rho^{\prime}\sin\varphi+[1-(\rho^{\prime})^{2}]\cos\varphi},{-2\rho^{\prime}\cos\varphi+[1-(\rho^{\prime})^{2}]\sin\varphi}\right).$

It follows from Theorem 1.5 that $X(s)$, is a non trivial self-similar
solution to the CF if and only if there exist $v\in\mathbb{R}_{1}^{3}$,
$c,a\in\mathbb{R}$ with $a>0$ such that $k(s)=c+a\langle T(s),v\rangle$.
Whenever $k(s)\neq 0$ it follows from Remark 1.1 and Proposition 2.1 that
$-Y(s)$ is a self-similar solution to the ICF. Since $k(s)$ has at most two
zeros (see Theorem 1.7) it follows that $-Y(s)$ may have at most three
connected components which are self-similar solutions to the ICF. In what
follows, we use (3.1) and the software Maple to plot $X(s)$ and $-Y(s)$ on
$Q_{+}^{2}$, for several choices of $c,\,a$ and $v$. In Figures 1-3, we
visualize soliton solutions ($c=0$) $X$ to the CF and $-Y$ solutions to the
ICF on $Q_{+}^{2}$, when $v$ is a timelike, lightlike and spacelike vector,
respectively. Similarly, in Figures 4-7 we visualize self-similar solutions
$X$ to the CF and $-Y$ solutions to the ICF for $c<0$ and in Figures 8-10 for
$c>0$.

By considering $a=1,\,c=0$ and the timelike vector $v=(1,0,0)$, in Figure
(1a), we visualize a soliton solution $X$ to the CF and Figure (1b) shows the
graph of its curvature. It follows from Lemmas 2.17 and 2.19 that $k(s)$ has a
unique zero and $\lim_{s\to+\infty}k(s)=0$. The curve $-Y$ associated to $X$
has two connected components visualized in Figures (1c) and (1d), which are
soliton solutions to the ICF.

(a)

(b)

(c)

(d)

Figure 1: Soliton solutions (a) to the CF and (c) (d) to ICF on $Q^{2}_{+}$
with $c=0$, $a=1$, $v=(1,0,0)$.

Proposition 2.31 provides explicit soliton solutions $-\tilde{Y}$ to the CF
and $\tilde{X}$ to the ICF. Considering $c=0$, $a=1$ and the lightlike vector
$v=(1,1,0)$, one can choose the initial conditions so that the constant $D=0$.
The soliton solutions $\tilde{Y}(s)$ and $\tilde{X}(s)$ are defined for $s<1$
and they are visualized in Figures (2a) and (2c) respectively. Figure 2b
provides the graph of the curvature of $\tilde{Y}(s)$.

(a)

(b)

(c)

Figure 2: Soliton solutions (a) to the CF and (c) to ICF on $Q^{2}_{+}$ with
$c=0$, $a=1$ and $v=(1,1,0)$.

In Figure 3a, we visualize a soliton solution $X$ to the CF, by choosing
$c=0$, $a=1$ and the spacelike vector $v=(0,0,1)$. Figure 3b shows the graph
of $k(s)$. Since the curvature vanishes at two points, the curve $-Y$
associated to $X$ has three connected components which are solutions to the
ICF and they are visualized in Figures 3c, 3d and 3e.

(a)

(b)

(c)

(d)

(e)

Figure 3: Soliton solutions to the CF and ICF on $Q^{2}_{+}$ with $c=0$,
$a=1$, $v=a(0,0,1)$.

When $c<0$ and $v=a(1,0,0)$ is a timelike vector, then Lemma 2.22 implies that
$c$ determines the singular point $p$ of $\Phi$ and the associated eigenvalues
are real if $4|c|\leq a$ and complex if $4|c|>a$. In Figures 4 and 5, we
visualize the self-similar solutions to the CF and to the ICF in each case.
Thus, $\psi(s)=p$, $s\in I$ is the singular solution of (2.13) and it
corresponds to a circle with radius $\displaystyle-1/(2c)$ in $Q^{2}_{+}$, and
its associated curve is a circle with radius $-c/2$ in $Q^{2}_{-}$. Moreover,
Lemma 2.23 implies that $\lim_{s\to+\infty}k(s)=c$ and $\psi(s)=p$, $s\in I$
is a global attractor solution of (2.13) in the set $H$.

