# Inhomogeneous domain walls in spintronic nanowires

L. Siemer111Universität Bremen<EMAIL_ADDRESS>Corresponding
author I. Ovsyannikov222Universität Hamburg, Lobachevsky State University of
Nizhny Novgorod J.D.M. Rademacher333Universität Bremen

###### Abstract

In case of a spin-polarized current, the magnetization dynamics in nanowires
are governed by the classical _Landau-Lifschitz_ equation with _Gilbert_
damping term, augmented by a typically non-variational _Slonczewski_ term.
Taking axial symmetry into account, we study the existence of domain wall type
coherent structure solutions, with focus on one space dimension and spin-
polarization, but our results also apply to vanishing spin-torque term. Using
methods from bifurcation theory for arbitrary constant applied fields, we
prove the existence of domain walls with non-trivial azimuthal profile,
referred to as _inhomogeneous_. We present an apparently new type of domain
wall, referred to as _non-flat_ , whose approach of the axial magnetization
has a certain oscillatory character. Additionally, we present the leading
order mechanism for the parameter selection of _flat_ and _non-flat_
inhomogeneous domain walls for an applied field below a threshold, which
depends on anisotropy, damping, and spin-transfer. Moreover, numerical
continuation results of all these domain wall solutions are presented.

##  1 Introduction

Magnetic domain walls (DWs) are of great interest both from a theoretical
perspective and for applications, especially in the context of innovative
magnetic storages [1]. Recent developments in controlled movement of DWs via
spin-polarized current pulses in nanomagnetic structures, in particular in
nanowires, are thought to lead to a new class of potential non-volatile
storage memories, e.g. racetrack memory [1, 2, 3, 4]. These devices make use
of the fact that spin-transfer driven effects can change the dynamics in
sufficiently small ferromagnetic structures (e.g. nanowires), where regions of
uniform magnetization, separated by DWs, can appear [5, 6, 7]. This motivates
further studies of the existence of magnetic domains and their interaction
with spin-polarized currents as a building block for the theory in this
context. In this paper we take a mathematical perspective and, in a model for
nanomagnetic wires, rigorously study the existence of DWs. This led us to
discover an apparently new kind of DWs with a certain inhomogeneous and
oscillatory structure as explained in more detail below.

The description of magnetization dynamics in nanomagnetic structures, governed
by the _Landau-Lifschitz-Gilbert_ (LLG) equation, is based on works by Berger
and Slonczewski assuming a spin-polarized current [8, 9]. In the presence of a
constant applied field and a spin-polarized current, the dynamics driven by
the joint action of magnetic field and spin torque can be studied by adding a
spin-transfer term in the direction of the current (current-perpendicular-to-
plane (CPP) configuration). In case of a spatially uniform magnetization, the
resulting _Landau-Lifschitz-Gilbert-Slonczewski_ (LLGS) equation for unit
vector fields
$(m_{1},m_{2},m_{3})=\boldsymbol{m}=\boldsymbol{m}(x,t)\in\mathbb{S}^{2}$ (cf.
Figure 1) reads

$\partial_{t}\boldsymbol{m}-\alpha\boldsymbol{m}\times\partial_{t}\boldsymbol{m}=-\boldsymbol{m}\times\boldsymbol{h}_{\textnormal{eff}}+\boldsymbol{m}\times\left(\boldsymbol{m}\times\boldsymbol{J}\right).$
(LLGS)

with effective field $\boldsymbol{h}_{\textnormal{eff}}$, Gilbert damping
factor $\alpha>0$, and the last term is the so-called polarized spin transfer
pseudotorque.

Note that the above equation reduces to the LLG equation for
$\boldsymbol{J}\equiv 0$, see §2 for more details.

In this paper we consider the axially symmetric case and set

$\boldsymbol{h}_{\textnormal{eff}}\coloneqq\partial^{2}_{x}\boldsymbol{m}+\boldsymbol{h}-\mu\,\textnormal{m}_{3}\,\boldsymbol{e}_{3}\,,\quad\boldsymbol{J}\coloneqq\frac{\beta}{1+c_{\textnormal{cp}}m_{3}}\boldsymbol{e}_{3}\,,$
(1)

where $\boldsymbol{h}=h\,\boldsymbol{e}_{3}$ with a uniform and time-
independent field strength $h\in\mathbb{R}$, and
$\textnormal{m}_{3}=\langle\boldsymbol{m},\boldsymbol{e}_{3}\rangle$,
$\boldsymbol{e}_{3}\in\mathbb{S}^{2}$. This effective field
$\boldsymbol{h}_{\textnormal{eff}}$ also includes the diffusive exchange term
$\partial_{x}^{2}\boldsymbol{m}$, the uniaxial anisotropy and demagnetization
field. The specific here with parameter $\mu\in\mathbb{R}$ derives from a
first order approximation in the thin film/wire limit for a uniformly
magnetized body [6, 10]. In the axially symmetric structure, $\beta\geq 0$ and
$c_{\textnormal{cp}}\in(-1,1)$ describe the strength of the spin-transfer and
the ratio of the polarization [7, 11]. The spin-transfer torque term may
provide energy to the system under certain conditions and counterbalance
dissipation associated to the Gilbert damping term, which gives rise to
coherent non-variational dynamics, see e.g. [12].

Notably, for $\beta=0$ one obtains the LLG-equation that does not account for
spin transfer effects. Moreover, as shown in [12], this also holds up to
parameter change in case $c_{\textnormal{cp}}=0$. Hence, solutions to the LLGS
equation for $\beta=0$ or $c_{\textnormal{cp}}=0$ are also solutions to the
LLG equation, so that all the analytical as well as numerical results for
$c_{\textnormal{cp}}=0$ in this paper directly transfer to the LLG equation.

(a)

(b)

(c)

Figure 1: Homogeneous DW profile ($q\equiv 0$) with
$\alpha=0.5,\beta=0.1,\mu=-1,h=50,c_{\textnormal{cp}}=0$. (a)
$(m_{2},m_{3})$-profile. (b) Projection onto $\mathbb{S}^{2}$. (c) Zoom-in on
$m_{1}$ (blue solid) and $m_{2}$ (red dashed).

A key ingredient for the separation of uniformly magnetized states in space
are interfaces between two magnetic domains. The most coherent form of such
interfaces in the uniaxial setting are relative equilibria with respect to
translation and rotation symmetry of the form

$\boldsymbol{m}(\xi,t)=\boldsymbol{m}_{0}(\xi)\mathrm{e}^{\mathrm{i}\varphi(\xi,t)},\quad\textnormal{where
}\xi=x-st\textnormal{ and }\varphi(\xi,t)\coloneqq\phi(\xi)+\Omega t,$

with _speed_ $s$ and _frequency_ $\Omega$. Here the complex exponential acts
on $\boldsymbol{m}_{0}\in\mathbb{S}^{2}$ by rotation about the
$\boldsymbol{e}_{3}$-axis, i.e., the azimuth, and in spherical coordinates we
can choose
$\boldsymbol{m}_{0}(\xi)=\left(\sin(\theta(\xi)),0,\cos(\theta(\xi))\right)$
with altitude angle $\theta$.

We refer to such solutions with
$\boldsymbol{m}_{0}(\xi)\to\pm\boldsymbol{e}_{3}$ as $\xi\to\pm\infty$ or
$\xi\to\mp\infty$ as _domain walls_. A first classification of DWs is based on
the _local wavenumber_ $q:=\phi^{\prime}$, which determines $\phi$ uniquely
due to the axial rotation symmetry and satisfies

$q(\xi)=\frac{\langle(m_{1}^{\prime},m_{2}^{\prime}),(-m_{2},m_{1})\rangle}{1-m_{3}^{2}}(\xi).$
(2)

######  Definition 1.

We call a DW with constant $\phi$ _homogeneous (hom)_ , i.e., $q\equiv 0$, and
_inhomogeneous_ otherwise.

Inhomogeneous DWs have a spatially inhomogeneous varying azimuthal angle,
compare Figures 1 and 2.

In the case of uniaxial symmetry and the LLG case $\beta=0$, an explicit
family of homogeneous DWs was discovered in [13] for applied fields with
arbitrary strength and time dependence, cf. Figure 1. Furthermore, for
constant applied fields and in case of $c_{\textnormal{cp}}\neq 0$ it was
shown in [12] that DWs cannot be homogeneous, and the existence of
inhomogeneous DWs was proven, whose spatial profile slowly converges to
$\pm\boldsymbol{e}_{3}$ and where $|s|\gg 1$. This latter type of DWs is
‘weakly localized’ and has large ‘width’ in the sense that the inverse slope
of $m_{3}$ at $m_{3}(0)=0$ tends to infinity as $|s|\to\infty$.

small applied field (a)

large applied field (b)

(c)

(d)

Figure 2: Shown are profiles of inhomogeneous DWs $\boldsymbol{m}(\xi)$
computed by parameter continuation, cf. §4, in $c_{\textnormal{cp}}$ to
$c_{\textnormal{cp}}=0.5$ with fixed $\alpha=0.5,\beta=0.1,\mu=-1$. (a,c)
codim 2 case $h=0.5,s=0.112027,\Omega=0.447173$, (b,d) codim 0 case
$h=50,s=19.92,\Omega=40.4$. (c) magnification of the $m_{1}$-component; note
the change of frequency for small vs. large $\xi$. (d) Magnification of
$m_{1}$ (blue solid) as well as $m_{2}$ (red dashed) component.

An apparently thus far unrecognized distinction of DWs is based on the
convergence behavior of $q$ as $\xi\to\pm\infty$.

######  Definition 2.

We call a DW _flat_ if $|q(\xi)|$ has a limit on $\mathbb{R}\cup\\{\infty\\}$
as $|\xi|\to\infty$ and _non-flat_ otherwise.

Note that homogeneous DWs are flat ones by definition (recall
$\phi^{\prime}=q$). Moreover, for all DWs $m_{0}(\xi)$ converges to
$\boldsymbol{e}_{3}$ or $-\boldsymbol{e}_{3}$ as $|\xi|\to\infty$.

The main result of this paper is an essentially complete understanding of the
existence and the type of DWs near the aforementioned explicit solution family
for a nanowire geometry, i.e., $\mu<0$. This includes the LLG case $\beta\cdot
c_{\textnormal{cp}}=0$, but our focus is on the spintronic case $\beta\cdot
c_{\textnormal{cp}}\neq 0$ for which these results pertain
$0<|c_{\textnormal{cp}}|\ll 1$ and any value of the (constant) applied field
$h$.

The different types of DWs occur in parameter regimes close to
$c_{\textnormal{cp}}=0$ in the (spatial) ODE which results from the coherent
structure ansatz. Since the parameters $\alpha$ and $\mu$ are material-
dependent we take the applied field strength $h$ as the primary parameter.

In brief, organized by stability properties of the steady states
$\pm\boldsymbol{e}_{3}$ in the ODE, this leads to the following cases and
existence results for localized DWs in nanowires ($\mu<0$):

  * •

‘codim-2’ ($h_{\ast}<h<h^{\ast}$) : existence of flat inhomogeneous DWs with
$s,\Omega$ selected by the other parameters,

  * •

‘center’ ($h=h_{\ast}$ or $h=h^{\ast}$) : existence of flat and non-flat
inhomogeneous DWs,

  * •

‘codim-0’ ($h<h_{\ast}$ or $h>h^{\ast}$) : existence of flat inhomogeneous
DWs,

where $h_{\ast}\coloneqq\beta/\alpha+\frac{2\mu}{\alpha^{2}}(1+\alpha^{2})$ as
well as $h^{\ast}\coloneqq\beta/\alpha-\frac{2\mu}{\alpha^{2}}(1+\alpha^{2})$.
Note that $h_{\ast}<h^{\ast}$ always (recall $\alpha>0$ and $\mu<0$). Due to
symmetry reasons, we mainly discuss the existence of right-moving DWs close to
the explicit solution family and thus focus on an applied field
$\beta/\alpha\leq h$ (cf. §3). The main results can be directly transferred to
the case of left-moving DWs ($h\leq\beta/\alpha$). Notably, the codim-0 case
occurs for ‘large’ magnetic field $h$ above a material dependent threshold. In
the center and codim-2 cases there is a selection of $s$ and $\Omega$ by the
existence problem.

The basic relation between the PDE and the ODE stability properties w.r.t. $h$
and $c_{\textnormal{cp}}$ are illustrated in Figure 3 for
$\alpha=1,\beta=0.5,\mu=-1$ fixed. Due to the fact that $s$ and $\Omega$ are
ODE parameters only, the diagram illustrates a slice in the four dimensional
parameter space with axes $h,c_{\textnormal{cp}},s$, and $\Omega$. Note that
homogeneous DWs (hom) can occur only on the line $c_{\textnormal{cp}}\equiv 0$
(see [12, Theorem 5] for details). The stability regions are defined as
follows. _monostable -_ (blue): $+\boldsymbol{e}_{3}$ unstable and
$-\boldsymbol{e}_{3}$ stable, _bistable_ (shaded blue): both
$+\boldsymbol{e}_{3}$ and $-\boldsymbol{e}_{3}$ stable, _monostable +_ (red):
$+\boldsymbol{e}_{3}$ stable and $-\boldsymbol{e}_{3}$ unstable, _unstable_
(shaded red): both $+\boldsymbol{e}_{3}$ and $-\boldsymbol{e}_{3}$ unstable.
For a more detailed stability discussion, see Remark 5. Note that the
transition from bistable to monostable in the PDE does not coincide with the
transition, of the homogeneous family, from codim-2 to codim-0 in the ODE. In
contrast, the analogous transitions occur simultaneously for example in the
well-known _Allen-Cahn_ or _Nagumo_ equation.

Figure 3: Stability diagram of homogeneous states $\pm\boldsymbol{e}_{3}$ in
$h$ and $c_{\textnormal{cp}}$ for $\alpha=1,\beta=0.5$, and $\mu=-1$. State
$+\boldsymbol{e}_{3}$ unstable to the left and stable to the right of
$\Gamma^{+}$, $-\boldsymbol{e}_{3}$ stable to the left and unstable to the
right of $\Gamma^{-}$. Homogeneous DWs (hom) exist only on the $h$-axis, i.e.,
$c_{\textnormal{cp}}\equiv 0$. See text for further explanations. Note that
also negative applied fields are shown.

In Figure 4 below, we present numerical evidence that inhomogeneous DWs are
indeed also dynamically selected states, especially for large applied fields,
also in the LLG case ($\beta\,,c_{\textnormal{cp}}=0$).

The understanding of DW selection by stability properties generally depends on
the existence problem discussed in this paper, which is therefore a
prerequisite for the _dynamical_ selection problem, cf. Remark 5.

(a)

(b)

(c)

Figure 4: Direct simulation of full PDE (LLGS) for
$\alpha=0.5,\beta=0.1,\mu=-1,h=50$, and $c_{\textnormal{cp}}=0$ with dynamical
selection of an inhomogeneous DW. Initial condition near homogeneous DW (9) in
codim-0 regime ($h^{\ast}=10.2,s_{0}=19.92$, and $\Omega_{0}=40.04$). (a)
Profile at $t=20$ projected onto the sphere. (b) Speed and frequency of DW
over time with asymptotic (selected) values $s=12.5$ and $\Omega=78.28$. (c)
Space-time plots of DW components (without co-moving frame), range as in black
box in (b). Final profile is heteroclinic connection in (7), cf. Proposition
2.

To our knowledge, existence results of DWs for $c_{\textnormal{cp}}\neq 0$ are
new. In more detail, the existence of localised inhomogeneous, i.e. flat as
well as non-flat, DWs for $c_{\textnormal{cp}}\neq 0$ and especially for
$c_{\textnormal{cp}}=0$ are new results. Indeed, the existence proof of non-
flat DWs is the most technical result and entails an existence proof of
heteroclinic orbits in an ODE between an equilibrium and a periodic orbit.
These solutions indicate the presence of DWs in other regimes of spin driven
phenomena and may be of interest for spin-torque transfer MRAM
(Magnetoresistive random-access memory) systems [14].

This paper is organized as follows. In §2, the LLGS equation and coherent
structures as well as first properties are discussed. Section 3 more precisely
introduces homogeneous and inhomogeneous as well as flat and non-flat DWs and
it also includes the main results of this paper (Theorem 1, 2, and 3). The
technical proofs of Theorem 2 as well as Theorem 3 are deferred to Appendix
6.1 and 6.2. Section 4 presents results of numerical continuation in parameter
$c_{\textnormal{cp}}$ for the three regimes of the applied field (codim-2,
center, and codim-0), where the center case is studied in more detail. We
conclude with discussion and outlook in §5.

### Acknowledgements

L.S. and J.R. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - Projektnummer 281474342/GRK2224/1. J.R. also
acknowledges support by DFG grant Ra 2788/1-1. I.O. acknowledges funding of a
previous position by Uni Bremen, where most of this paper was written, as well
as support by the recent Russian Scientific Foundation grant 19-11-00280.

