# A new integrable model of long wave-short wave interaction and linear
stability spectra

Marcos Caso-Huerta
Department of Mathematics, Physics and Electrical Engineering
Northumbria University, Newcastle upon Tyne, UK
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Antonio Degasperis
Dipartimento di Fisica
“Sapienza” Università di Roma, Rome, Italy
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Sara Lombardo
Department of Mathematical Sciences, School of Science
Loughborough University, Loughborough, UK
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Matteo Sommacal
Department of Mathematics, Physics and Electrical Engineering
Northumbria University, Newcastle upon Tyne, UK
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###### Abstract

We consider the propagation of short waves which generate waves of much longer
(infinite) wave-length. Model equations of such long wave-short wave resonant
interaction, including integrable ones, are well-known and have received much
attention because of their appearance in various physical contexts,
particularly fluid dynamics and plasma physics. Here we introduce a new long
wave-short wave integrable model which generalises those first proposed by
Yajima-Oikawa and by Newell. By means of its associated Lax pair, we carry out
the linear stability analysis of its continuous wave solutions by introducing
the _stability spectrum_ as an algebraic curve in the complex plane. This is
done starting from the construction of the eigenfunctions of the linearised
long wave-short wave model equations. The geometrical features of this
spectrum are related to the stability/instability properties of the solution
under scrutiny. Stability spectra for the plane wave solutions are fully
classified in the parameter space together with types of modulational
instabilities.

##  1 Integrable models of long wave-short wave resonant interaction

In several physical applications, waves are represented as solutions of
nonlinear partial differential equations. Finding out the dynamical behaviour
of waves usually poses mathematical problems, which are very rarely solvable
by analytical methods. More frequently, approximate solutions may be found by
treating nonlinear terms of the equations of motion as small perturbations of
the linear part. In this framework, investigating how the linear superposition
of two or more plane waves changes due to nonlinear effects leads to
predictions on wave-wave interaction. A well known approach of this kind, the
multiscale method (see e.g [1], [2] and the references quoted therein),
requires that Fourier wave amplitudes be small, say
$\mathcal{O}(\varepsilon)$, and be functions only of appropriately
$\varepsilon$-rescaled space and time coordinates, where $\varepsilon$ is a
small dimensionless parameter. In this way, one obtains partial differential
equations in the rescaled variables which are generally simpler, and sometimes
even integrable. When they are, these approximate wave equations, even if
nonlinear, are exactly solvable by spectral techniques.

The best known example of such model equations is the nonlinear Schrödinger
(NLS) equation in a one-dimensional space,

$iS_{t_{2}}+S_{\xi\xi}-2\sigma S^{\ast}S^{2}=0,\quad\sigma=\pm 1\;,$ (1)

where $\xi=\varepsilon(x-vt)$ and $t_{2}=\varepsilon^{2}t$ are the rescaled
space and time coordinates, and the asterisk indicates complex conjugation.
This equation follows via the multiscale method from almost any real
propagation equation and provides the lowest order effect of the nonlinear
terms on the linear solution

$\varepsilon S(\xi,t_{2})e^{i(kx-\omega t)}+\varepsilon
S^{\ast}(\xi,t_{2})e^{-i(kx-\omega t)},\quad
t_{2}=\varepsilon^{2}t\,,\quad\xi=\varepsilon(x-vt),$

given by the sum of two plane waves with amplitudes $S$ and $S^{\ast}$, and
with wave number $k$, frequency $\omega$ and group velocity $v$.

If more plane waves are superimposed, cross-interaction takes place if the
wave numbers and frequencies satisfy certain resonance conditions. Among the
many cases of interest and physical significance, we consider here wave
equations which model the interaction of two small-amplitude plane waves: one
with a very long wave length ($k=0$) and a real amplitude $L$, and a second
one with a much shorter wave length and a complex amplitude $S$. Equations of
this kind have been derived and proposed with various motivations, which
mainly come from plasma physics [3] and fluid dynamics [4, 5].

On the mathematical side, long wave-short wave (LS) equations have been
differently introduced because they are, at the same time, _close_ to physical
equations and _integrable_. This second distinctive property allows the
construction of a broad range of solutions with relevant and useful insight
into experimental observations. The focus of the present work is on LS models
which are integrable.