We now consider self-similar solution with $c<0$. In Figure 4a, we visualize a
self-similar solution $X$ to the CF on $Q_{+}^{2}$ for $c=-{1}/{4}$,
$a={1}/{6}$ and $v=a(1,0,0)$ is a timelike vector. Figure 4b shows the graph
of $k(s)$, which vanishes at one point. In Figure 4c, we visualize one
component of the self-similar solution to the ICF on $Q_{+}^{2}$, associated
to $X$, that corresponds to $k(s)<0$.

(a)

(b)

(c)

Figure 4: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with
$c=-\frac{1}{4}$, $a=\frac{1}{6}$, $v=a(1,0,0)$.

In Figure 5a, we visualize a self-similar solution $X$ to the CF on
$Q_{+}^{2}$ for $c=-{1}/{4}$, $a=5$ and $v=a(1,0,0)$ is a timelike vector.
Figure 5b shows the graph of $k(s)$, which vanishes at one point. In Figure 5c
we visualize one component of the self-similar solution to the ICF on
$Q_{+}^{2}$ that corresponds to $k(s)<0$.

(a)

(b)

(c)

Figure 5: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with
$c=-\frac{1}{4}$, $a=5$ and $v=a(1,0,0)$.

In Figure 6a, we visualize a self-similar solution $X$ to the CF in
$Q_{+}^{2}$ when $c=-2$, $a={1}/{2}$ and $v=a(1,1,0)$ is a lightlike vector.
Figure 6b shows the graph of $k(s)$, which vanishes at one point. It follows
from Lemma 2.23 that $\lim_{s\to+\infty}k(s)=c=-2$. In Figure 6c, we visualize
one component of the self-similar solution to the ICF on $Q_{+}^{2}$,
associated to $X$, that corresponds to $k(s)<0$.

(a)

(b)

(c)

Figure 6: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with $c=-2$,
$a=\frac{1}{2}$ and $v=a(1,1,0)$.

In Figure 7a, we visualize a self-similar solution $X$ to the CF on
$Q_{+}^{2}$ when $c=-1$, $a=1$ and $v=(0,0,1)$ is a spacelike vector. Figure
7b shows the graph of $k(s)$ which vanishes at two points. In Figures 7c, 7d
and 7e, we visualize the three connected components of the self-similar
solution to the ICF on $Q_{+}^{2}$, associated to $X$.

(a)

(b)

(c)

(d)

(e)

Figure 7: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with $c=-1$,
$a=1$ and $v=(0,0,1)$.

We conclude considering self-similar solutions with $c>0$. In Figure 8a, we
visualize a self-similar solution $X$ to the CF on $Q_{+}^{2}$ when $c=4$,
$a=1$ and $v=(1,0,0)$ is a timelike vector. Figure 8b shows the graph of
$k(s)$, which vanishes at one point. Lemma 2.26 implies that
$\lim_{s\to+\infty}k(s)=0.$ Finally, in Figures 8c and 8d we visualize the two
components of the self-similar solution to the ICF on $Q_{+}^{2}$ associated
to $X$.

(a)

(b)

(c)

(d)

Figure 8: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with $c=4$,
$a=1$ and $v=(1,0,0)$.

In Figure 9a, we visualize a self-similar solution $X$ to the CF on
$Q_{+}^{2}$ when $c=4$, $a=1$ and $v=(1,1,0)$ is a lightlike vector. Figure 9b
shows the graph of $k(s)$, which vanishes at one point. Lemma 2.27 implies
that when $s$ tends to $+\infty$ then $k(s)$ tends to $0$ or $c$. In Figures
9c and 9d we visualize the two components of the self-similar solutions to the
ICF on $Q_{+}^{2}$ associated to $X$.