##  2 Model equations and coherent structure form

The classical model for magnetization dynamics was proposed by Landau and
Lifschitz based on gyromagnetic precession, and later modified by Gilbert [15,
16]. See [17] for an overview. The _Landau-Lifschitz-Gilbert_ equation for
unit vector fields $\boldsymbol{m}(x,t)\in\mathbb{S}^{2}$ in one space
dimension $x\in\mathbb{R}$ and in terms of normalized time in dimensionless
form is

$\partial_{t}\boldsymbol{m}-\alpha\boldsymbol{m}\times\partial_{t}\boldsymbol{m}=-\boldsymbol{m}\times\boldsymbol{h}_{\textnormal{eff}}.$
(LLG)

Here $\boldsymbol{m}=\boldsymbol{M}/M_{S}$ represents the normalized
magnetization,
$\boldsymbol{h}_{\textnormal{eff}}=\boldsymbol{H}_{\textnormal{eff}}/M_{S}$
the effective field, i.e. the negative variational derivative of the total
magnetic free energy with respect to $\boldsymbol{m}$, both normalized by the
spontaneous magnetization $M_{S}$. For gyromagnetic ratio $\gamma$ and
saturation magnetization $M_{S}$ the time is measured in units of $(\gamma
M_{S})^{-1}$, and it is assumed that the temperature of the magnetic body is
constant and below the _Curie_ temperature [5]. Finally, _Gilbert_ damping
$\alpha>0$ turns $\boldsymbol{m}$ towards $\boldsymbol{h}_{\textnormal{eff}}$
and both vectors are parallel in the static solution.

In modern spin-tronic applications, e.g. Spin-Transfer Torque Magnetoresistive
Random Access Memories (MRAM), the spin of electrons is flipped using a spin-
polarized current. To take these effects into account, the LLG equation is
supplemented by an additional spin transfer torque term. Using a semiclassical
approach, Slonczewski derived an extended effective field

$\mathcal{H}_{\textnormal{eff}}=\boldsymbol{h}_{\textnormal{eff}}-\boldsymbol{m}\times\boldsymbol{J},$

where $\boldsymbol{J}=\boldsymbol{J}(\boldsymbol{m})$ depends on the
magnetization and the second term is usually called Slonczewski term [9]. In
contrast to the LLG equation, which can be written as the gradient of free
ferromagnetic energy, this generalized form is no longer variational and the
energy is no longer a Lyapunov functional.

As to the specific form of $\mathcal{H}_{\textnormal{eff}}$, including a
leading order form of exchange interaction, uniaxial crystal anisotropy in
direction $\boldsymbol{e}_{3}$, and _Zeeman_ as well as stray-field
interactions with an external magnetic field, see e.g. [6], gives the well
known form (1).

In this paper we consider a constant applied magnetic field $h\in\mathbb{R}$
along $\boldsymbol{e}_{3}$ and uniaxial anisotropy with parameter
$\mu\in\mathbb{R}$, for which the anisotropy energy density is rotationally
symmetric w.r.t. $\boldsymbol{e}_{3}$. According to the energetically
preferred direction in the uniaxial case, minima of the anisotropy energy
density correspond to _easy_ directions, whereas saddles or maxima correspond
to _medium-hard_ or _hard_ directions, respectively. Therefore, one refers to
$\mu<0$ as _easy-axis_ anisotropy and $\mu>0$ as _easy-plane_ , both with
regard to $\boldsymbol{e}_{3}$.

As mentioned before, the LLG equation with its variational structure appears
as a special case of (LLGS) for $\beta=0$ or $c_{\textnormal{cp}}=0$. While
our main focus is the non-variational spintronic case $\beta\cdot
c_{\textnormal{cp}}\neq 0$, all results contain the case $\beta\cdot
c_{\textnormal{cp}}=0$ and thus carry over to (LLG).

It is well-known that (LLGS) admits an equivalent form as an explicit
evolution equation of quasilinear parabolic type in the form, see e.g. [12],

$\partial_{t}\boldsymbol{m}=\partial_{x}\left(A(\boldsymbol{m})\partial_{x}\boldsymbol{m}\right)+B\left(\boldsymbol{m},\partial_{x}\boldsymbol{m}\right).$

As a starting point, we briefly note the existence of spatially homogeneous
equilibrium solutions of (LLGS) for which $\boldsymbol{m}(x,t)$ is constant in
$x$ and $t$.

######  Remark 1.

The only $($spatially$)$ homogeneous equilibria of (LLGS) for $\beta>0$ are
the constant up- and down magnetization states $\pm\boldsymbol{e}_{3}$.
Indeed, setting $\partial_{t}\boldsymbol{m}=\partial_{x}^{2}\boldsymbol{m}=0$
in $($LLGS$)$, for $\beta\neq 0$ the last equation implies that
$m_{1}=m_{2}=0$ and thus the only solutions
$\boldsymbol{m}_{\pm}^{\ast}\in\mathbb{S}^{2}$ are
$\boldsymbol{m}_{\pm}^{\ast}=(0,0,\pm 1)^{\mkern-1.2mu\mathsf{T}}$.

######  Remark 2.

In case $\beta=0$ as well as $|h/\mu|<1$ there exist a family of additional
homogeneous solutions of (LLGS) given by
$\boldsymbol{m}^{\ast}=(m_{1},m_{2},h/\mu)^{\mkern-1.2mu\mathsf{T}},$ with
$m^{2}_{1}+m^{2}_{2}=1-(h/\mu)^{2}$. Note that similar cases occur for
symmetry axis being $\boldsymbol{e}_{1}$ and $\boldsymbol{e}_{2}$,
respectively $($cf. Brown’s equations$)$.

###  2.1 Coherent structure ODE

Due to the rotation symmetry around the $\boldsymbol{e}_{3}$-axis of (LLGS),
it is natural to use spherical coordinates

$\boldsymbol{m}=\begin{pmatrix}\cos(\varphi)\sin(\theta)\\\
\sin(\varphi)\sin(\theta)\\\ \cos(\theta)\end{pmatrix},$

where $\varphi=\varphi(x,t)$ and $\theta=\theta(x,t)$. This changes (LLGS) to

$\displaystyle\begin{split}\begin{pmatrix}\alpha&-1\\\
1&\alpha\end{pmatrix}\begin{pmatrix}\partial_{t}\varphi\sin(\theta)\\\
-\partial_{t}\theta\end{pmatrix}=&\begin{pmatrix}2\partial_{x}\varphi\partial_{x}\theta\cos(\theta)\\\
-\partial_{x}^{2}\theta\end{pmatrix}\\\
&+\sin(\theta)\begin{pmatrix}\partial_{x}^{2}\varphi+\beta/\left(1+c_{\textnormal{cp}}\cos(\theta)\right)\\\
(\partial_{x}\varphi)^{2}\cos(\theta)+h-\mu\cos(\theta)\end{pmatrix}\end{split}$
(3)

Note that the rotation symmetry has turned into the shift symmetry in the
azimutal angle $\varphi$, as (3) depends on derivatives of $\varphi$ only.

Recall that DW solutions spatially connect the up and down magnetization
states $\pm\boldsymbol{e}_{3}$ in a coherent way as relative equilibria with
respect to the translation symmetry in $x$ and $\phi$, which yields the ansatz

$\xi\coloneqq x-st,\,\theta=\theta(\xi),\,\varphi=\phi(\xi)+\Omega t.$ (4)

Such solutions are generalized travelling waves that move with constant speed
$s\in\mathbb{R}$ in space and rotate pointwise with a constant frequency
$\Omega\in\mathbb{R}$ around the $\boldsymbol{e}_{3}$-axis; solutions with
$\Omega=0$ are classical travelling waves.

As in [12], applying ansatz (4) to (3) leads to the so-called _coherent
structure ODE_

$\begin{split}\begin{pmatrix}\alpha&-1\\\
1&\alpha\end{pmatrix}\begin{pmatrix}(\Omega-s\phi^{\prime})\sin(\theta)\\\
s\theta^{\prime}\end{pmatrix}=&\begin{pmatrix}2\phi^{\prime}\theta^{\prime}\cos(\theta)\\\
-\theta^{\prime\prime}\end{pmatrix}\\\
&+\sin(\theta)\begin{pmatrix}\phi^{\prime\prime}+\beta/\left(1+c_{\textnormal{cp}}\cos(\theta)\right)\\\
(\phi^{\prime})^{2}\cos(\theta)+h-\mu\cos(\theta)\end{pmatrix},\end{split}$
(5)

where ${}^{\prime}=d/d\xi$. This system of two second-order ODEs does not
depend on $\phi$ and thus reduces to three dynamical variables
$(\theta,\psi=\theta^{\prime},q=\phi^{\prime})$. Following standard
terminology for coherent structures, we refer to $q$ as the _local
wavenumber_.

Writing (5) as a first-order three-dimensional system gives

$\begin{array}[]{rl}\theta^{\prime}&=\psi\\\
\psi^{\prime}&=\sin(\theta)\left[h-\Omega+sq+(q^{2}-\mu)\cos(\theta)\right]-\alpha
s\psi\\\
q^{\prime}&=\alpha\Omega-\beta/(1+c_{\textnormal{cp}}\cos(\theta))-\alpha
sq-\frac{s+2q\cos(\theta)}{\sin(\theta)}\psi\end{array},$ (6)

and DWs in the original PDE are in 1-to-1-correspondence with the ODE
solutions connecting $\theta=0$ and $\theta=\pi$.

####  2.1.1 Blow-up charts and asymptotic states

As in [12], the singularities at zeros of $\sin(\theta)$ in (6) can be removed
by the singular coordinate change $\psi:=p\sin(\theta)$, which is a blow-up
transformation mapping the poles of the sphere $\pm\boldsymbol{e}_{3}$ to
circles thus creating a cylinder. The resulting _desingularized_ system reads

$\begin{array}[]{rl}\theta^{\prime}&=\sin(\theta)p\\\
p^{\prime}&=h-\Omega-\alpha sp+sq-(p^{2}-q^{2}+\mu)\cos(\theta)\\\
q^{\prime}&=\alpha\Omega-\beta/(1+c_{\textnormal{cp}}\cos(\theta))-sp-\alpha
sq-2pq\cos(\theta).\end{array}$ (7)

The coherent structure system (6) is equivalent to the desingularized system
(7) for $\theta\neq n\pi\,,n\in\mathbb{Z}$ and therefore also for
$\boldsymbol{m}$ away from $\pm\boldsymbol{e}_{3}$. Furthermore, the planar
_blow-up charts_ $\theta=0$ and $\theta=\pi$ are invariant sets of (7), which
are mapped to the single points $\boldsymbol{e}_{3}$ and
$-\boldsymbol{e}_{3}$, respectively by the blow-down transformation. System
(7) has a special structure (cf. Figure 6) that will be relevant for the
subsequent DW analysis. In the remainder of this section we analyze this in
some detail.

######  Lemma 1.

Consider the equations for $p$ and $q$ in (7) for an artificially fixed value
of $\theta$. In terms of $z\coloneqq p+\mathrm{i}q$ this subsystem can be
written as the complex (scalar) ODE

$z^{\prime}=Az^{2}+Bz+C,$ (8)

where
$A\coloneqq-\cos(\theta)\,,B\coloneqq-(\alpha+\mathrm{i})s\,,\textnormal{ and
}C^{\theta}\coloneqq
h-\Omega+A\mu+\mathrm{i}\left(\alpha\Omega-\frac{\beta}{1-Ac_{\textnormal{cp}}}\right)$.

For $A\neq 0$ the solution with $z_{0}=z(\xi_{0})$ away from the equilibria
$z^{\theta}_{+}=-\frac{B}{2A}+\mathrm{i}\frac{\gamma^{\theta}}{2A}$ and
$z^{\theta}_{-}=-\frac{B}{2A}-\mathrm{i}\frac{\gamma^{\theta}}{2A}$, with
$\gamma^{\theta}=\gamma(\theta)\coloneqq\sqrt{\textstyle{4AC^{\theta}-B^{2}}}$
, reads

$z(\xi)=\frac{\gamma^{\theta}}{2A}\tan\left(\frac{\gamma^{\theta}}{2}\xi+\delta_{0}\right)-\frac{B}{2A}\,,$
(9)

where

$\delta_{0}=\arctan\left(\frac{2Az_{0}+B}{\gamma^{\theta}}\right)-\frac{\gamma^{\theta}}{2}\xi_{0}.$

For $A=0$, the solution away from the equilibrium $z^{\pi/2}=-C^{\pi/2}/B$ is
given by

$z(\xi)=\left(z_{0}+\frac{C^{\pi/2}}{B}\right)\mathrm{e}^{B(\xi-\xi_{0})}-\frac{C^{\pi/2}}{B}.$

Clearly, the solution of (8) relates only to those solutions of (7) for which
$\theta$ is constant, i.e., $\theta=0,\pi$. Although we are mostly interested
in the dynamics on the blow-up charts, we consider $\theta$ as a parameter in
order to demonstrate the special behaviour of (7) for $\theta$ artificially
fixed. Notably, the equilibria $z_{\pm}^{\theta}$ of (8) for $\theta\neq
0,\pi$ are not equilibria in the full dynamics, due to the fact that (7) is
only invariant for $\theta$ at the blow-up charts.

###### Proof.

We readily verify the claimed form of the ODE and directly check the claimed
solutions. ∎

######  Remark 3.

Lemma 1 states in particular that the desingularized ODE system (7) can be
solved explicitly on the invariant blow-up charts, where $\theta=0,\pi$ and
thus $A=-1,1$, respectively. System (7) possesses two real equilibria on each
blow-up chart,
$Z^{0}_{\pm}\coloneqq(0,p^{0}_{\pm},q^{0}_{\pm})^{\mkern-1.2mu\mathsf{T}}$ and
$Z^{\pi}_{\pm}\coloneqq(\pi,p^{\pi}_{\pm}\,,q^{\pi}_{\pm})^{\mkern-1.2mu\mathsf{T}}$.
Here
$p^{\theta}_{\sigma}\coloneqq\mathrm{Re}(z^{\theta}_{\sigma})\,,q^{\theta}_{\sigma}\coloneqq\mathrm{Im}(z^{\theta}_{\sigma})$
for $\theta=0,\pi$, $\ \sigma=\pm$ and

$z^{0}_{+}\coloneqq 1/2(B-\mathrm{i}\gamma^{0})\,,\quad z^{0}_{-}\coloneqq
1/2(B+\mathrm{i}\gamma^{0})$

and analogously

$z^{\pi}_{+}\coloneqq 1/2(-B+\mathrm{i}\gamma^{\pi})\,,\quad
z^{\pi}_{-}\coloneqq 1/2(-B-\mathrm{i}\gamma^{\pi})\,,$

where we set

$\gamma^{0}\coloneqq\gamma\big{|}_{A=-1}=\sqrt{-4C^{0}-B^{2}}\quad\textnormal{and}\quad\gamma^{\pi}\coloneqq\gamma\big{|}_{A=1}=\sqrt{4C^{\pi}-B^{2}}$

with $C^{0}\coloneqq C\big{|}_{A=-1}$ and $C^{\pi}\coloneqq C\big{|}_{A=1}$.

Due to the analytic solution (9), we obtain the following more detailed result
in case $\theta\neq\pi/2$ (cf. Figure 6).

######  Lemma 2.

For each given $0\leq\theta\leq\pi$ with $\theta\neq\pi/2$ as a parameter, the
fibers of (7) with constant $\theta$ consists entirely of heteroclinic orbits
between $z^{\theta}_{-}$ and $z^{\theta}_{+}$ in case
$\mathrm{Im}(\gamma^{\theta})\neq 0$, or $\gamma^{\theta}\neq 0$, except for
the equilibrium states. In case $\mathrm{Im}(\gamma^{\theta})=0$ and
$\mathrm{Re}(\gamma^{\theta})\neq 0$, the fiber at fixed $\theta$ is filled
with periodic orbits away from the invariant line $\\{q=\frac{s}{2A}\\}$, for
which the period of solutions close to it tends to infinity.