Two very well know examples of integrable LS models are the Yajima-Oikawa (YO)
equation [3]

$iS_{t}+S_{xx}-LS=0\,,\quad L_{t}=2\left(|S|^{2}\right)_{x}\,,$ (2)

and the alternative integrable Newell (N) wave system, introduced in [5],

$iS_{t}+S_{xx}+\left(iL_{x}+L^{2}-2\sigma|S|^{2}\right)S=0\,,\quad
L_{t}=2\sigma\left(|S|^{2}\right)_{x}\,,\quad\sigma^{2}=1\,,$ (3)

where, in addition to a long wave-short wave coupling, the short wave has the
same self-interaction as the NLS equation, see (1). The two systems (2) and
(3) are related by a Miura transformation, see [6].

In [7, 8, 2] the YO equation has also been derived via multiscale method
applied to a generic real wave equation, with the resonance condition that the
the group velocity $v_{S}=\omega^{\prime}(k_{S})$ of the short wave equals the
group velocity $v_{L}=\omega^{\prime}(0)$ of the long wave, say
$v_{S}=v_{L}=v$, where $\omega(k)$ is the linear dispersion relation,
$\omega^{\prime}(k)=\mathrm{d}\omega(k)/\mathrm{d}k$, and $k_{S}$ is the non-
vanishing wave number of the short wave.

The integrability of both the YO and N equations has been exploited to
investigate extensions in various directions. Thus these LS wave equations
have been generalised to multi-component short waves, either in vector form,
as in [9], or in matrix form as in [10] requiring a higher rank matrix Lax
pair. A third LS wave scalar equation, which reads

$iS_{t}+S_{xx}+i\left(LS\right)_{x}-2|S|^{2}S=0\,,\quad
L_{t}=2\left(|S|^{2}\right)_{x}\,,$ (4)

has been derived in [11] from a $3\times 3$ matrix Lax pair, which is however
different from the Lax pair investigated in the present paper (see Sections 2
and 3), and is not discussed here.

In Section 3, we show that the YO and N equations, (2) and (3), do not need to
be treated separately. Indeed, these two model equations can be combined in
just one system, namely

$iS_{t}+S_{xx}+\left(i\alpha L_{x}+\alpha^{2}L^{2}-\beta
L-2\alpha|S|^{2}\right)S=0\,,\quad L_{t}=2\left(|S|^{2}\right)_{x}\,,$ (5)

which we refer to as YON equations. This system coincides with the YO equation
(2) for $\alpha=0$, $\beta=1$, while it reads as the N equation (3) by setting
$\alpha=\sigma$, $\beta=0$ and by substituting the field $L$ with $\sigma L$
($\sigma=\pm 1$). In fact, as we will show in Section 3, the novel system (5)
is integrable for _any_ real value of $\alpha$ and $\beta$. Thus it is likely
to be relevant to applications as it is more flexible in modelling the long
wave-short wave interaction. Moreover, this unifying result makes our present
overall analysis and discussion of the two model equations (2) and (3) simpler
and more compact.

Other LS wave integrable equations may be sorted out via transformations of
the wave fields. Indeed, the two integrable LS equations (5) and (4) may be
given an equivalent, even simpler, form by performing the gauge transformation
[5, 11]

$S(x,t)=e^{i\phi(x,t)}\hat{S}(x,t)\;,\quad
L(x,t)=\hat{L}(x,t),\quad\phi_{x}=\mu
L,\quad\phi_{t}=2\mu\left(|S|^{2}\right)\,.$ (6)

This transformation, which introduces the extra real parameter $\mu$,
originates from the conservative form of the evolution equation for the long
wave amplitude $L$, in both the systems (5) and (4). Thus gauge transforming
the YON system (5) yields the three parameter family of LS wave resonance
equations

$i\hat{S}_{t}+\hat{S}_{xx}+2i\mu\hat{L}\hat{S}_{x}+\left\\{\left(\alpha+\mu\right)\left[i\hat{L}_{x}+\left(\alpha-\mu\right)\hat{L}^{2}-2|\hat{S}|^{2}\right]-\beta\hat{L}\right\\}\hat{S}=0\,,\quad\hat{L}_{t}=2(|\hat{S}|^{2})_{x}\,,$
(7)

while the same transformation, as applied to the integrable equations (4),
leads to the LS wave system