(a)

(b)

(c)

(d)

Figure 9: Self-similar solutions to the CF and ICF on $Q^{2}_{+}$ with $c=4$,
$a=1$ and $v=(1,1,0)$.

Finally, in Figure 10a, we visualize a self-similar solution $X$ to the CF on
$Q_{+}^{2}$ when $c=3$, $a=1$ and $v=(0,0,1)$ is a spacelike vector. Figure
10b shows the graph of $k(s)$, which vanishes at two points. Finally, in
Figures 10c, 10d and 10e we visualize the three components of the self-similar
solutions to the ICF on $Q_{+}^{2}$ associated to $X$.

(a)

(b)

(c)

(d)

(e)

Figure 10: Self-similar solutions to the CF and ICF in $Q^{2}_{+}$ with $c=3$,
$a=1$ and $v=(0,0,1)$.

## References

  * [1] Abresch, U.; Langer, J. The normalized curve shortening flow and homothetic solutions, Journal Differential Geometry 23, n. 2, p. 175–196 (1986).
  * [2] Andrews, B. Classification of limiting shapes for isotropic curve flows, Journal of the American Mathematical Society, v. 16, 443-459 (2002).
  * [3] Angenent, S. B. On the formation of singularities in the curve shortening flow, Journal of Differential Geometry, v. 33, 601-633 (1991).
  * [4] Bourni, T., Langford, M.; Tinaglia, G. Convex ancient solutions to curve shortening flow, Calc. Var. 59, 133 (2020).
  * [5] Dos Reis, H. F. S.; Tenenblat, K. Soliton solutions to the curve shortening flow on the sphere, Proc. Amer. Math. Soc., v. 147, 4955-4967 (2019).
  * [6] Drugan, G.; Lee, H.; Wheeler, G. Solitons for the inverse mean curvature flow Pacific Journal of Mathematics, v. 284, no. 2, 309–326 (2016).
  * [7] Epstein, C.L.; Weinstein, M.I. A stable manifold theorem for the curve shortening equation, Communications on Pure and Applied Mathematics., v. 40, 119-139 (1987).
  * [8] Gage, M. E., Curve shortening makes convex curves circular, Inventiones mathematicae, v. 76, n.2, p. 357–364 (1984).
  * [9] Gage, M. E., Curve shortening on surfaces, Annales scientifiques de École Normale Supérieure, v.23, no 2, p. 229-256 (1990).
  * [10] Giga, Y., Surface evolutions equations. A level set approach, Monographs in Mathematics, vol. 99, Birkhauser, Basel, 2006.
  * [11] Grayson, M. A. The heat equation shrinks embedded plane curves to round a points, Journal Differential Geometry, v.26, p. 285-314 (1987).
  * [12] Grayson, M. A. Shortening embedded curves, Annals of Mathematics, v. 129, n.1, p. 71–111 (1989).
  * [13] Halldorsson, H. P. Self-similar solutions to the curve shortening flow, Transactions of the American Mathematical Society, v. 364, n. 10, p. 5285–5309 (2012).
  * [14] Halldorsson, H. P. Self-similar solutions to the mean curvature flow in the Minkowski plane $\mathbb{R}^{1,1}$, Journal für die reine und angewandte Mathematik (Crelles Journal), v. 704, 209–243 (2015).
  * [15] Liu, H. Curves in the lightlike cone, Beiträge zur Algebra und Geometrie, v. 45 (2004).
  * [16] Ma, L.; Chen, D. Curve shortening in a Riemannian manifold, Annali di Matematica Pura ed Applicata v. 186, p. 663-684 (2007).
  * [17] Silva, F. N.; Tenenblat, K. Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane, arXiv:2101.07916v2.
  * [18] Urbas, J. An expansion of convex hypersurfaces, Journal of Differential Geometry, v.33, p. 91-125 (1991).
  * [19] Urbas, J. Convex curves moving homothetically by neagtive powers of their curvature, Asian Journal of Mathematics, v.3, p. 635-658 (1999).