###### Proof.

For $\theta$ fixed in (8), consider the case $\mathrm{Re}(\gamma^{\theta})=0$
and also $\mathrm{Im}(\gamma^{\theta})\neq 0$ for $A\neq 0$ which leads to

$\displaystyle z(\xi)$
$\displaystyle=\mathrm{i}\frac{\mathrm{Im}(\gamma^{\theta})}{2A}\cdot\tan\bigg{(}\mathrm{i}\Big{(}\underbrace{\frac{\mathrm{Im}(\gamma^{\theta})}{2}\xi+\mathrm{Im}(\delta_{0})}_{\eqqcolon\check{\xi}}\Big{)}+\mathrm{Re}(\delta_{0})\bigg{)}-\frac{B}{2A}$
$\displaystyle=\frac{\mathrm{Im}(\gamma^{\theta})}{2A}\cdot\frac{\mathrm{i}\sin\left(2\mathrm{Re}(\delta_{0})\right)-\sinh\left(2\check{\xi}\right)}{\cos\left(2\mathrm{Re}(\delta_{0})\right)+\cosh\left(2\check{\xi}\right)}-\frac{B}{2A}\,.$

For $\mathrm{Re}(\gamma^{\theta})\neq 0$ as well as
$\mathrm{Im}(\gamma^{\theta})\neq 0$, we obtain

$\displaystyle z(\xi)$
$\displaystyle=\frac{\gamma^{\theta}}{2A}\tan\bigg{(}\underbrace{\frac{\mathrm{Re}(\gamma^{\theta})}{2}\xi+\mathrm{Re}(\delta_{0})}_{\eqqcolon\tilde{\xi}}+\mathrm{i}\frac{\mathrm{Im}(\gamma^{\theta})}{2}\xi+\mathrm{i}\mathrm{Im}(\delta_{0})\bigg{)}-\frac{B}{2A}$
$\displaystyle=\frac{\gamma^{\theta}}{2A}\cdot\frac{\sin(2\tilde{\xi})+\mathrm{i}\sinh\left(2\left(\frac{\mathrm{Im}(\gamma^{\theta})}{\mathrm{Re}(\gamma^{\theta})}\tilde{\xi}-\frac{\mathrm{Im}(\gamma^{\theta})}{\mathrm{Re}(\gamma^{\theta})}\mathrm{Re}(\delta_{0})+\mathrm{Im}(\delta_{0})\right)\right)}{\cos(2\tilde{\xi})+\cosh\left(2\left(\frac{\mathrm{Im}(\gamma^{\theta})}{\mathrm{Re}(\gamma^{\theta})}\tilde{\xi}-\frac{\mathrm{Im}(\gamma^{\theta})}{\mathrm{Re}(\gamma^{\theta})}\mathrm{Re}(\delta_{0})+\mathrm{Im}(\delta_{0})\right)\right)}-\frac{B}{2A}\,,$

The asymptotic states are

$\mathrm{Im}\left(\gamma^{\theta}\right)>0:\qquad\lim_{\xi\to-\infty}z(\xi)=-\mathrm{i}\frac{\gamma^{\theta}}{2A}-\frac{B}{2A}\,,\quad\lim_{\xi\to+\infty}z(\xi)=\mathrm{i}\frac{\gamma^{\theta}}{2A}-\frac{B}{2A}\,,$

as well as

$\mathrm{Im}\left(\gamma^{\theta}\right)<0:\qquad\lim_{\xi\to-\infty}z(\xi)=\mathrm{i}\frac{\gamma^{\theta}}{2A}-\frac{B}{2A}\,,\quad\lim_{\xi\to+\infty}z(\xi)=-\mathrm{i}\frac{\gamma^{\theta}}{2A}-\frac{B}{2A}\,,$

which simplify in case $\mathrm{Re}(\gamma^{\theta})=0$ to

$\mathrm{Im}\left(\gamma^{\theta}\right)>0:\qquad\lim_{\xi\to-\infty}z(\xi)=\frac{\mathrm{Im}(\gamma^{\theta})-B}{2A}\,,\quad\lim_{\xi\to+\infty}z(\xi)=-\frac{\mathrm{Im}(\gamma^{\theta})+B}{2A}\,,$

as well as

$\mathrm{Im}\left(\gamma^{\theta}\right)<0:\qquad\lim_{\xi\to-\infty}z(\xi)=-\frac{\mathrm{Im}(\gamma^{\theta})+B}{2A}\,,\quad\lim_{\xi\to+\infty}z(\xi)=\frac{\mathrm{Im}(\gamma^{\theta})-B}{2A}\,.$

Note that the asymptotic states coincide if $\gamma^{\theta}=0$.

The last case to consider is $\mathrm{Re}\left(\gamma^{\theta}\right)\neq 0$
and $\mathrm{Im}\left(\gamma^{\theta}\right)=0$, where the solutions are

$z(\xi)=\frac{\mathrm{Re}\left(\gamma^{\theta}\right)}{2A}\cdot\frac{\sin(2\hat{\xi})+\mathrm{i}\sinh(2\mathrm{Im}(\delta_{0}))}{\cos(2\hat{\xi})+\cosh(2\mathrm{Im}(\delta_{0}))}-\frac{B}{2A}\,,$

with
$\hat{\xi}\coloneqq\frac{\mathrm{Re}(\gamma^{\theta})}{2}\xi+\mathrm{Re}(\delta_{0})$
and which leads to periodic solutions of (8) iff

$\displaystyle\mathrm{Im}(\delta_{0})\neq 0$ $\displaystyle\Leftrightarrow
2A\mathrm{Im}(z_{0})+\mathrm{Im}(B)\neq 0\Leftrightarrow
q_{0}\neq\frac{s}{2A}\,,$

where $z_{0}=p_{0}+\mathrm{i}q_{0}$ and recall that $B=-(\alpha+\mathrm{i})s$.
∎

Based on Lemma 2, explicitly on the blow-up chart $\theta=0$ the heteroclinic
orbits are from $z_{-}^{0}$ to $z_{+}^{0}$ in case
$\mathrm{Im}(-4C^{0}-B^{2})>0$, or $\mathrm{Im}(-4C^{0}-B^{2})=0$ and
$\mathrm{Re}(-4C^{0}-B^{2})\leq 0$, and from $z_{+}^{0}$ to $z_{-}^{0}$ if
$\mathrm{Im}(-4C^{0}-B^{2})<0$ . For $\theta=\pi$, if
$\mathrm{Im}(4C^{\pi}-B^{2})>0$, or $\mathrm{Im}(4C^{\pi}-B^{2})=0$ and
$\mathrm{Re}(4C^{\pi}-B^{2})\leq 0$ they are connections from $z_{-}^{\pi}$ to
$z_{+}^{\pi}$, and if $\mathrm{Im}(4C^{\pi}-B^{2})<0$ from $z_{+}^{\pi}$ to
$z_{-}^{\pi}$.

For $A\neq 0$, the case $s=0$ is a special situation, which will be also
discussed in the context of DWs in § 3 later. It turns out that on the blow-up
charts $\theta=0$ (or $\theta=\pi$), the solution with appropriate initial
conditions has a limit as $|\xi|\to\infty$ if and only if
$\mathrm{Im}(\sqrt{\textstyle{-C^{0}}})\neq 0$
($\mathrm{Im}(\sqrt{\textstyle{C^{\pi}}})\neq 0$). In terms of the parameters
in (7) and with

$\boldsymbol{\beta}^{-}\coloneqq\frac{\beta}{1-c_{\textnormal{cp}}}\qquad\textnormal{and}\qquad\boldsymbol{\beta}^{+}\coloneqq\frac{\beta}{1+c_{\textnormal{cp}}}\,,$

this leads to the conditions for $\theta=0$ given by:

$\Omega\neq\frac{\boldsymbol{\beta}^{+}}{\alpha}\quad\textnormal{or}\quad\Omega=\frac{\boldsymbol{\beta}^{+}}{\alpha}\textnormal{
and }\Omega\leq h-\mu\,,$ (10)

and for $\theta=\pi$ given by:

$\Omega\neq\frac{\boldsymbol{\beta}^{-}}{\alpha}\quad\textnormal{or}\quad\Omega=\frac{\boldsymbol{\beta}^{-}}{\alpha}\textnormal{
and }\Omega\geq h+\mu\,,$ (11)

In case $c_{\textnormal{cp}}=0$, i.e. for the LLG equation, the conditions in
(10) and (11) reduce to

$\Omega\neq\frac{\beta}{\alpha}\quad\textnormal{or}\quad\Omega=\frac{\beta}{\alpha}\textnormal{
and }2\mu\leq\frac{\beta}{\alpha}\,,$

where the latter inequality always holds in case of a nanowire geometry
($\mu<0$). Hence standing domain walls in nanowires in case
$c_{\textnormal{cp}}=0$ can only connect equilibria, if they exist.

Lemma 2 also states that the equilibria on the blow-up charts
$\theta\in\\{0,\pi\\}$ are surrounded by periodic orbits in case
$\mathrm{Im}(\gamma^{0})=0$ and $\mathrm{Re}(\gamma^{0})\neq 0$
($\mathrm{Im}(\gamma^{\pi})=0$ and $\mathrm{Re}(\gamma^{\pi})\neq 0$). In
fact, system (7) is Hamiltonian (up to rescaling) on the blow-up charts for
certain frequencies $\Omega$, as follows

(a)

(b)

Figure 5: (a) Phase plane streamplot with Mathematica of (14) around the
equilibrium $z^{\pi}_{-}$, i.e., (7) at $\theta=\pi$, for
$\alpha=0.5,\beta=0.1,\mu=-1,h=10.2,s=4,\Omega=8.2$ and
$c_{\textnormal{cp}}=0$, which leads to
$\left(p^{\pi}_{-},q^{\pi}_{-}\right)^{\mkern-1.2mu\mathsf{T}}=\left(1,0\right)^{\mkern-1.2mu\mathsf{T}}$.
The red solid line marks the trajectory with initial condition
$(p_{0},q_{0})=(7/4,0)$ (cf. plot of solutions in b). (b) Plot of the profile
for the solution highlighted in (a), where $p$ (solid blue line) and $q$
(dashed red line) are given by (9) for the parameter set as in (a).

######  Proposition 1.

The dynamics of (7) on the invariant blow-up chart $\theta=0$ in case
$\Omega=\frac{\boldsymbol{\beta}^{+}}{\alpha}-\frac{s^{2}}{2}$ possesses the
invariant line $\\{q=-\frac{s}{2}\\}$ and, after time-rescaling, for
$q\neq-\frac{s}{2}$ the Hamiltonian

$H^{0}(p,q)=-\frac{p^{2}+q^{2}+\alpha
sp+sq-h+\boldsymbol{\beta}^{+}/\alpha+\mu}{q+\frac{s}{2}}$

along solutions of (8). Analogously on the chart $\theta=\pi$, in case

$\Omega=\frac{\boldsymbol{\beta}^{-}}{\alpha}+\frac{s^{2}}{2}$ (12)

possesses the invariant line $\\{q=\frac{s}{2}\\}$ and for $q\neq s/2$ the
Hamiltonian

$H^{\pi}(p,q)=\frac{p^{2}+q^{2}-\alpha sp-
sq+h-\boldsymbol{\beta}^{-}/\alpha+\mu}{q-\frac{s}{2}}.$ (13)

Moreover, each half plane $\\{\theta=0,q\leq-\frac{s}{2}\\}$,
$\\{\theta=0,q\geq-\frac{s}{2}\\}$
$\left(\\{\theta=\pi,q\leq\frac{s}{2}\\}\right.$ and
$\left.\\{\theta=\pi,q\geq\frac{s}{2}\\}\right)$ is filled with periodic
orbits encircling the equilibria at $z_{\pm}^{0}$ $\left(z^{\pi}_{\pm}\right)$
if additionally $\Omega>h-\mu+\frac{s^{2}}{4}(\alpha^{2}-1)$
$\left(\Omega<h+\mu+\frac{s^{2}}{4}(1+\alpha^{2})\right)$.

###### Proof.

For the sake of clarity, we will only present the computation for the blow-up
chart $\theta=\pi$; the computation on $\theta=0$ can be done in the same
manner. With respect to the parameters of equation (7), the condition
$\mathrm{Im}(4C-B^{2})=0$ is equivalent to
$\Omega=\frac{\boldsymbol{\beta}^{-}}{\alpha}+\frac{s^{2}}{2}$ and in this
case the system (7) on $\\{\theta=\pi\\}$ is given by

$\begin{array}[]{rl}p^{\prime}&=p^{2}-q^{2}-\alpha
sp+sq+h+\mu-\frac{\boldsymbol{\beta}^{-}}{\alpha}-\frac{s^{2}}{2}\\\
q^{\prime}&=2pq-sp-\alpha sq+\frac{\alpha s^{2}}{2}\end{array}$ (14)

We readily compute that for solutions of this

$\frac{dH^{\pi}}{d\xi}=\frac{\partial H^{\pi}}{\partial
p}p^{\prime}+\frac{\partial H^{\pi}}{\partial
q}q^{\prime}=\frac{q^{\prime}\cdot p^{\prime}-p^{\prime}\cdot
q^{\prime}}{(q-\frac{s}{2})^{2}}=0\,,$

which shows the canonical Hamiltonian structure of (14) up to time rescaling.
If additionally $\Omega<h+\mu+\frac{s^{2}}{4}(1+\alpha^{2})$, it follows from
Lemma 2 that each half plane is filled with periodic orbits. ∎

Proposition 1 concerns the special case that $c_{\textnormal{cp}}\in(-1,1)$
and $\beta,\Omega$ are such that (12) holds, which is henceforth referred to
as the center-case. In particular, each orbit except the line $q\equiv s/2$ on
the blow-up chart $\theta=\pi$ can by identified via the quantity (13), and
each equilibrium $z^{\pi}_{\pm}$ has a neighborhood filled with periodic
orbits if additionally
$h>\frac{\boldsymbol{\beta}^{-}}{\alpha}-\mu+\frac{s^{2}}{4}(1-\alpha^{2})$
(cf. Figure 5). Note the relation between the conditions (10) and (11) and the
conditions in Proposition 1 in case $s=0$.

Based on Lemma 2, we also state the following uniqueness result.

######  Proposition 2.

For $\Omega<\frac{\boldsymbol{\beta}^{+}}{\alpha}-\frac{s^{2}}{2}$
$\left[\Omega>\frac{\boldsymbol{\beta}^{+}}{\alpha}-\frac{s^{2}}{2}\right]$,
or $\Omega=\frac{\boldsymbol{\beta}^{+}}{\alpha}-\frac{s^{2}}{2}$ and
$\Omega\leq h-\mu+\frac{s^{2}}{4}(\alpha^{2}-1)$ there is a unique orbit with
$(\theta,p,q)^{\mkern-1.2mu\mathsf{T}}(\xi)$ with $\theta(\xi)\to 0$ as
$\xi\to-\infty$, and it holds that $(p+\mathrm{i}q)(\xi)\to z^{0}_{-}$
$\left[(p+\mathrm{i}q)(\xi)\to z_{+}^{0}\right]$ as $\xi\to-\infty$ .

###### Proof.

The conditions on $\Omega$ are equivalent to those in Lemma 2. If the
statement were false, it nevertheless follows from Lemma 2 that
$(p+\mathrm{i}q)(\xi)\to z^{0}_{-}$ as $\xi\to-\infty$. However, transverse to
the blow-up chart, the equilibrium state $Z^{0}_{-}$ is stable for increasing
$\xi$ and thus repelling for decreasing $\xi$. This contradicts the
requirement $\theta(\xi)\to 0$ as $\xi\to-\infty$. Together with the fact that
$Z^{0}_{-}$ has a one-dimensional unstable manifold uniqueness follows.
Analogously in case
$\Omega>\frac{\boldsymbol{\beta}^{+}}{\alpha}-\frac{s^{2}}{2}$. ∎

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Phase plane streamplots (with Mathematica) in blow-up charts near
the equilibrium $z_{-}^{\pi}=(1,0)$ for
$\alpha=0.5\,,\beta=0.1\,,\mu=-1\,,c_{\textnormal{cp}}=0$, i.e., the second
and third equation of (7). (a-c) $\theta=0$ and (d-f) $\theta=\pi$. (a,d)
codim-2 regime, (b,e) center case, where $\Omega=\beta/\alpha+s^{2}/2$ holds
on the chart $\theta=\pi$, and (c,f) codim-0 regime. The remaining parameters
and equilibria in (a,d): $h=0.5$, $s_{0}=0.12$, $\Omega_{0}=0.44$, and
$z_{+}^{0}=-1.06-0.12\mathrm{i}$, $z_{+}^{\pi}=-0.94+0.12\mathrm{i}$. In
(b,e): $h=10.2$, $s_{0}=4$, $\Omega_{0}=8.2$, and $z_{+}^{0}=-3-4\mathrm{i}$,
$z_{+}^{\pi}=1+4\mathrm{i}$. In (c,f): $h=50.0$, $s_{0}=19.92$,
$\Omega_{0}=40.04$, and $z_{+}^{0}=-10.96-19.92\mathrm{i}$,
$z_{+}^{\pi}=8.96+19.92\mathrm{i}$.

Domain walls are heteroclinic orbits between the blow-up charts and decisive
for their bifurcation structure are the dimensions (and directions) of
un/stable manifolds of the equilibria on these charts. Hence, we next discuss
the equilibria $Z^{0}_{\pm}$ and $Z^{\pi}_{\pm}$ and their stability.

Transverse to the blow-up charts in $\theta$-direction we readily compute the
linearization $\partial_{\theta}(\sin(\theta)p)=\cos(\theta)p$, i.e., the
transverse eigenvalue is $-A^{\theta}\cdot\mathrm{Re}(z^{\theta}_{\pm})$ at
$\theta=0$ and $\pi$, respectively. The eigenvalues within the blow-up charts
are determined by $\pm\mathrm{i}\gamma$. With $\sigma=\pm$, respectively, the
eigenvalues for $Z^{0}_{\sigma}$ are

$\nu^{0}_{1,\sigma}=-\sigma\mathrm{i}\gamma^{0}\,,\quad\nu^{0}_{2,\sigma}=\overline{\nu^{0}_{1,\sigma}}\,,\quad\nu^{0}_{3,\sigma}=\mathrm{Re}(z^{0}_{\sigma})$
(15)

and for $Z^{\pi}_{\sigma}$

$\nu^{\pi}_{1,\sigma}=\sigma\mathrm{i}\gamma^{\pi}\,,\quad\nu^{\pi}_{2,\sigma}=\overline{\nu^{\pi}_{1,\sigma}}\,,\quad\nu^{\pi}_{3,\sigma}=-\mathrm{Re}(z^{\pi}_{\sigma}).$
(16)

Therefore, the signs of the real parts within each blow-up chart are opposite
at $Z^{\pi}_{+}$ compared to $Z^{\pi}_{-}$ and determined by the sign of
$\mathrm{Re}\left(\nu^{0,\pi}_{1,+}\right)$. Hence, within the blow-up charts
each equilibrium is either two-dimensionally stable, unstable or a linearly
neutral center point.