$i\hat{S}_{t}+\hat{S}_{xx}+i\mu\hat{L}\hat{S}_{x}+\left(\mu+1\right)\left[i\left(\hat{L}\hat{S}\right)_{x}-\left(\mu\hat{L}^{2}+2|\hat{S}|^{2}\right)\hat{S}\right]=0\,,\quad\hat{L}_{t}=2(|\hat{S}|^{2})_{x}\,.$
(8)

The integrability of (2), (3) and (4), and of their transformed versions via
(6), allows the construction of special exact solutions of these model
equations, such as solitons and rogue waves (e.g., see [12, 13, 14, 15, 9]).

Integrability is also a key property to address one of the basic issues in
nonlinear wave dynamics, namely that of linear stability against small changes
of the initial conditions (e.g., see [16] and the references quoted there).
For instance, the modulational instability of a periodic wave train, as in
fluid dynamics and optics, has been well exemplified via the NLS equation (1)
with focusing self-interaction ($\sigma=-1$). Indeed, the NLS equation has
deserved special interest as it proves to yield, for focusing interaction, a
simple description of the instability of a regular wave train, a phenomenon
first observed by Talanov [17] in optics and by Benjamin-Feir [18] in a water
tank. For this particular equation, understanding modulational instability of
continuous wave solutions may be achieved via standard Fourier analysis.
However this analysis fails if the unperturbed solution is not just a plane
wave (see [19]), or if several nonlinear wave trains have a resonant
interaction. In general, when dealing with integrable systems, the stability
of interacting plane waves is better treated by means of integrability
techniques rather than by a Fourier approach. Indeed, the Lax pair plays a
key-role as it allows to extend the linear stability analysis to other wave
solutions, such as those constructed from plane waves by Darboux dressing
methods [20].

The integrability method has been used to unveil the stability properties of
plane wave solutions of two coupled NLS equations [16, 21]. For these model
equations, instabilities have been fully classified in terms of coupling
constants, amplitudes and wave numbers, including instability effects due to
defocusing self- and cross-interactions. As for the long wave-short wave
resonance interaction, in addition to orbital stability [22] and transverse
stability [23], also linear stability has been considered for solutions of
particular models [5].

On the wake of those results, we investigate here the linear stability of
continuous wave solutions of the YON equations (5). Although the presentation
is intended to be self-contained, we refer the reader to [16, 21] for specific
proofs and further details on the general method. The main target of our
approach is the computation of the _stability spectrum_ ($\mathbb{S}$),
associated to the given plane wave solution. For a multicomponent system such
as (5), this spectrum does not generically coincide with the ordinary Lax
spectrum. It turns out to be a piece-wise continuous curve in the complex
plane of the spectral variable $\lambda$. In each point of this curve, one or
more eigenmode solutions of the linearised equation of motion are well
defined, see Section 2. The relevant stability-instability properties of the
wave solution are readily conveyed by the geometrical properties of the
spectral curve $\mathbb{S}$ itself. Additional relevant targets are the
linearised equation eigenfunction frequencies and their associated _gain_
functions, which tell us whether the wave number instability bands are of
_passband_ or _baseband_ type [24].

This paper is organised as follows. In Section 2, we review our approach to
linear stability; in particular, we sketch the role that integrability plays
in investigating the linear stability of a given solution. In Section 3, the
integrability of the unified YON model is established by showing its
corresponding Lax pair. Moreover, we consider the plane wave solution of the
YON model equations, and study its linear stability by constructing and
classifying the corresponding stability spectra. We compute the gain function,
characterise the instabilities and show that the plane wave solution is
unstable for a generic choice of the physical parameters. We conclude with
some remarks and outlooks in Section 4.