For completeness, we next note that the equilibria on both blow-up charts can
be neutral centers simultaneously (cf. Figure 3). However, this requires a
negative spin polarization and a small Gilbert damping factor, and is not
further studied in this paper.

######  Remark 4.

The equilibria of both blow-up charts are centers simultaneously, if and only
if $\mathrm{Im}(\pm\gamma^{0,\pi})=0$ and $\gamma^{0,\pi}\neq 0$ (compare
Lemma 2). For example if
$\alpha=0.5,\beta=0.1,\mu=-1,c_{\textnormal{cp}}=-0.99,h=10$

$s^{2}=\frac{3960}{199}\,,\quad\Omega=\frac{\beta/\alpha}{1-c_{\textnormal{cp}}}+\frac{s^{2}}{2}=\frac{2000}{199}\,,$

we obtain

$\gamma^{0}=3.33551\,,\quad\gamma^{\pi}=3.27469\,.$

##  3 Domain Walls

All domain walls between $\pm\boldsymbol{e}_{3}$ that we are aware of are of
coherent structure type, and thus in one-to-one correspondence to heteroclinic
connections between the blow-up charts $\\{\theta=0\\}$ and $\\{\theta=\pi\\}$
in (7). Typically we expect these to be heteroclinics between equilibria
within the charts, but this is not necessary. Based on the previous analysis,
there are three options for heteroclinics between the charts: point-to-point,
point-to-cycle, and cycle-to-cycle. We study the first two in this section,
for which Proposition 2 implies uniqueness of the DW (up to
translations/rotations) for a given set of parameters. The third case can
occur at most in a relatively small set of parameters (see Remark 4). Its
analysis is beyond the scope of this paper.

Note that in case of an existing connection between an equilibrium and a
periodic orbit (see Proposition 1), the domain wall is automatically an
inhomogeneous non-flat one. Moreover, via the singular coordinate change any
such heteroclinic solution is heteroclinic between $\theta=0,\pi$ in (6) and
through the spherical coordinates it is a heteroclinic connection between
$\pm\boldsymbol{e}_{3}$ in the sphere, possibly with unbounded $\varphi$.

###  3.1 Homogeneous Domain Walls

It is known from [13] in case $\beta=0$ and from [12] in case
$c_{\textnormal{cp}}=0$ (and arbitrary $\beta$) that (7) admits for $\mu<0$ a
family of explicit homogeneous DWs $\boldsymbol{m}_{0}$ given by

$\begin{pmatrix}\theta_{0}\\\ p_{0}\\\
q_{0}\end{pmatrix}=\begin{pmatrix}2\arctan\left(\mathrm{e}^{\sigma\sqrt{-\mu}\xi}\right)\\\
\sigma\sqrt{-\mu}\\\ 0\end{pmatrix}$ (17)

and parameterized by $\Omega=\frac{h+\alpha\beta}{1+\alpha^{2}}$,
$s^{2}=-\frac{(\beta-\alpha h)^{2}}{\mu(1+\alpha^{2})^{2}}>0$, and $\sigma=1$
for positive speed $s$ and $\sigma=-1$ for negative $s$; the family extends to
$s=0$ in the limit $h\to\frac{\beta}{\alpha}$ with scaling of the frequency by
$\Omega=\frac{\beta}{\alpha}+\frac{\sqrt{-\mu}}{\alpha}s$. For $s=0$
(standing) fronts with both orientations exist simultaneously
$c_{\textnormal{cp}}=0$ and are given by

$\begin{pmatrix}\theta_{0}\\\ p_{0}\\\
q_{0}\end{pmatrix}=\begin{pmatrix}2\arctan\left(\mathrm{e}^{\pm\sqrt{-\mu}\xi}\right)\\\
\pm\sqrt{-\mu}\\\ 0\end{pmatrix}.$

Hence, the branches of left and right moving walls as parametrized by $s$ each
have termination point at $s=0$ (cf. Figure 13).

The family of explicit DWs (17) have domain wall width $\sqrt{-\mu}$, a
profile independent of the applied field $h$ and propagate along a nanowire
($\mu<0$) with velocity $s$ while precessing with azimuthal velocity $\Omega$.
Since these are unique up to spatial reflection symmetry, the direction of
motion is related to the spatial direction of connecting
$\pm\boldsymbol{e}_{3}$ through $\sigma$,

$\displaystyle\theta(-\infty)$ $\displaystyle=0$
$\displaystyle\theta(+\infty)$ $\displaystyle=\pi$
$\displaystyle\Leftrightarrow s$ $\displaystyle>0\textnormal{ (wall moves to
the right)}$ (18) $\displaystyle\theta(-\infty)$ $\displaystyle=\pi$
$\displaystyle\theta(+\infty)$ $\displaystyle=0$ $\displaystyle\Leftrightarrow
s$ $\displaystyle<0\textnormal{ (wall moves to the left)}.$

To simplify some notation we will focus on the case of right-moving walls
including standing walls ($s\geq 0$) and thus make the standing assumptions
that $h\geq\beta/\alpha$ as well as $\mu<0$. We therefore have a 1-to-1
relation of parameters $(\alpha,\beta,h,\mu)$ and right-moving DWs from

$\boldsymbol{m}(\xi,t)=\boldsymbol{m}_{0}(\xi,t;\alpha,\beta,h,\mu)$

with speed and frequency given by

$s_{0}=s_{0}(\alpha,\beta,h,\mu):=\frac{\alpha
h-\beta}{\sqrt{-\mu}(1+\alpha^{2})}\;,\quad\Omega_{0}=\Omega_{0}(\alpha,\beta,h,\mu):=\frac{h+\alpha\beta}{1+\alpha^{2}}$
(19)

where the subindex $0$ emphasizes that $c_{\textnormal{cp}}=0$. Since $s_{0}$
is surjective on $\mathbb{R}_{\geq 0}$ any velocity can be realised. Spatial
reflection covers the case $h\leq\beta/\alpha$.

Based on Lemma 1 as well as Remark 3 for $c_{\textnormal{cp}}=0$ and
(homogeneous) speed and frequency (19), one readily finds the asymptotic
states of (7) given by

$E^{0}\coloneqq
Z^{0}_{-}\big{|}_{(s_{0},\Omega_{0})}=\left(0,\sqrt{-\mu},0\right)^{\mkern-1.2mu\mathsf{T}}\quad\textnormal{and}\quad
E^{\pi}\coloneqq
Z^{\pi}_{-}\big{|}_{(s_{0},\Omega_{0})}=\left(\pi,\sqrt{-\mu},0\right)^{\mkern-1.2mu\mathsf{T}},$

with (spatial) eigenvalues (15), (16) given by

$\begin{split}\nu_{k,-}^{0}&:=-\alpha
s_{0}-2\sqrt{-\mu}-(-1)^{k}\,\mathrm{i}s_{0}\,,\quad\nu^{0}_{3,-}=\sqrt{-\mu}\,,\\\
\nu_{k,-}^{\pi}&:=-\alpha
s_{0}+2\sqrt{-\mu}-(-1)^{k}\,\mathrm{i}s_{0}\,,\quad\nu^{\pi}_{3,-}=-\sqrt{-\mu}\,,\end{split}$
(20)

where $k=1,2$. Note that the above equilibria cannot be centers simultaneously
(recall $\mu<0$), hence a cycle-to-cycle connection can not exist close to it
(see Remark 4 for details). For this reason, we focus on point-to-point as
well as point-to-cycle connections.

###  3.2 Inhomogeneous Domain Walls

Homogeneous DWs exist only in case $c_{\textnormal{cp}}=0$ [12, Theorem 5],
are explicitly given by (17) and completely characterized by (19). By [12,
Theorem 6], fast inhomogeneous DW solutions with $|s|\gg 1$ exist for any
$c_{\textnormal{cp}}\in(-1,1)$, but in contrast to (17), the gradient of these
profiles is of order $1/|s|$ and thus have a large ‘width’. The natural
question arises what happens for any $s$ in case $c_{\textnormal{cp}}\neq 0$.

This section contains the main results of this paper: the existence, parameter
selection and structure of inhomogeneous DW solutions in case of small
$|c_{\textnormal{cp}}|$ for any value of the applies field $h$, and thus any
speed $s$. This will be achieved by perturbing away from the explicit solution
$\boldsymbol{m}_{0}$ given by (17), where the bifurcation structure is largely
determined by comparing the dimensions of the un/stable eigenspaces at the
asymptotic equilibrium states, which are determined by (20).

Let $W^{0}_{s/u}$ and $W^{\pi}_{s/u}$ denote the stable and unstable manifolds
associated to $E^{0}$ and $E^{\pi}$, respectively, and $w_{s/u}^{0}$ as well
as $w_{s/u}^{\pi}$ be the dimension of these manifolds so that
$w^{0}_{s}+w^{0}_{u}=w^{\pi}_{s}+w^{\pi}_{u}=3$. Notably $w^{0}_{s}=2$ and
$w^{0}_{u}=1$ for all values of the parameters, and $w^{\pi}_{s}$ is either
$1$ or $3$. Recall the standing assumption $s_{0}\geq 0$.

If $w^{\pi}_{s}=1$, the heteroclinic connection of $E^{0}$ and $E^{\pi}$
generically has _codimension-2_ , while for $w^{\pi}_{s}=3$ it has
_codimension-0_ , and we refer to the transition point between these cases,
following the discussion in § 2.1.1, as the _center case_. From (20) we have
$w^{\pi}_{s}=1\Leftrightarrow 0\leq s_{0}<\frac{2\sqrt{-\mu}}{\alpha}$,
$w^{\pi}_{s}=3\Leftrightarrow s_{0}>\frac{2\sqrt{-\mu}}{\alpha}$ and the
center case at $s_{0}=\frac{2\sqrt{-\mu}}{\alpha}$.

Hence, within the family of homogeneous DWs given by (17) and satisfying (19),
the different bifurcation cases have speed and frequency relations

codim-0: $\displaystyle
s_{0}>\frac{2\sqrt{-\mu}}{\alpha}\quad\textnormal{and}\quad\Omega_{0}>\frac{\beta}{\alpha}-\frac{2\mu}{\alpha^{2}}\,,$
center: $\displaystyle
s_{0}=\frac{2\sqrt{-\mu}}{\alpha}\quad\textnormal{and}\quad\Omega_{0}=\frac{\beta}{\alpha}-\frac{2\mu}{\alpha^{2}}\,,$
codim-2: $\displaystyle 0\leq
s_{0}<\frac{2\sqrt{-\mu}}{\alpha}\quad\textnormal{and}\quad\frac{\beta}{\alpha}\leq\Omega_{0}<\frac{\beta}{\alpha}-\frac{2\mu}{\alpha^{2}}.$

Using (19) these can be written in terms of the parameters of (LLGS), which
gives the characterization mentioned in the introduction §1.

######  Remark 5.

The case distinction is also related to the spectral stability of the
asymptotic states $\boldsymbol{m}=\pm\boldsymbol{e}_{3}$ in the dynamics of
the full PDE (LLGS) which is beyond the scope of this paper, but see Figure 3
for an illustration. In short, it follows from, e.g., [12, Lemma 1] that
$\boldsymbol{e}_{3}$ is $L^{2}$-stable for $h>\beta/\alpha$, while
$-\boldsymbol{e}_{3}$ is $L^{2}$-stable for $h<\beta/\alpha-\mu$ and unstable
for $h>\beta/\alpha-\mu$. Based on this, the stability curves in Figure 3 are
defined as follows

$\Gamma^{+}\coloneqq\frac{\beta/\alpha}{h-\mu}-1,\qquad\Gamma^{-}\coloneqq
1-\frac{\beta/\alpha}{h+\mu},$

which intersect at

$h=\frac{\beta}{2\alpha}+\sqrt{\frac{\beta^{2}}{4\alpha^{2}}+\mu^{2}}.$

Since the destabilisation of $-\boldsymbol{e}_{3}$ if $\beta>0$ corresponds to
a Hopf-instability of the $($purely essential$)$ spectrum, it is effectively
invisible in the coherent structure ODE, which detects changes in the
linearization at zero temporal eigenvalue only. Visible from the PDE stability
viewpoint is a transition of absolute spectrum through the origin in the
complex plane of temporal eigenmodes, cf. [18]. Now in the center case, the
state $-\boldsymbol{e}_{3}$ is already $L^{2}$-unstable since $\alpha>0$ as
well as $\mu<0$ and $h>\beta/\alpha$ implies

$h=h^{\ast}=\frac{\beta}{\alpha}-\frac{2\mu}{\alpha^{2}}(1+\alpha^{2})>\frac{\beta}{\alpha}-\mu$

and therefore that $\Gamma^{-}$ never intersects the line
$c_{\textnormal{cp}}\equiv 0$ at $h=h^{\ast}$.

Moreover, it was shown in [19] that the family of explicit homogeneous DWs (9)
is ($linearly$) stable for sufficiently small applied fields, actually for
$h<-\mu/2$, in case $\beta=0$, hence in the bi-stable case where
$\pm\boldsymbol{e}_{3}$ are $L^{2}$-stable. As mentioned before, $\beta=0$ is
equivalent to $c_{\textnormal{cp}}=0$ in the LLGS equation with an additional
shift in $h$ and $\beta$, which leads to the (LLG) case. We expect these DWs
are also stable for small perturbations in $c_{\textnormal{cp}}$, due to the
properties of the operator established in [19], but further analysis also on
the transition from convective/transient to absolute instability will be done
elsewhere.

With these preparations, we next state the main results, which concern
existence of DWs in the three regimes.

######  Theorem 1.

For any parameter set $(\alpha_{0},\beta_{0},h_{0},\mu_{0})$ in the codim-0
case, i.e., $\mu_{0}<0$ and
$h_{0}>\beta_{0}/\alpha_{0}-2\mu_{0}-2\mu_{0}/\alpha_{0}^{2}$, the following
holds. The explicit homogeneous DWs $\boldsymbol{m}_{0}$ in (9) lies in a
smooth family $\boldsymbol{m}_{c_{\textnormal{cp}}}$ of DWs parameterized by
$(c_{\textnormal{cp}},\alpha,\beta,h,\mu,s,\Omega)$ near
$(0,\alpha_{0},\beta_{0},h_{0},\mu_{0},s_{0},\Omega_{0})$ with $s_{0}$,
$\Omega_{0}$ from (19) evaluated at $(\alpha_{0},\beta_{0},h_{0},\mu_{0})$.
Moreover, in case $c_{\textnormal{cp}}=0$ and
$(s,\Omega)\neq(s_{0},\Omega_{0})$ evaluated at $(\alpha,\beta,h,\mu)$, or
$c_{\textnormal{cp}}\neq 0$, these are inhomogeneous flat DWs.

###### Proof.

As mentioned, in the codim-0 case we have $w_{s}^{\pi}=3$ and for all
parameters $w_{u}^{0}=1$. Due to the existence of the heteroclinic orbit (17),
this means $W^{0}_{u}$ intersects $W^{\pi}_{s}$ transversely and non-trivially
for $c_{\textnormal{cp}}=0$ in a unique trajectory. Therefore, this DW
perturbs to a locally unique family by the implicit function theorem for
perturbations of the parameters in (7).

For $c_{\textnormal{cp}}\neq 0$ sufficiently small these are inhomogeneous DWs
since the derivative of the third equation, the $q$-equation, in (7) with
respect to $c_{\textnormal{cp}}$ is nonzero in this case; hence already the
equilibrium states move into the inhomogeneous regime.

For $c_{\textnormal{cp}}=0$ but $(s,\Omega)\neq(s_{0},\Omega_{0})$ at
$(\alpha,\beta,h,\mu)$, it follows from [12, Theorem 5] that these DWs cannot
be homogeneous. ∎

Next we consider the center case, where
$h=h^{\ast}=\beta/\alpha-2\mu-2\mu/\alpha^{2}$. We start with a result that
follows from the same approach used in the codim-2 case and give refined
results below.

######  Corollary 1.

The statement of Theorem 1 also holds for a parameter set in the center case
if the perturbed parameters
$(c_{\textnormal{cp}},\alpha,\beta,h,\mu,s,\Omega)$ satisfy
$\Omega>s^{2}/2+\boldsymbol{\beta}^{-}/\alpha$. If
$\Omega=s^{2}/2+\boldsymbol{\beta}^{-}/\alpha$ and
$\Omega<h+\mu+\frac{s^{2}}{4}(1+\alpha^{2})$ the same holds except the DW is
possibly non-flat.