##  2 Integrable equations and small variation of their solutions

Our approach to the linear stability of solutions of integrable partial
differential equations starts from the associated Lax pair, a characteristic
feature of integrable systems:

$\Psi_{x}=X\,\Psi\,,\quad\Psi_{t}=T\,\Psi\,.$ (9)

In the present context, the solution $\Psi(x,t,\lambda)$ and the coefficients
$X(x,t,\lambda)$, $T(x,t,\lambda)$ are $3\times 3$ matrices. The latter two
matrices are assumed to have the following polynomial dependence on the
complex _spectral variable_ $\lambda$:

$X(\lambda)=i\lambda\Sigma+Q\,,\quad T(\lambda)=(i\lambda)^{2}A+i\lambda
B+C\,,$ (10)

where $\Sigma$ is the constant, traceless, diagonal matrix

$\Sigma=\textrm{diag}\\{1\,,\,0\,,\,-1\\}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&-1\end{array}\right),$ (11)

and $Q(x,t)$ is $\lambda$-independent and off-diagonal, namely $Q_{jj}=0$,
while its off-diagonal entries $Q_{jm}(x,t)$, $j\neq m$, are generically six
complex-valued functions of $x$ and $t$; $A$ is a constant matrix, while $B$
and $C$ can be written in terms of commutators and anti-commutators of
$\Sigma$, $Q(x,t)$ and their powers. Then, the compatibility condition

$X_{t}-T_{x}+[X\,,\,T]=0$ (12)

leads to an evolution equation for the matrix $Q$ in the form of a system of
six coupled nonlinear wave equations for the entries of $Q$. Among other
reductions, this system can be reduced to just two equations describing the
resonant interaction of two waves of physical interest (see next section).

Before looking at any particular reduction, let us first turn our attention to
the way integrability plays an essential role in investigating the linear
stability of a given solution $Q(x,t)$. In order to make this paper self-
contained, we first briefly report the main ingredients and features of the
method (see [16] and references therein). The starting point is the following
theorem (see [16] for a proof), which provides the quite remarkable and well
known property of the matrix solution $\Psi(x,t,\lambda)$ of the two ordinary
differential equations (9):

###### Theorem 2.1

Let $Q(x,t)$ be a given solution of the compatibility equation (12) and let
$\Psi(x,t,\lambda)$ be a corresponding fundamental solution of (9), that is
$\det\Psi\neq 0$, then the matrix

$F(x,t,\lambda)=\left[\Sigma\,,\,\Psi(x,t,\lambda)\,M(\lambda)\,\Psi^{-1}(x,t,\lambda)\right]\,,$
(13)

where $M(\lambda)$ is an arbitrary $(x,\,t)$-independent matrix, is a solution
of the _linear_ equation obtained by linearising around $Q(x,t)$ the nonlinear
evolution equation resulting from (12).

In other words, suppose that $Q(x,t)$ is such a given solution, and that
$Q(x,t)+\delta Q(x,t)$ is a second solution of the same evolution equation.
Suppose moreover that the entries of $\delta Q(x,t)$ are sufficiently small
(this is certainly so at small initial times $t$, if the initial condition
$\delta Q(x,0)$ is small enough), then the linearised evolution equation for
$\delta Q(x,t)$ is obtained by neglecting all the nonlinear terms. This
evolution equation takes the form

$\delta Q_{t}(x,t)=\mathcal{L}(\delta Q(x,t))\,,$ (14)

where $\mathcal{L}(V)$ is a matrix-valued linear function (with
$(x,t)$-dependent coefficients) of $V$ and its $x$-derivatives. Then Theorem
2.1 states that the matrix $F(x,t,\lambda)$ (13) is a $\lambda$-dependent
solution of the same linear equation (14), namely

$F_{t}(x,t,\lambda)=\mathcal{L}(F(x,t,\lambda))\,,$ (15)

for any complex $\lambda$.

This is a system of six linear partial differential equations; its solutions
$F(x,t,\lambda)$, which depend on the spectral variable $\lambda$, play the
role of Fourier-like modes if the values of $\lambda$ lie in an appropriate
subset $\mathbb{S}$ of the complex plane, defined below as _stability
spectrum_ because of its key relevance to finding out whether a given solution
$Q(x,t)$ is linearly stable or unstable. This spectrum, which depends on the
particular solution $Q(x,t)$, is defined so as to guarantee the boundedness of
each solution $F(x,t,\lambda)$, at any fixed $t$, on the entire $x$-axis. By
keeping in mind this analogy with the Fourier analysis of solutions of linear
partial differential equations with constant coefficients, we assume that the
set of solutions $F(x,t,\lambda)$, see (13), of the equation (15) allows the
representation