###### Proof.

It follows from Proposition 1 and the discussion before that
$\Omega>s^{2}/2+\boldsymbol{\beta}^{-}/\alpha$ for the parameter perturbation
implies that the eigenvalues of the perturbed equilibrium $Z^{\pi}_{-}\approx
E^{\pi}$ satisfy $\mathrm{Re}(\nu_{k,-}^{\pi})<0$, $k=1,2$. Hence, the stable
manifold at the target equilibrium is two-dimensional and lies in a smooth
family with the center-stable manifold at the transition point. Then the proof
is the same as in the codim-0 case. If the perturbation has
$\Omega=s^{2}/2+\boldsymbol{\beta}^{-}/\alpha$ then we consider as target
manifold the three-dimensional stable manifold of a neighborhood of
$Z^{\pi}_{-}$ within the blow-up chart $\theta=0$. This neighborhood consists
of periodic orbits by Proposition 1 if
$\Omega<h+\mu+\frac{s^{2}}{4}(1+\alpha^{2})$. By dimensionality the
intersection with the unstable manifold of $Z^{0}_{-}$ persists and yields a
heteroclinic orbit from the perturbed equilibrium at $\theta=0$ to the blow-up
chart at $\theta=\pi$. Perturbing $c_{\textnormal{cp}}$ away from zero moves
the left-asymptotic state into the inhomogeneous regime and thus generates an
inhomogeneous DWs. Note from Proposition 1 that the right-asymptotic state is
either an equilibrium with $q\neq 0$ or a periodic orbit along which $q=0$
happens at most at two points. ∎

Next we present a refined result in which we show that typical perturbations
indeed give non-flat DWs, i.e., heteroclinic connections with right-asymptotic
state being a periodic orbit. The existence of flat DWs for
$c_{\textnormal{cp}}\neq 0$ is severely constrained, but not ruled out by this
result. Our numerical results, such as those presented in §4, always lead to a
selected solution with a periodic asymptotic state.

In addition, attempts to perform numerical continuation (see §4) of flat DWs
to $c_{\textnormal{cp}}\neq 0$ failed. Here we added the constraint
$\widetilde{H}=0$ and allowed adjustment the parameters $h$ and $s$, but the
continuation process did not converge, which confirms numerically the generic
selection of a periodic orbit.

As mentioned before, the right asymptotic state is $\boldsymbol{e}_{3}$ in
either case in the PDE coordinates; the difference between flat and non-flat
lies in the finer details of how the profile approaches $\boldsymbol{e}_{3}$
in term of $p$ and also $q$, which relates to $\boldsymbol{m}$ through (2).

######  Theorem 2.

Consider the smooth family of DWs from Corollary 1 with
$c_{\textnormal{cp}}=0$ for parameters satisfying (12) with fixed $\alpha>0$,
$\beta\geq 0$, and $\mu<0$. Then there is a neighborhood
$(c_{\textnormal{cp}},s,h)$ of $(0,s_{0},h^{*})$ such that the following
holds. Flat DWs occur at most on a surface in the
$(c_{\textnormal{cp}},s,h)$-parameter space and, for $\beta\neq 0$, satisfy
$|h-h_{0}|^{2}+|s-s_{0}|^{2}=\mathcal{O}(|c_{\textnormal{cp}}|^{3})$, more
precisely (31) holds, where $h_{0}=h^{\ast}$ and $s_{0}=2\sqrt{-\mu}/\alpha$.
Otherwise DWs are non-flat, in particular all DWs not equal to
$\boldsymbol{m}_{0}$ for $c_{\textnormal{cp}}=0$ or $\beta=0$ are non-flat.

Due to its more technical nature, the proof of this theorem is deferred to
Appendix 6.1.

It remains to consider the codim-2 case.

######  Theorem 3.

For any parameter set $(\alpha_{0},\beta_{0},h_{0},\mu_{0})$ in the codim-2
case, i.e., $\mu_{0}<0$ and $\beta_{0}/\alpha_{0}\leq
h_{0}<\beta_{0}/\alpha_{0}-2\mu_{0}-2\mu_{0}/\alpha_{0}^{2}$, the following
holds. The explicit homogeneous DWs $\boldsymbol{m}_{0}$ in (9) lies in a
smooth family of DWs parameterized by
$(c_{\textnormal{cp}},\alpha,\beta,h,\mu)$ near
$(0,\alpha_{0},\beta_{0},h_{0},\mu_{0})$. Here the values of $(s,\Omega)$ are
functions of the parameters $(c_{\textnormal{cp}},\alpha,\beta,h,\mu)$ and lie
in a neighbourhood of $(s_{0},\Omega_{0})$ from (19). This family is locally
unique near $\boldsymbol{m}_{0}$ and for $c_{\textnormal{cp}}\neq 0$ consists
of inhomogeneous flat DWs.

The proof of Theorem 3 is presented in Appendix 6.2 and is based on the
Melnikov method for perturbing from $\boldsymbol{m}_{0}$. As the unperturbed
heteroclinic orbit has codimension two, the bifurcation is studied in a three-
parametric family with perturbation parameters
$\eta\coloneqq\left(c_{\textnormal{cp}},s,\Omega\right)^{\mkern-1.2mu\mathsf{T}}$,
which yields a two-component splitting function $M(\eta)$ that measures the
mutual displacement of the manifolds $W^{0}_{u}$ and $W^{\pi}_{s}$.

Due to the fact that $\mathrm{Re}(\nu^{\pi}_{j,-})<0$ also for $\beta=0$ in
the codim-0 regime and the case $\beta=0$ is included in Theorem 2 as well as
Theorem 3, we immediately get the following result.

######  Corollary 2.

Inhomogeneous flat DWs also exist in the LLG equation $(\beta=0)$, which can
be flat or non-flat, respectively.

Theorem 3 completes the existence study of DWs. Therefore, for any value of
the applied field $h$ there exists a heteroclinic connection between the blow-
up charts with $c_{\textnormal{cp}}\neq 0$ and $q\not\equiv 0$, thus an
inhomogeneous (typically flat) DW. Recall that we have focused on right moving
DWs, but all results are also valid for left moving walls due to symmetry.
Therefore inhomogeneous DWs exist with $c_{\textnormal{cp}}\neq 0$ for any
value of the applied field $h\in\mathbb{R}$.

##  4 Numerical Results

Numerical continuation for ordinary differential equations is an established
tool for bifurcation analysis in dynamical system. In this section we present
continuation results to illustrate the analytical results discussed in § 3. In
particular, we will focus on continuation in the parameter
$c_{\textnormal{cp}}$ in the range of $(-0.5,0.5)$ as this perturbs away from
the known family $\boldsymbol{m}_{0}$ from (17) (cf. Figure 1(b)) with speed
and frequency determined by (19) for a given applied field. Note that we also
focus only on right-moving fronts in this section for reasons of clarity. All
results were produced by continuation in AUTO-07P and graphics were created
with Mathematica as well as MATLAB.

Heteroclinic orbits were detected as solutions to the boundary value problem
given by the desingularized system (7) plus a phase condition and boundary
conditions at $\xi=-L$ and $\xi=L$ taken from the analytic equilibrium states
in $p$ and $q$ on the blow-up charts (Remark 3). In the codim-2 case, the four
required conditions are the $p,q$ values at the charts. In the center case,
the three required conditions are: (1,2) the two $p,q$ values at the left
chart and (3) the energy difference determined by the function (13). In the
codim-0 case, the two required conditions are the $p$ values at both charts.
Moreover, we found $L=50$ was sufficiently large.

In order to relate to (LLGS), we plot most of the profiles after blowing down
to the sphere rather than using the ODE phase space.

(a)

(b)

Figure 7: DWs obtained from continuation of $\boldsymbol{m}_{0}$ in system (7)
in the codim-2 regime $h=0.5$ ($h^{\ast}=10.2$) with initial speed and
frequency $s_{0}=0.12$ as well as $\Omega_{0}=0.44$, and
$(c_{\textnormal{cp}},s,\Omega)=(-0.5,0.11221,0.44077)$. (a) Projection onto
the sphere. (b) Zoom-in of corresponding $q$-profile (red) and $m_{1}$
component (blue).

Following the standing assumption on positive speeds and using
$c_{\textnormal{cp}}$ as well as $h$ as the main parameters, we keep the other
parameters fixed with values

$\alpha=0.5,\beta=0.1,\mu=-1.$

The value of the applied field for the center case, given the fixed
parameters, is $h^{\ast}=10.2$ (cf. § 3.2), which leads to $s_{0}=4.0$ as well
as $\Omega_{0}=8.2$ (cf. (19)).

###  4.1 Codim-2 case

The lower boundary for values of the applied field $h$ lies in the codim-2
regime and is given by $h=\beta/\alpha=0.2$. As a first numerical example we
consider the slightly larger value $h=0.5$. The results upon continuation in
the negative as well as positive direction of $c_{\textnormal{cp}}$ are
presented in Figures 2(a), 2(c), and 7. The inhomogeneous nature of these
solutions ($c_{\textnormal{cp}}\neq 0$) is reflected in the significantly
varying azimuthal angles, also visible in the oscillatory nature of the
$m_{1}$ component in Figures 2(c) as well as 7b.

The linear part of the splitting function (33) (see Theorem 3), which predicts
the direction of parameter variation for the existence of inhomogeneous DWs
($c_{\textnormal{cp}}\neq 0$) to leading order, reads in this example

$M(c_{\textnormal{cp}},s,\Omega)=\begin{pmatrix}-0.00147567&-0.499245&0.245945\\\
-0.000577908&-0.245945&-0.499245\end{pmatrix}\cdot\begin{pmatrix}c_{\textnormal{cp}}\\\
s\\\ \Omega\end{pmatrix},$

so that $M=(0,0)^{\mkern-1.2mu\mathsf{T}}$ for
$(s,\Omega)=\left(-0.00283744\cdot c_{\textnormal{cp}},0.000240252\cdot
c_{\textnormal{cp}}\right)$. For the parameter values in Figure 7 and 2(a) we
obtain, respectively,

$M(-0.5,-0.007788,0.000771)=\left(0.00481558,0.00181945\right)^{\mkern-1.2mu\mathsf{T}},$
$M(0.5,-0.007973,0.007173)=\left(0.00648248,-0.00133122\right)^{\mkern-1.2mu\mathsf{T}}.$

Note that here the splitting of the (1-dimensional) unstable manifold of the
left equilibrium and the (1-dimensional) stable manifold of the right
equilibrium differ, i.e., are in opposite directions (signs) in frequency and
speed for variations in $c_{\textnormal{cp}}$.

(a)

(b)

Figure 8: DWs obtained from continuation of $\boldsymbol{m}_{0}$ in system (7)
projected onto the sphere in the codim-2 regime $h=10.1$ ($h^{\ast}=10.2$)
with initial speed and frequency $s_{0}=3.96$ and $\Omega_{0}=8.12$. (a)
$(c_{\textnormal{cp}},s,\Omega)=(-0.5,3.99541,8.05973)$. (b)
$(c_{\textnormal{cp}},s,\Omega)=(0.5,4.08089,8.22402)$.

In addition note the decrease in frequency in the $m_{1}$ component, and thus
also in the $m_{2}$ as a result of the increase of the $q$ component towards
zero, cf. Figure 7b. Here, the azimuthal angle decreases since $\phi=\int q$
and $q<0$.

As a further example in the codim-2 regime, we consider $h=10.1<10.2=h^{\ast}$
near the upper boundary of the codim-2 regime in terms of the applied field
$h$. The results of the continuation in $c_{\textnormal{cp}}$ are presented in
Figure 8. The linear approximation of the splitting in this case is given by
$M(-0.5,0.03541,-0.06027)=\left(-0.000175537,-0.00104378\right)^{\mkern-1.2mu\mathsf{T}}$
as well as
$M(0.5,0.12089,0.10402)=\left(-0.00149519,0.0014975\right)^{\mkern-1.2mu\mathsf{T}}$
in (a) and (b), respectively. Note that the direction of splitting of the two
components in this case is also dependent on the polarity sign, as in the
previous example. In both cases ($h=0.5$ and $h=10.1$), the continuation
results look basically the same in the codim-2 regime, where the solution is,
roughly speaking, constantly spiraling down from the north to the south pole.

###  4.2 Center case

(a)

(b)

Figure 9: DWs obtained from continuation of $\boldsymbol{m}_{0}$ in system (7)
in the center regime $h=h^{\ast}=10.2$ with $s=s_{0}=4,\Omega=\Omega_{0}=8.2$,
and $c_{\textnormal{cp}}=0.5$. (a) Projection onto the sphere. (b) Profile of
corresponding $q$-component.

We perform computations in the center case with applied field
$h=h^{\ast}=10.2$ and fixed frequency
$\Omega=\boldsymbol{\beta}^{-}/\alpha+s^{2}/2$ (see Proposition 1 and its
discussion details). Theorem 2 shows that the right asymptotic state is
generically a periodic orbit and more precisely that in case
$c_{\textnormal{cp}}=0$, no constellation of $h$ and $s$ exists, both not
equal to zero, for which the right asymptotic state is the (shifted)
equilibrium. The results of continuation in $c_{\textnormal{cp}}$ projected on
the sphere look quite the same, which is why only the result for
$c_{\textnormal{cp}}=0.5$ is presented in Figure 9. The fact that the right
asymptotic state is a periodic orbit on the blow-up chart $\theta=\pi$ is
reflected by the nearly constant oscillations in the $q$-profile for $\xi$
close to the right boundary (cf. Figure 9b).

(a)

(b)

(c)

Figure 10: Continuation of $\boldsymbol{m}_{0}$ in system (7) in the center
case with applied field $h=h^{\ast}=10.2$ and fixed frequency
$\Omega=\boldsymbol{\beta}^{-}/\alpha+s^{2}/2$; here $s_{0}=4$. Shown is the
energy difference (solid blue line) between the equilibrium and asymptotic
state from continuation on right boundary against the continuation parameter
$c_{\textnormal{cp}}$ in (a), $h$ in (b), and $s$ in (c). The red dashed curve
in (b) and (c) is the quadratic approximation (21).

That the right state is not the equilibrium in the blow-up chart is further
corroborated by computing the difference in energy $\widetilde{H}$ between
this equilibrium state (see Remark 3) and the approximate right asymptotic
state obtained from continuation. The analytic prediction of this difference
up to second order is given by (31), which reads, for the chosen parameters,

$-0.006612+0.00673s-0.00183s^{2}-0.00134h-0.000086h^{2}+0.00077hs\,.$ (21)

As this analytic prediction is independent of $c_{\textnormal{cp}}$ the
dependence of $\widetilde{H}$ on $c_{\textnormal{cp}}\approx 0$ is of cubic or
higher power. Indeed, the results plotted in Figure 10a suggest an at least
quartic dependence since a maximum lies at $c_{\textnormal{cp}}=0$. The
asymmetric nature of the graph suggests that odd powers appear in the
expansion beyond our analysis, but also note the order of ${10}^{-6}$ in
$\widetilde{H}$. In addition to the dependence on $c_{\textnormal{cp}}$,
continuations for $c_{\textnormal{cp}}=0$ of $\widetilde{H}$ in $h$ with fixed
$s=s_{0}=4$ and in $s$ with fixed $h=h^{\ast}=10.2$ are plotted in Figure 10b
and 10c, respectively. Here we also plot the quadratic prediction (21).

###  4.3 Codim-0 case

Next, we consider an applied field $h=10.3$ in the codim-0 regime, just above
the applied field value for the center case $h^{*}=10.2$. The results of
continuation in $c_{\textnormal{cp}}$ are plotted on the sphere in Figure 11.
The azimuthal profile in $\phi$ and hence in $q$ are non-trivial as predicted
for inhomogeneous DWs.

In the ODE, $q$ possesses an oscillating profile and has a monotonically
decreasing amplitude in both cases. This is a consequence of the proximity to
the center case and the convergence to equilibria (see § 3.2 for details).
Recall that the speed and frequency are not selected by the existence problem
during continuation in $c_{\textnormal{cp}}$, but are taken as the fixed
parameters $(s_{0},\Omega_{0})$ defined in (19).

(a)

(b)

Figure 11: DWs obtained from continuation of $\boldsymbol{m}_{0}$ in system
(7) projected onto the sphere in the codim-0 regime $h=10.3$ ($h^{\ast}=10.2$)
and initial speed and frequency $s_{0}=4.04$ and $\Omega_{0}=8.28$. (a)
$c_{\textnormal{cp}}=-0.5$. (b) $c_{\textnormal{cp}}=0.5$.

The final example is for a relatively large applied field $h=50$ in the
codim-0 regime, far away from the center case, and the results of continuation
in $c_{\textnormal{cp}}$ projected on the sphere are presented in Figure 2(b)
as well as Figure 12a. Moreover, the corresponding $m_{1}$ and $m_{2}$
profiles for $c_{\textnormal{cp}}=0.5$ are presented in Figure 2(d), and for
$c_{\textnormal{cp}}=-0.5$, in Figure 12b. As in the previous example, the
inhomogeneous nature is visible in the non-trivial azimuthal profile.

(a)

(b)

Figure 12: DWs obtained from continuation of $\boldsymbol{m}_{0}$ in system
(7) in the codim-0 regime $h=50$ ($h^{\ast}=10.2$) with initial speed and
frequency $s_{0}=19.92,\Omega_{0}=40.04$, and $c_{\textnormal{cp}}=-0.5$. (a)
Projection onto the sphere. (b) Zoom-in of the corresponding $m_{1}$ (solid
blue) and $m_{2}$ (dashed red) component.

In summary, switching on the parameter $c_{\textnormal{cp}}$ leads to a
variety of inhomogeneous flat as well as non-flat DW solutions, but also in
case $c_{\textnormal{cp}}=0$ there exist inhomogeneous DWs (cf. Figure 4)
which are much more complex than the homogeneous one given by (9) (cf. Figure
1(b)).

Finally, recall from § 3.1 that in the explicit family (9), the right moving
DWs terminate at $s=0$. The question arises what happens for
$c_{\textnormal{cp}}\neq 0$ along the parameter $s$. To study this, we
performed a continuation in the parameter $s$ for different values of
$c_{\textnormal{cp}}$. For decreasing $s$ we found that numerical continuation
failed at some $s>0$ for $c_{\textnormal{cp}}\neq 0$ (cf. Figure 13). The
details of this apparent existence boundary are beyond the scope of this
paper. Note that the special role of $s$ is reflected in the splitting
function (34). In case $s_{0}=s=0$, the first column of (32) is zero (see
(34)) and thus the parameterization of $c_{\textnormal{cp}}$ can not be
written as a function in $s$.

In detail, we continued the analytic solution (9) in $c_{\textnormal{cp}}$
away from zero for different initial values of $\beta/\alpha<h<h^{\ast}$. This
led to inhomogeneous ($c_{\textnormal{cp}}\neq 0$) DWs, which we in turn
continued in the parameter $s$ towards zero for different fixed values of
$c_{\textnormal{cp}}$ until the continuation process fails to converge. Based
on this, there is numerical evidence that DWs with opposite speed sign
(counter-propagating fronts) can only exist simultaneously for $s=0$ (standing
fronts). We took a polynomial fit on these points as an approximation of the
existence boundary (cf. blue curve in Figure 13a). The continuation process
towards the boundary is indicated by red arrows in Figure 13a for positive
$c_{\textnormal{cp}}$. Additionally, the corresponding results in the
parameter space $\Omega$ and $c_{\textnormal{cp}}$ is presented in 13b.

(a)

(b)

Figure 13: Continuation results in $s$ and $\Omega$ for fixed
$c_{\textnormal{cp}}$ in the codim-2 parameter regime. Further parameters are
$\alpha=0.5,\beta=0.1,\mu=-1$, and $h$ free. Blue solid line represents
interpolated termination boundary for $s$ in (a) and $\Omega$ in (b),
respectively. Red arrows indicate continuation approach towards boundary.

As a last point, we briefly describe the numerical method for time-integration
near DWs, including freezing of speed and frequency (cf. Figure 4). All
calculations were done with the (free) software package pde2path which is
based on a Finite Element Method (FEM), cf. [20] and the references therein.
Time-integration in pde2path with the so-called ‘freezing’ method is discussed
in [21]. In addition to the phase condition for the speed we added a phase
condition for the rotation and time-integrated via a semi-implicit Euler
scheme.

##  5 Discussion and Outlook

We have presented results pertaining the existence of different types of
domain walls for the LLG as well as LLGS equation. Our main focus has been on
a nonzero polarisation parameter $c_{\textnormal{cp}}\neq 0$ for any value of
the applied field, including the high-field case, and thus for any domain wall
speed. These results extend what is known in particular for inhomogeneously
structured DWs, and we have discovered an apparently new type of DWs with
certain oscillatory tails, referred to as non-flat here.

In detail, we have provided a classification of DWs based on co-dimension
properties in a reduced (spatial) coherent structure ODE, which relates to
stability and selection properties that we review next. First, we have proven
the existence of inhomogeneous flat DWs in case $c_{\textnormal{cp}}=0$ as
well as $c_{\textnormal{cp}}\neq 0$ for an applied field above a certain
threshold, which is mainly material depending. To our knowledge, the only
previous existence result for $c_{\textnormal{cp}}\neq 0$ with ’large’ applied
fields concerns less relevant non-localized DWs [12]. Here the existence
problem does not select speed and frequency.

Second, we have discussed the so-called center case, which is characterized by
non-hyperbolic equilibria in the underlying coherent structure ODE. In this
case, we have shown the existence of inhomogeneous DWs including the leading
order selection mechanism. These solutions are non-flat in case
$c_{\textnormal{cp}}=0$ and generically also non-flat for
$c_{\textnormal{cp}}$ away from zero, which was substantiated by numerical
results. The fundamental observation has been the existence of a Hamiltonian
function in a certain parameter regime in the corresponding coherent structure
ODE.

Third, we have proven the existence of inhomogeneous DWs in the so-called
codim-2 regime, which is a range of values for the applied field in which the
speed $s$ is between zero and the center case speed. In this regime, each
solution in case $c_{\textnormal{cp}}\neq 0$ is uniquely determined by its
speed as well as frequency. Here we have also presented the leading order
selection function in the coherent structure ODE variables $p$ and $q$, which
depends on the speed $s$, the frequency $\Omega$, and is independent of
$c_{\textnormal{cp}}$ for standing fronts.

We believe that these results are not only interesting and relevant from a
theoretical and mathematical viewpoint, but also from an application
viewpoint. They could help to better understand the interfaces between
different magnetic domains in nanostructures, e.g. in the development of
racetrack memories, which are a promising prospective high density storage
unit that utilize a series of DWs by shifting at high speed along magnetic
nanowires through nanosecond current pulses.

In order to illustrate and corroborate these theoretical results, we have
presented numerical computations for a variety of values for the applied field
in § 4. On the one hand, the examples in essence show that large applied
fields lead to more complex profiles of the DWs in case
$c_{\textnormal{cp}}\neq 0$. On the other hand, while in the center case the
DWs projected on the sphere appear similar to those for small applied fields,
these solutions approach the poles in a qualitatively different ‘non-flat’
manner – as predicted by our analysis. Moreover, we compared the numerical and
analytical results of the selection mechanism in the center case, showing that
the analytical leading order approximation predicts the effect of small
perturbations in the parameters. Notably, for applied fields above a certain
threshold, where the existence analysis does not provide a selection of speed
and frequency, numerically the DWs selected in the PDE dynamics are in the
center case, both for $c_{\textnormal{cp}}=0$ as well as
$c_{\textnormal{cp}}\neq 0$. Hence, it might be possible to detect these
solutions in a high-field regime in real materials.

One question concerning existence beyond our analysis is whether inhomogeneous
(flat or non-flat) solutions exist for any value of
$c_{\textnormal{cp}}\in(-1,1)$, and whether this class could be utilized in
applications.

A natural step towards the understanding of domain wall motion in nanowires
beyond the question of existence concerns the dynamic stability. For
inhomogeneous solutions there appears to be no rigorous result in this
direction. In particular for larger applied fields, stability results would be
an essential step towards understanding the selection mechanism of solutions
in terms of speed and frequency; our first numerical investigations show that
solutions in the center parameter regime are selected, i.e., inhomogeneous
non-flat DWs.

Moreover, preliminary analytic results, for $c_{\textnormal{cp}}=0$ as well as
$c_{\textnormal{cp}}\neq 0$, show that selection mechanism is mainly
determined by the value of the applied field, where in the bi-stable case
($\pm\boldsymbol{e}_{3}$ linearly stable) homogeneous DWs are selected, and in
the mono-stable case inhomogeneous non-flat DWs are selected, which will be
studied in detail in an upcoming work.

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##  6 Appendix

###  6.1 Proof of Theorem 2

We use the notation

$u=u(\xi;\eta,\alpha,\beta,\mu)=\left(\theta(\xi;\eta,\alpha,\beta,\mu),p(\xi;\eta,\alpha,\beta,\mu),q(\xi;\eta,\alpha,\beta,\mu)\right)^{\mkern-1.2mu\mathsf{T}}$

and bifurcation parameters
$\eta=\left(c_{\textnormal{cp}},s,h\right)^{\mkern-1.2mu\mathsf{T}}$, where
$s_{0}$ and $h_{0}=h^{\ast}$ are defined below (see § 3.2 for details). The
starting point for our perturbation analysis are the unperturbed parameters
and explicit heteroclinic solution in the center case (12), where the
frequency is $\Omega_{0}=s_{0}^{2}/2+\beta/\alpha$. These are given by

$\eta_{0}:=\begin{pmatrix}c_{\textnormal{cp}_{0}}\\\ s_{0}\\\
h_{0}\end{pmatrix}:=\begin{pmatrix}0\\\ \frac{2\sqrt{-\mu}}{\alpha}\\\
\frac{\beta}{\alpha}-2\mu-\frac{2\mu}{\alpha^{2}}\end{pmatrix}$

as well as

$u_{0}=u_{0}(\xi;\eta_{0},\alpha,\beta,\mu):=\begin{pmatrix}\theta_{0}(\xi;\eta_{0},\alpha,\beta,\mu)\\\
p_{0}(\xi;\eta_{0},\alpha,\beta,\mu)\\\
q_{0}(\xi;\eta_{0},\alpha,\beta,\mu)\end{pmatrix}:=\begin{pmatrix}2\arctan\left(\exp(\sqrt{-\mu}\;\xi)\right)\\\
\sqrt{-\mu}\\\ 0\end{pmatrix}.$

Unless stated otherwise, we suppress the explicit dependence of $u$ on
$\alpha,\beta$, and $\mu$ in the following discussion. Let us write
$Z^{\pi}:=Z^{\pi}_{-}$ with the notation from Remark 3 so that the unperturbed
right asymptotic state is given by

$Z^{\pi}(\eta_{0})=\left(\pi,\frac{\alpha
s_{0}}{2},\frac{s_{0}}{2}-\sqrt{-\frac{\mu}{\alpha^{2}}}\right)^{\mkern-1.2mu\mathsf{T}}=\left(\pi,\sqrt{-\mu},0\right)^{\mkern-1.2mu\mathsf{T}}$

and its derivative with respect to $\eta$ is given by

$Z^{\pi}_{\eta}(\eta_{0})=\begin{pmatrix}0&0&0\\\ 0&\frac{\alpha}{2}&0\\\
\frac{\beta}{2\sqrt{-\mu}}&\frac{2+\alpha^{2}}{2}&-\frac{\alpha}{2\sqrt{-\mu}}\end{pmatrix}.$

We write system (7) for brevity as

$u^{\prime}=f(u;\eta)\,,$ (22)

so $f(u;\eta)$ denotes the right side of (7). The linearization w.r.t. $\eta$
in the unperturbed heteroclinic connection $u_{0}$, given by (17), is the non-
autonomous linear equation

$u_{\eta}^{\prime}=f_{u}(u_{0};\eta_{0})u_{\eta}+f_{\eta}(u_{0};\eta_{0})\eta,$
(23)

where
$u_{\eta}=\left(\theta_{\eta},p_{\eta},q_{\eta}\right)^{\mkern-1.2mu\mathsf{T}}$.
Its homogeneous part is

$\begin{array}[]{rl}\theta^{\prime}_{\eta}&=\sqrt{-\mu}\cos(\theta_{0})\theta_{\eta}+p_{\eta}\sin(\theta_{0})\\\
p^{\prime}_{\eta}&=-\left(\alpha
s_{0}+2\sqrt{-\mu}\cos(\theta_{0})\right)p_{\eta}+s_{0}q_{\eta}\\\
q^{\prime}_{\eta}&=-s_{0}p_{\eta}-\left(\alpha
s_{0}+2\sqrt{-\mu}\cos(\theta_{0})\right)q_{\eta}\end{array},$ (24)

with $\theta_{0}(\xi)=2\arctan\left(\exp(\sqrt{-\mu}\,\xi)\right)$ due to
(17). We next solve (24) and determine its fundamental solution matrix.

The first obvious vector-solution of it is
$U_{1}=u_{0}^{\prime}=(\theta^{\prime}_{0},0,0)$ since the second and the
third equation of (24) do not depend on $\theta_{\eta}$. The other solutions
can be obtained from $U_{1}$ and the result of Lemma 1. Changing to polar
coordinates

$p_{\eta}=r\cos\varphi\,,\quad q_{\eta}=r\sin\varphi\,,$

the equations for $p_{\eta}$ and $q_{\eta}$ become

$\begin{array}[]{rl}r^{\prime}&=-\left(\alpha
s_{0}+2\sqrt{-\mu}\cos(\theta_{0})\right)r\\\
\varphi^{\prime}&=-s_{0}\,,\end{array}$

whose general solution can be written as

$\begin{array}[]{rl}p_{\eta}&=r_{0}r(\xi)\cos(-s_{0}\xi+\varphi_{0})\\\
q_{\eta}&=r_{0}r(\xi)\sin(-s_{0}\xi+\varphi_{0})\end{array}$

where

$r(\xi)=\exp\left(-\alpha
s_{0}\xi-2\sqrt{-\mu}\int\limits_{\xi}\cos(\theta_{0}(\tau))d\tau\right)=\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)^{2}\,\mathrm{e}^{(-2\sqrt{-\mu}-\alpha
s_{0})\xi},$

and $r_{0},\varphi_{0}$ are arbitrary integration constants corresponding to
suitable initial conditions. Note that
$\lim\limits_{\xi\to\pm\infty}r(\xi)=\infty$ for $0\leq
s_{0}<2\sqrt{-\mu}/\alpha$.

Next, the values of the integration constants have to be selected in order for
the second and the third vector-solutions

$U_{2}=\begin{pmatrix}\theta^{1}_{1}\\\
r_{1}r(\xi)\cos(-s_{0}\xi+\varphi_{1})\\\
r_{1}r(\xi)\sin(-s_{0}\xi+\varphi_{1})\end{pmatrix}\,,\quad
U_{3}=\begin{pmatrix}\theta^{2}_{1}\\\
r_{2}r(\xi)\cos(-s_{0}\xi+\varphi_{2})\\\
r_{2}r(\xi)\sin(-s_{0}\xi+\varphi_{2})\end{pmatrix}$ (25)

to be linearly independent. Here $\theta^{1}_{1},\theta^{2}_{1}$ are not
relevant for what follows. The determinant of the fundamental matrix reads

$\det\Phi(\xi)=\det\left(U_{1}(\xi),U_{2}(\xi),U_{3}(\xi)\right)=r_{1}r_{2}r^{2}(\xi)\theta^{\prime}_{0}(\xi)\sin(\varphi_{2}-\varphi_{1}),$

which is non-zero for $r_{1}=r_{2}=1$, $\varphi_{1}=0$ and
$\varphi_{2}=\pi/2$, i.e. $\det\Phi(\xi)=r^{2}(\xi)\theta^{\prime}_{0}(\xi)$.
Together, we get the fundamental solution matrix of the homogeneous part as

$\Phi(\xi)=\begin{pmatrix}\theta^{\prime}_{0}(\xi)&\theta^{1}_{1}(\xi)&\theta^{2}_{1}(\xi)\\\
0&r(\xi)\cos(-s_{0}\xi)&-r(\xi)\sin(-s_{0}\xi)\\\
0&r(\xi)\sin(-s_{0}\xi)&r(\xi)\cos(-s_{0}\xi)\end{pmatrix}.$ (26)

The derivative of (22) with respect to $\eta$ is given by (23) and from the
variation of constants formula we get for some $\xi_{0}$ that

$u_{\eta}(\xi)=\Phi_{\xi,\xi_{0}}u_{\eta}(\xi_{0})+\int\limits_{\xi_{0}}^{\xi}{\Phi_{\xi,\tau}f_{\eta}(u_{0}(\tau);\eta_{0})d\tau}\,,$

where $\Phi_{\xi,\tau}=\Phi(\xi)\cdot\Phi^{-1}(\tau)$ is the evolution
operator. Using (26) we find

$\Phi_{\xi,\tau}(\xi,\tau;\eta)=\begin{pmatrix}\Theta_{1}&\Theta_{2}&\Theta_{3}\\\
0&\frac{r(\xi)}{r(\tau)}\cos\left(-s_{0}(\xi-\tau)\right)&-\frac{r(\xi)}{r(\tau)}\sin\left(-s_{0}(\xi-\tau)\right)\\\
0&\frac{r(\xi)}{r(\tau)}\sin\left(-s_{0}(\xi-\tau)\right)&\frac{r(\xi)}{r(\tau)}\cos\left(-s_{0}(\xi-\tau)\right)\end{pmatrix},$
(27)

where the explicit forms of the functions $\Theta_{1,2,3}(\xi)$ are not
relevant for the remainder of this proof. Since $u_{\eta}(\xi)$ tends to
$\partial_{\eta}Z^{0}_{-}$ for $\xi\to-\infty$ the hyperbolicity of
$Z^{0}_{-}$ (more precisely the resulting exponential dichotomy) implies
$\Phi_{\xi,\xi_{0}}u_{\eta}(\xi_{0})\to 0$ as $\xi_{0}\to-\infty$ and so

$u_{\eta}(\xi)=\int\limits_{-\infty}^{\xi}{\Phi_{\xi,\tau}f_{\eta}(u(\tau;\eta_{0});\eta_{0})d\tau}\,.$
(28)

Regarding the limiting behavior as $\xi\to\infty$, recall that Corollary 1
states that the right asymptotic limit of the perturbed heteroclinic orbit is
either the perturbed equilibrium $Z^{\pi}(\eta)$ or a periodic orbit around it
in the blow-up chart at $\theta=\pi$. The integral (28) distinguishes these
case in the sense that either it has a limit as $\xi\to+\infty$ so the
heteroclinic orbit connects the two equilibria, or it does not and the
heteroclinic orbit connects to a periodic solution.

We next determine $u_{\eta}(\xi)$ componentwise

$u_{\eta}(\xi)=v\coloneqq\begin{pmatrix}v_{11}&v_{12}&v_{13}\\\
v_{21}&v_{22}&v_{23}\\\ v_{31}&v_{32}&v_{33}\end{pmatrix},$

where $v_{ij}$ are the components of (28) and index $i=1,2,3$ relates to
$\theta,p,q$ as well as $j=1,2,3$ to $c_{\textnormal{cp}},s,h$.

Towards this, we compute

$f_{\eta}(u_{0}(\tau),\eta_{0})=\begin{pmatrix}0&0&0\\\
-\beta/\alpha&-\frac{\sqrt{-\mu}}{\alpha}(2+\alpha^{2})&1\\\
\frac{2\beta}{1+\mathrm{e}^{2\sqrt{-\mu}\;\tau}}&\sqrt{-\mu}&0\end{pmatrix},$

and together with (28) and (27) we obtain

$v_{21}=-\frac{\beta}{\alpha}I_{C}-2\beta J_{S}\,,$ | $v_{22}=-\frac{\sqrt{-\mu}}{\alpha}(2+\alpha^{2})I_{C}-\sqrt{-\mu}I_{S}\,,$ | $v_{23}=I_{C}\,,$
---|---|---
$v_{31}=-\frac{\beta}{\alpha}I_{S}+2\beta J_{C}\,,$ | $v_{32}=-\frac{\sqrt{-\mu}}{\alpha}(2+\alpha^{2})I_{S}+\sqrt{-\mu}I_{C}$ | $v_{33}=I_{S}\,,$

where

$I_{C}=I_{C}(\xi)\coloneqq\int\limits_{-\infty}^{\xi}\frac{\left(1+\exp\left(-2\sqrt{-\mu}\;\xi\right)\right)^{2}}{\left(1+\exp\left(-2\sqrt{-\mu}\;\tau\right)\right)^{2}}\cos\left(-s_{0}(\xi-\tau)\right)d\tau\,,$
$I_{S}=I_{S}(\xi)\coloneqq\int\limits_{-\infty}^{\xi}\frac{\left(1+\exp\left(-2\sqrt{-\mu}\;\xi\right)\right)^{2}}{\left(1+\exp\left(-2\sqrt{-\mu}\;\tau\right)\right)^{2}}\sin\left(-s_{0}(\xi-\tau)\right)d\tau\,,$
$J_{C}=J_{C}(\xi)\coloneqq\int\limits_{-\infty}^{\xi}\frac{\exp\left(-2\sqrt{-\mu}\;\tau\right)\left(1+\exp\left(-2\sqrt{-\mu}\;\xi\right)\right)^{2}}{\left(1+\exp\left(-2\sqrt{-\mu}\;\tau\right)\right)^{3}}\cos\left(-s_{0}(\xi-\tau)\right)d\tau\,,$
$J_{S}=J_{S}(\xi)\coloneqq\int\limits_{-\infty}^{\xi}\frac{\exp\left(-2\sqrt{-\mu}\;\tau\right)\left(1+\exp\left(-2\sqrt{-\mu}\;\xi\right)\right)^{2}}{\left(1+\exp\left(-2\sqrt{-\mu}\;\tau\right)\right)^{3}}\sin\left(-s_{0}(\xi-\tau)\right)d\tau\,.$

Note that we do not provide explicit formulas for $v_{11}$, $v_{12}$ and
$v_{13}$, because they are not needed for further computations. This is the
reason why we neglected the explicit expressions of $\Theta_{1}\,,\Theta_{2}$,
and $\Theta_{3}$ before. We now introduce the following complex-valued
integrals for further computations:

$I(\xi)\coloneqq
I_{C}(\xi)+\mathrm{i}I_{S}(\xi)=\int\limits_{-\infty}^{\xi}\frac{\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}\tau}\right)^{2}}\,\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}(\xi-\tau)\right)d\tau\,,$

for $I_{C}$ and $I_{S}$ as well as

$J(\xi)\coloneqq
J_{C}(\xi)+\mathrm{i}J_{S}(\xi)=\int\limits_{-\infty}^{\xi}{\frac{\mathrm{e}^{-2\sqrt{-\mu}\tau}\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}\tau}\right)^{2}}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}(\xi-\tau)\right)d\tau}\,$

for $J_{C}$ and $J_{S}$.

$\xi$$\xi_{0}$$\frac{\mathrm{i}\pi}{2\sqrt{-\mu}}$$I_{1}$$I_{2}$$I_{3}$$I_{4}$
Figure 14: Contour $\mathbf{C}$ in $\mathbb{C}$ for the integrals $I$ and $J$.

We extend the above integrals to the complex plane and integrate along the
counter-clockwise oriented rectangular contour $\mathbf{C}$ as illustrated in
Figure 14, and let $\xi_{0}\to-\infty$. We will provide the details of the
computation of $I$ only, as $J$ can be calculated in a fully analogous way.

The complex integrand of $I$ is

$g(z;\xi)\coloneqq\frac{\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}z}\right)^{2}}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}(\xi-z)\right)\,,$

with singularities in $\mathbb{C}$ at the points
$\frac{\mathrm{i}(\pi+2k\pi)}{2\sqrt{-\mu}}$, $k\in\mathbb{Z}$, one of which
lies in the interior of $\mathbf{C}$, namely
$z_{0}:=\frac{\mathrm{i}\pi}{2\sqrt{-\mu}}$. The contour integral $I$ can now
be written via the residue theorem as

$I_{1}(\xi)+I_{2}(\xi)+I_{3}(\xi)+I_{4}(\xi)=2\pi\mathrm{i}\sum\limits_{\textnormal{int}\mathbf{C}}\textnormal{Res}\,g\left(z;\xi\right)\,,$

where $I_{1},\ldots,I_{4}$ are given by

  1. $I_{1}:$

$z=x$,

$\displaystyle I_{1}(\xi_{0},\xi)$
$\displaystyle=\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\int\limits_{\xi_{0}}^{\xi}{\frac{\exp\left(\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}x\right)}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}x}\right)^{2}}dx}$
$\displaystyle I(\xi)=\lim_{\xi_{0}\to-\infty}I_{1}(\xi_{0},\xi)$
$\displaystyle=\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\int\limits_{-\infty}^{\xi}{\frac{\exp\left(\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}x\right)}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}x}\right)^{2}}dx},$

  2. $I_{2}:$

$z=\xi+\mathrm{i}y$,

$I_{2}(\xi)=\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\int\limits_{0}^{\frac{\pi}{\sqrt{-\mu}}}{\frac{\mathrm{i}\exp\left(-\frac{2\sqrt{-\mu}}{\alpha}y\right)}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}(\xi+\mathrm{i}y)}\right)^{2}}dy},$

  3. $I_{3}:$

$z=x+\frac{\pi}{\sqrt{-\mu}}\mathrm{i}$,

$\displaystyle I_{3}(\xi_{0},\xi)$
$\displaystyle=\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\mathrm{e}^{-\frac{2\pi}{\alpha}}\int\limits_{\xi}^{\xi_{0}}{\frac{\exp\left(\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}x\right)}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}dx}$
$\displaystyle=-\mathrm{e}^{-\frac{2\pi}{\alpha}}I_{1}(\xi_{0},\xi)\,,$

  4. $I_{4}:$

$z=\xi_{0}+\mathrm{i}y$,

$\displaystyle I_{4}(\xi_{0},\xi)$
$\displaystyle=\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}(\xi_{0}-\xi)\right)\int\limits_{\frac{\pi}{\sqrt{-\mu}}}^{0}{\frac{\mathrm{i}\exp\left(-\frac{2\sqrt{-\mu}}{\alpha}y\right)}{\left(1+\mathrm{e}^{-2\sqrt{-\mu}(\xi_{0}+iy)}\right)^{2}}dy,}$
$\displaystyle\lim_{\xi_{0}\to-\infty}I_{4}(\xi_{0},\xi)$ $\displaystyle=0\,.$

Utilizing the Laurent series of $g$ we obtain

$\textnormal{Res}\,g\left(z;\xi\right)|_{z=z_{0}}=\frac{\alpha+\mathrm{i}}{2\alpha\sqrt{-\mu}}\mathrm{e}^{-\frac{\pi}{\alpha}}\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\,,$

which leads to

$I(\xi)=\left(1-\mathrm{e}^{-\frac{2\pi}{\alpha}}\right)^{-1}\left(\frac{\pi\mathrm{i}(\alpha+\mathrm{i})}{\alpha\sqrt{-\mu}}\mathrm{e}^{-\frac{\pi}{\alpha}}\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}\exp\left(-\mathrm{i}\frac{2\sqrt{-\mu}}{\alpha}\xi\right)-I_{2}(\xi)\right).$

Now we can write

$I_{C}(\xi)=\mathrm{Re}\,I(\xi)$
$=\frac{\pi\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{\alpha\sqrt{-\mu}\left(\mathrm{e}^{\frac{\pi}{\alpha}}-\mathrm{e}^{-\frac{\pi}{\alpha}}\right)}\left[-\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)-\alpha\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\right]-\frac{1}{1-\mathrm{e}^{-\frac{2\pi}{\alpha}}}I_{2}^{r}(\xi)$

as well as

$I_{S}(\xi)=\mathrm{Im}\,I(\xi)$
$=\frac{\pi\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{\alpha\sqrt{-\mu}\left(\mathrm{e}^{\frac{\pi}{\alpha}}-\mathrm{e}^{-\frac{\pi}{\alpha}}\right)}\left[\alpha\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)-\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\right]-\frac{1}{1-\mathrm{e}^{-\frac{2\pi}{\alpha}}}I_{2}^{i}(\xi)\,,$

where $I_{2}^{r}(\xi)$ and $I_{2}^{i}(\xi)$ are the real and imaginary part of
$I_{2}(\xi)$, respectively.

Studying the integral $J$ in a similar fashion, we obtain

$J_{C}(\xi)=\mathrm{Re}\,J(\xi)$
$=\frac{\pi\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{2\alpha^{2}\sqrt{-\mu}\left(\mathrm{e}^{\frac{\pi}{\alpha}}-\mathrm{e}^{-\frac{\pi}{\alpha}}\right)}\left[\alpha\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)-\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\right]-\frac{1}{\left(1-\mathrm{e}^{-\frac{2\pi}{\alpha}}\right)}J_{2}^{r}(\xi)$

as well as

$J_{S}(\xi)=\mathrm{Im}\,J(\xi)$
$=\frac{\pi\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)^{2}}{2\alpha^{2}\sqrt{-\mu}\left(\mathrm{e}^{\frac{\pi}{\alpha}}-\mathrm{e}^{-\frac{\pi}{\alpha}}\right)}\left[\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)+\alpha\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\right]-\frac{1}{\left(1-\mathrm{e}^{-\frac{2\pi}{\alpha}}\right)}J_{2}^{i}(\xi)\,,$

where also here $J_{2}^{r}(\xi)$ and $J_{2}^{i}(\xi)$ are the real and
imaginary part of $J_{2}(\xi)$. Direct computations show also that

$\lim_{\xi\to+\infty}I_{2}^{i}(\xi)=\frac{\alpha}{2\sqrt{-\mu}}\left(1-\exp\left(-\frac{2\pi}{\alpha}\right)\right)$

and

$\lim\limits_{\xi\to+\infty}I_{2}^{r}(\xi)=\lim\limits_{\xi\to+\infty}J_{2}^{r}(\xi)=\lim\limits_{\xi\to+\infty}J_{2}^{i}(\xi)=0\,.$

Summing up, the second and third component of $u_{\eta}(\xi)\cdot\eta$ for
sufficiently large $\xi$ are

$\begin{pmatrix}\frac{\alpha}{2}s\\\
\frac{\beta}{2\sqrt{-\mu}}c_{\textnormal{cp}}+\frac{2+\alpha^{2}}{2}s-\frac{\alpha}{2\sqrt{-\mu}}h\end{pmatrix}+$
$\frac{\pi}{\rho}\begin{pmatrix}\left(-\frac{1}{\alpha\sqrt{-\mu}}h+\frac{2}{\alpha^{2}}s\right)\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)+\left(-\frac{1}{\sqrt{-\mu}}h+\frac{(3+\alpha^{2})}{\alpha}s\right)\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\\\
\left(\frac{1}{\sqrt{-\mu}}h-\frac{(3+\alpha^{2})}{\alpha}s\right)\cos\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)+\left(-\frac{1}{\alpha\sqrt{-\mu}}h+\frac{2}{\alpha^{2}}s\right)\sin\left(-\frac{2\sqrt{-\mu}}{\alpha}\xi\right)\end{pmatrix}+$
$+\mathcal{O}(\mathrm{e}^{-2\sqrt{-\mu}\xi}),$

where $\rho\coloneqq\exp(\pi/\alpha)-\exp(-\pi/\alpha)$. One readily verifies
that the oscillatory part in the expression above vanishes if and only if $s$
and $h$ are zero and thus we infer that the heteroclinic connection cannot be
between equilibria to first order in the parameters.

In order to detect cancellations of these oscillatory parts for higher orders
of $s$ and $h$, we next consider the behavior of the quantity (13) with
respect to parameter perturbations. With slight abuse of notation, for
$u=(\theta,p,q)^{\mkern-1.2mu\mathsf{T}}$ we write $H(u;\eta):=H(p,q)$
evaluated at parameters $\eta$, and other parameters at some fixed value, and
we always consider the heteroclinic solutions from Corollary 1.

Our strategy in the following steps is as follows: we utilize the quantity $H$
because $\lim_{\xi\to+\infty}H$ always exists along these solutions. In order
to distinguish whether this limit is an equilibrium or a periodic orbit, we
consider

$\widetilde{H}(u;\eta)\coloneqq H(u;\eta)-H(Z^{\pi};\eta),$

i.e., the difference of the $H$-values of the (parameter dependent)
equilibrium $Z^{\pi}$ and the limit of $u$ as $\xi$ tends to infinity.
Expanding $\widetilde{H}$ in the limit $\xi\to\infty$ with respect to the
parameter $\eta$ yields conditions for periodic asymptotics. In the following,
subindices of $H$ denote partial derivatives, e.g. $H_{u}=\partial_{u}H$.

Clearly, $H(u_{0};\eta_{0})=H(Z^{\pi}(\eta_{0});\eta_{0})$, thus
$\widetilde{H}_{0}=0$ and, since equilibria are critical points of $H$, we
have
$\widetilde{H}_{u}(u_{0};\eta_{0})=\widetilde{H}_{\eta}(u_{0};\eta_{0})=(0,0,0)^{\mkern-1.2mu\mathsf{T}}$.
The second derivative is given by

$\frac{d^{2}}{d\eta^{2}}\widetilde{H}=\langle
u_{\eta},\widetilde{H}_{uu}u_{\eta}\rangle+\langle
u_{\eta},\widetilde{H}_{u\eta}\rangle+\langle\widetilde{H}_{\eta
u},u_{\eta}\rangle\,,$ (29)

since $\widetilde{H}_{\eta\eta}$ is the zero matrix, $\widetilde{H}_{u}$ the
zero vector, and $\widetilde{H}_{u\eta}=\widetilde{H}_{\eta
u}^{\mkern-1.2mu\mathsf{T}}$. Thus

$\begin{split}\widetilde{H}(u_{0}+u_{\eta}\eta;\eta)&=\frac{1}{2}\left(u_{\eta}\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{uu}(u_{0};\eta_{0})(u_{\eta}\eta)+(u_{\eta}\eta)^{\mkern-1.2mu\mathsf{T}}H_{u\eta}(u_{0};\eta_{0})\eta\\\
&\quad-\frac{1}{2}\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{uu}\left(Z^{\pi}(\eta_{0});\eta_{0}\right)\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)\\\
&\quad-\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{u\eta}\left(Z^{\pi}(\eta_{0});\eta_{0}\right)\eta+\mathcal{O}\left(\|\eta\|^{3}\right)\,.\end{split}$
(30)

With the derivatives $H_{uu},H_{u\eta}$ in (30) given by

$\displaystyle H_{uu}(u;\eta)$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&\frac{2}{q-s/2}&-\frac{2p-\alpha s}{(q-s/2)^{2}}\\\ 0&-\frac{2p-\alpha
s}{(q-s/2)^{2}}&2\frac{p^{2}-\alpha
sp+h-\beta^{-}/\alpha+\mu-s^{2}/4}{(q-s/2)^{3}}\end{pmatrix},$ $\displaystyle
H_{u\eta}(u;\eta)$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&\frac{p-\alpha
q}{(q-s/2)^{2}}&0\\\
\frac{\beta/\alpha}{(1-c_{\textnormal{cp}})^{2}(q-s/2)^{2}}&-\frac{p^{2}-\alpha
pq-\frac{\alpha
s}{2}p-\frac{s}{2}q+h-\beta^{-}/\alpha+\mu}{(q-s/2)^{3}}&-\frac{1}{(q-s/2)^{2}}\end{pmatrix},$

for the right hand side of (30) in the limit $\xi\to+\infty$ we obtain

$\displaystyle\frac{1}{2}\left(u_{\eta}\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{uu}(u_{0};\eta_{0})\left(u_{\eta}\eta\right)+\left(u_{\eta}\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{u\eta}(u_{0};\eta_{0})\eta=-\frac{\alpha\beta^{2}}{4\mu\sqrt{-\mu}}c_{\textnormal{cp}}^{2}$
$\displaystyle~{}+\frac{(4+5\alpha^{2}+\alpha^{4})(\alpha^{2}\rho^{2}-4(1+\alpha^{2})\pi^{2})}{4\alpha^{3}\rho^{2}\sqrt{-\mu}}(s-s_{0})^{2}-\frac{\alpha^{4}\rho^{2}-4(1+\alpha^{2})\pi^{2}}{4\alpha\rho^{2}\mu\sqrt{-\mu}}(h-h_{0})^{2}$
$\displaystyle-\frac{\alpha\beta(2+\alpha^{2})}{2\mu}c_{\textnormal{cp}}(s-s_{0})+\frac{\alpha^{2}\beta}{2\mu\sqrt{-\mu}}c_{\textnormal{cp}}(h-h_{0})$
$\displaystyle~{}+\frac{(2+\alpha^{2})(\alpha^{4}\rho^{2}-4(1+\alpha^{2})\pi^{2})}{2\alpha^{2}\rho^{2}\mu}(s-s_{0})(h-h_{0})\,,$

as well as

$\displaystyle\frac{1}{2}\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{uu}(Z^{\pi}(\eta_{0});\eta_{0})\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)+\left(Z^{\pi}_{\eta}(\eta_{0})\eta\right)^{\mkern-1.2mu\mathsf{T}}H_{u\eta}(Z^{\pi}(\eta_{0});\eta_{0})\eta=$
$\displaystyle~{}-\frac{\alpha\beta^{2}}{4\mu\sqrt{-\mu}}c_{\textnormal{cp}}^{2}+\frac{\alpha(4+5\alpha^{2}+\alpha^{4})}{4\sqrt{-\mu}}(s-s_{0})^{2}-\frac{\alpha^{3}}{4\mu\sqrt{-\mu}}(h-h_{0})^{2}$
$\displaystyle\quad-\frac{\alpha\beta(2+\alpha^{2})}{2\mu}c_{\textnormal{cp}}(s-s_{0})+\frac{\alpha^{2}\beta}{2\mu\sqrt{-\mu}}c_{\textnormal{cp}}(h-h_{0})+\frac{\alpha^{2}(2+\alpha^{2})}{2\mu}(s-s_{0})(h-h_{0}).$

Therefore, the expansion in the limit $\xi\to+\infty$ is independent of
$c_{\textnormal{cp}}$ and reads

$\begin{split}\lim\limits_{\xi\to\infty}\widetilde{H}(u_{0}+u_{\eta}\eta;\eta)&=-\frac{(1+\alpha^{2})^{2}(4+\alpha^{2})\pi^{2}}{\alpha^{3}\rho^{2}\sqrt{-\mu}}(s-s_{0})^{2}\\\
&\quad-\frac{2(1+\alpha^{2})(2+\alpha^{2})\pi^{2}}{\alpha^{2}\rho^{2}\mu}(s-s_{0})(h-h_{0})\\\
&\quad+\frac{(1+\alpha^{2})\pi^{2}}{\alpha\rho^{2}\mu\sqrt{-\mu}}(h-h_{0})^{2}+\mathcal{O}\left(\|\eta-\eta_{0}\|^{3}\right)\,.\end{split}$
(31)

Recall $\rho=\exp(\pi/\alpha)-\exp(-\pi/\alpha)$. One readily verifies that
the resulting (binary) quadratic form of (31) is negative definite for all
$\alpha>0$ so the only solution to the leading order problem

$\frac{d^{2}}{d\eta^{2}}\widetilde{H}(u_{0};\eta_{0})=0$

is the trivial one $(s,h)=(0,0)$, and any non-trivial solution satisfies
$|s-s_{0}|^{2}+|h-h_{0}|^{2}=\mathcal{O}(|c_{\textnormal{cp}}|^{3})$.

In particular, for $c_{\textnormal{cp}}=0$ there is a neighborhood of
$(s_{0},h_{0})$ such that the only solution is the trivial one, which is
therefore also the case in the LLG equation. In case $c_{\textnormal{cp}}\neq
0$, higher orders may lead to a solution with non-zero $s$ and/or $h$, but
there is numerical evidence that such solutions do not exist (see § 4 for
details).

###  6.2 Proof of Theorem 3

The idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to
determine a bifurcation equation whose solutions are in one-to-one
correspondence with heteroclinic connections between equilibria (7) near one
of the explicit solutions $u_{0}$ from (17) connecting the equilibria
$Z^{0}_{-}$ and $Z^{\pi}_{-}$. In the present context this is known as
Melnikov’s method, see for example [22].

Recall $u_{0}$ corresponds to a homogeneous DW for $c_{\textnormal{cp}}=0$
with speed $s_{0}$ and rotation frequency $\Omega_{0}$ given by (19). In the
present codim-2 parameter regime we will show that the bifurcation equation
defines a codimension two bifurcation curve in the three-dimensional parameter
space $(c_{\textnormal{cp}},s,\Omega)$, which passes through the point
$(0,s_{0},\Omega_{0})$. The main part of the proof is to show the existence of
certain integrals for the considered parameter set. These integrals are almost
identical to the ones studied within the proof of Theorem 2 and we use the
same approach.

In this section we denote the parameter vector by
$\eta\coloneqq(c_{\textnormal{cp}},s,\Omega)^{\mkern-1.2mu\mathsf{T}}\in\mathbb{R}^{3}$,
with initial value $\eta_{0}=(0,s_{0},\Omega_{0})^{\mkern-1.2mu\mathsf{T}}$
corresponding to the unperturbed values. The solutions of the perturbed system
close to $u_{0}$ has the form
$u(\xi;\eta)=u_{0}(\xi)+u_{\eta}(\xi;\eta_{0})(\eta-\eta_{0})+\mathcal{O}(\|\eta-\eta_{0}\|^{2})$,
where
$u_{\eta}=(\theta_{\eta},p_{\eta},q_{\eta})^{\mkern-1.2mu\mathsf{T}}=\mathcal{O}(\|\eta-\eta_{0}\|)$.

As discussed in Appendix 6.1, the linearization (23) of system (7) around the
unperturbed heteroclinic connection $u_{0}$ has the fundamental solution
matrix $\Phi(\xi)$ as defined in (26). In the present codim-2 case with
$\dim(W^{0}_{u})=\dim(W^{\pi}_{s})=1$ in $\mathbb{R}^{3}$, the bifurcation
equation $M(\eta)=0$ entails two equations. Here $M(\eta)$ measures the
displacement of the manifolds $W^{0}_{u}$ and $W^{\pi}_{s}$, and we will
choose this to be near the point
$u_{0}(0)=\left(\frac{\pi}{2},\sqrt{-\mu},0\right)^{\mkern-1.2mu\mathsf{T}}$
in the directions given by vectors $v_{1}(0)$ and $v_{2}(0)$ from adjoint
solutions as detailed below. From the Taylor expansion
$M(\eta)=M_{\eta}(\eta_{0})(\eta-\eta_{0})+\mathcal{O}(\|\eta-\eta_{0}\|^{2})$
we infer by the implicit function theorem that a full rank of
$M_{\eta}(\eta_{0})$ implies a one-to-one correspondence of solutions to the
bifurcation equation with elements in the kernel of $M(\eta_{0})$.

In order to compute $M_{\eta}(\eta_{0})$ and its rank, we project onto the
transverse directions to $u_{0}$, which means to project the inhomogeneous
part of equation (23) onto two linearly independent bounded solutions
$v_{1},v_{2}$ of the adjoint variational equation
$v^{\prime}=-A^{\mkern-1.2mu\mathsf{T}}\cdot v$, where

$A^{\mkern-1.2mu\mathsf{T}}=\begin{pmatrix}\sqrt{-\mu}\cos(\theta_{0})&0&0\\\
\sin(\theta_{0})&-\alpha s_{0}-2\sqrt{-\mu}\cos(\theta_{0})&-s_{0}\\\
0&s_{0}&-\alpha s_{0}-2\sqrt{-\mu}\cos(\theta_{0})\end{pmatrix}.$

The solutions are given in terms of (25) by

$v_{1}=\frac{U_{1}\times U_{2}}{\det\Phi}=\begin{pmatrix}0\\\
-\frac{1}{r(\xi)}\sin(-s_{0}\xi)\\\
\frac{1}{r(\xi)}\cos(-s_{0}\xi)\end{pmatrix}\quad\textnormal{and}\quad
v_{2}=\frac{U_{3}\times U_{1}}{\det\Phi}=\begin{pmatrix}0\\\
\frac{1}{r(\xi)}\cos(-s_{0}\xi)\\\
\frac{1}{r(\xi)}\sin(-s_{0}\xi)\end{pmatrix}\,.$

Implementing the projection onto these, we obtain the so-called _Melnikov
integral_

$\begin{split}M_{\eta}(\eta_{0})&\coloneqq\int\limits_{-\infty}^{+\infty}{\left(v_{1},v_{2}\right)^{\mkern-1.2mu\mathsf{T}}\cdot
f_{\eta}(u_{0};\eta_{0})}\,d\xi\\\ &=\begin{pmatrix}\beta
I_{CC}&\alpha\sqrt{-\mu}I_{S}-\sqrt{-\mu}I_{C}&I_{S}+\alpha I_{C}\\\ \beta
I_{CS}&-\alpha\sqrt{-\mu}I_{C}-\sqrt{-\mu}I_{S}&-I_{C}+\alpha
I_{S}\end{pmatrix},\end{split}$ (32)

where

$\displaystyle I_{CC}$
$\displaystyle:=\int\limits_{-\infty}^{+\infty}{\frac{\left(1-\mathrm{e}^{2\sqrt{-\mu}\xi}\right)\mathrm{e}^{\alpha
s_{0}\xi+2\sqrt{-\mu}\xi}\cos(-s_{0}\xi)}{\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)^{3}}}d\xi\,,$
$\displaystyle I_{CS}$
$\displaystyle:=\int\limits_{-\infty}^{+\infty}{\frac{\left(1-\mathrm{e}^{2\sqrt{-\mu}\xi}\right)\mathrm{e}^{\alpha
s_{0}\xi+2\sqrt{-\mu}\xi}\sin(-s_{0}\xi)}{\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)^{3}}}d\xi\,,$
$\displaystyle I_{C}$
$\displaystyle:=\int\limits_{-\infty}^{+\infty}{\frac{\mathrm{e}^{\alpha
s_{0}\xi}\cos(-s_{0}\xi)}{\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)\cdot\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)}}d\xi\,,$
$\displaystyle I_{S}$
$\displaystyle:=\int\limits_{-\infty}^{+\infty}{\frac{\mathrm{e}^{\alpha
s_{0}\xi}\sin(-s_{0}\xi)}{\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)\cdot\left(1+\mathrm{e}^{-2\sqrt{-\mu}\xi}\right)}}d\xi\,.$

We next show that the second and third columns in (32) have non-vanishing
determinant so that the rank is always 2, in particular also for $\beta=0$.

For brevity, we present the calculations of $I_{CC}$ and $I_{CS}$ only, which
are based on the same idea as the computations in Appendix 6.1. The solutions
for $I_{C}$ and $I_{S}$ can be computed in an analogous way.

From Appendix 6.1 we know that the following integral would not exist in case
$s_{0}=2\sqrt{-\mu}/\alpha$ and one readily verifies the existence for
$s_{0}=0$. Therefore, we first assume $0<s_{0}<2\sqrt{-\mu}/\alpha$ for the
moment and discuss the case $s_{0}=0$ later. We set

$I\coloneqq\int\limits_{-\infty}^{+\infty}{g(\xi)}d\xi\,,\quad
g(\xi)\coloneqq\frac{\left(1-\mathrm{e}^{2\sqrt{-\mu}\xi}\right)\mathrm{e}^{(\alpha
s_{0}+2\sqrt{-\mu})\xi}\mathrm{e}^{-\mathrm{i}s_{0}\xi}}{\left(1+\mathrm{e}^{2\sqrt{-\mu}\xi}\right)^{3}}\,,$

so that $I_{CC}=\mathrm{Re}(I)$ and $I_{CS}=\mathrm{Im}(I)$. Utilizing the
same idea for the contour integral as in Appendix 6.1 (cf. Figure 14) and the
residue theorem, we obtain

$I-\mathrm{e}^{\alpha
s_{0}\frac{\mathrm{i}\pi}{\sqrt{-\mu}}}\cdot\mathrm{e}^{s_{0}\frac{\pi}{\sqrt{-\mu}}}\cdot
I=2\pi\mathrm{i}\sum{\mathrm{Res}(g)}\,.$

The function $g$ has a pole of order three at
$\frac{\mathrm{i}\pi}{2\sqrt{-\mu}}$ and the _Laurent_ series gives

$\mathrm{Res}(g)=-\frac{(\alpha-\mathrm{i})^{2}s^{2}_{0}}{8\mu\sqrt{-\mu}}\exp\left((\alpha-\mathrm{i})\frac{\mathrm{i}\pi
s_{0}}{2\sqrt{-\mu}}\right),$

and therefore

$\displaystyle I$ $\displaystyle=\frac{2\pi s_{0}^{2}\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}\left[-2\alpha\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)+\left(\alpha^{2}-1\right)\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)\right]}{8\mu\sqrt{-\mu}\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha
s_{0}}{\sqrt{-\mu}}\right)-\mathrm{i}\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\sin\left(\frac{\pi\alpha
s_{0}}{\sqrt{-\mu}}\right)\right)}$ $\displaystyle\quad+\frac{\mathrm{i}2\pi
s_{0}^{2}\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}\left[\left(1-\alpha^{2}\right)\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)-2\alpha\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)\right]}{8\mu\sqrt{-\mu}\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha
s_{0}}{\sqrt{-\mu}}\right)-\mathrm{i}\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\sin\left(\frac{\pi\alpha
s_{0}}{\sqrt{-\mu}}\right)\right)}\,.$

Separating the real and imaginary parts of $I$ we get

$I_{CC}=\frac{\pi s_{0}^{2}\sqrt{-\mu}\,\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}}{4\mu^{2}}\cdot\frac{2\alpha\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)+\left(1-\alpha^{2}\right)\left(1+\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)}{1+\mathrm{e}^{\frac{2\pi
s_{0}}{\sqrt{-\mu}}}-2\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha s_{0}}{\sqrt{-\mu}}\right)}$
$I_{CS}=\frac{\pi s_{0}^{2}\sqrt{-\mu}\,\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}}{4\mu^{2}}\cdot\frac{\left(1-\alpha^{2}\right)\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)+2\alpha\left(1+\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)}{1+\mathrm{e}^{\frac{2\pi
s_{0}}{\sqrt{-\mu}}}-2\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha s_{0}}{\sqrt{-\mu}}\right)}\,.$

As mentioned before, one analogously gets

$I_{C}=\frac{\pi s_{0}\,\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}}{2\mu}\cdot\frac{\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)-\alpha\left(1+\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)}{1+\mathrm{e}^{\frac{2\pi
s_{0}}{\sqrt{-\mu}}}-2\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha s_{0}}{\sqrt{-\mu}}\right)}$
$I_{S}=\frac{\pi s_{0}\,\mathrm{e}^{\frac{\pi
s_{0}}{2\sqrt{-\mu}}}}{2\mu}\cdot\frac{\alpha\left(1-\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\cos\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)+\left(1+\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\right)\sin\left(\frac{\pi\alpha
s_{0}}{2\sqrt{-\mu}}\right)}{1+\mathrm{e}^{\frac{2\pi
s_{0}}{\sqrt{-\mu}}}-2\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\cos\left(\frac{\pi\alpha s_{0}}{\sqrt{-\mu}}\right)}\,.$

Having the explicit expressions for $I_{C}$, $I_{S}$, $I_{CC}$, as well as
$I_{CS}$, we can study the rank of (32) in case $0<s_{0}<2\sqrt{-\mu}/\alpha$.
The determinant of the second and third column of (32) simplifies to

$\left(\alpha I_{S}-I_{C}\right)^{2}+\left(I_{S}+\alpha
I_{C}\right)^{2}=\left(1+\alpha^{2}\right)^{2}\pi^{2}s_{0}^{2}\mathrm{e}^{\frac{\pi
s_{0}}{\sqrt{-\mu}}}\neq 0\,,\quad\forall s_{0}\neq 0\,.$

Therefore, $M_{\eta}(\eta_{0})$ has full rank and we obtain

$M(\eta)=\begin{pmatrix}\beta
I_{CC}&\alpha\sqrt{-\mu}I_{S}-\sqrt{-\mu}I_{C}&I_{S}+\alpha I_{C}\\\ \beta
I_{CS}&-\alpha\sqrt{-\mu}I_{C}-\sqrt{-\mu}I_{S}&-I_{C}+\alpha
I_{S}\end{pmatrix}\cdot(\eta-\eta_{0})+\mathcal{O}\left(\|\eta-\eta_{0}\|^{2}\right).$
(33)

In the remaining case $s_{0}=0$, which means $h=\beta/\alpha$, we have
$I_{CC}=I_{CS}=I_{S}=0$, $I_{C}=\frac{1}{2\sqrt{-\mu}}$, and thus

$M_{\eta}(\eta_{0})=\begin{pmatrix}0&-\frac{1}{2}&\frac{\alpha}{2\sqrt{-\mu}}\\\
0&-\frac{\alpha}{2}&-\frac{1}{2\sqrt{-\mu}}\end{pmatrix},$ (34)

whose rank is always 2. Thus the splitting directions are independent of
$c_{\textnormal{cp}}$ in first order in case $h=\beta/\alpha$. Moreover, note
that the splitting in $s$ is independent of the anisotropy $\mu$ to first
order in case $h=\beta/\alpha$. This completes the proof of Theorem 2.