$\delta Q(x,t)=\int_{\mathbb{S}}\mathrm{d}\lambda\,F(x,t,\lambda)\,,$ (16)

for an appropriate choice of $M(\lambda)$, and in an appropriate functional
space of initial data: for instance, we have in mind the physical assumption
that the entries of the initial condition $\delta Q(x,0)$ be localised wave
packets. Although the proof [16] of the theorem above requires only local
properties (differentiability up to sufficiently high order in $x$), the
integral representation (16) should be justified by spectral methods. To our
present purpose, that is assessing the linear stability of $Q(x,t)$, we do not
need to enter into this matter, while we devote our next section to detail the
properties of the stability spectrum $\mathbb{S}$, for a particularly simple
choice of $Q(x,t)$.

Before going into this investigation, we give an explicit and general
construction of the matrix eigenfunctions $F(x,t,\lambda)$ by taking advantage
of the properties of the wedge product of 3-dimensional vectors. Consider
first the matrix solution $\Psi$ of the Lax pair (9) and its three column
vectors $\psi^{(j)}$, $j=1,2,3$:

$\Psi=\left(\begin{array}[]{ccc}\psi^{(1)}&\psi^{(2)}&\psi^{(3)}\end{array}\right)\;\;.$
(17)

Without loss of generality, we assume that this fundamental solution has unit
determinant, say $\det\,\Psi=\psi^{(1)}\cdot\psi^{(2)}\wedge\psi^{(3)}=1$.
This implies that we can give the defining expression (13) of the matrix
eigenmode the alternative expression

$F(x,t,\lambda)=\left[\Sigma\,,\,\Psi(x,t,\lambda)\,M(\lambda)\,\Psi^{A}(x,t,\lambda)\right]\,,$
(18)

where $\Psi^{A}$ is the adjugate matrix (or classical adjoint) of $\Psi$. In
turn, the matrix of co-factors $\Psi^{A}$ can be expressed through its rows,
namely

$\Psi^{A}=\left(\begin{array}[]{ccc}{\psi^{A(1)}}^{T}\\\ {\psi^{A(2)}}^{T}\\\
{\psi^{A(3)}}^{T}\end{array}\right)\;\;,$ (19)

where the superscript $T$ stands for transposition that takes a column vector
$\psi^{A(j)}$ into a row vector, so that the vectors $\psi^{A(j)}$ are the
columns of ${\Psi^{A}}^{T}$. The computation of $\Psi^{A}$ is then provided by
the expression (19) with

$\psi^{A(j)}=\psi^{(m)}\wedge\psi^{(n)}\,,\quad\\{j,m,n\\}=\text{cyclic
permutations of}\;\\{1,2,3\\}\,.$ (20)

As we need six eigenfunctions for each value of $\lambda$ if $Q(x,t)$ is
generic (non-reduced), we choose to compute the eigenfunctions (18) by setting
$M$ equal to one of the six basis matrices $M^{(jm)}$,

$M^{(jm)}_{ab}=\delta_{ja}\delta_{mb}\,,\quad j,m=1,2,3\,,\quad j\neq m\,,$
(21)

of the space of the off-diagonal $3\times 3$ matrices (here $\delta_{jm}=1$
for $j=m$ and $\delta_{jm}=0$ for $j\neq m$), namely
$F^{(jm)}=\left[\Sigma\,,\,\Psi M^{(jm)}\Psi^{A}\right]$. The convenience of
this choice stems from the algebraic identity

$\Psi(x,t,\lambda)\,M^{(jm)}\,\Psi^{A}(x,t,\lambda)=\psi^{(j)}\left(\psi^{(j)}\wedge\psi^{(n)}\right)^{T}\;\epsilon_{jnm}\;\;,$
(22)

where $\epsilon_{jnm}$ is the parity of the permutation $\\{j,n,m\\}$ of
$\\{1,2,3\\}$, that is $\epsilon_{jnm}=1$ if $\\{j,n,m\\}$ is a cyclic
permutation of $\\{1,2,3\\}$, and $\epsilon_{jnm}=-1$ otherwise. Thus, the six
eigenfunctions finally take the expression

$F^{(jm)}(x,t,\lambda)=\left[\Sigma\,,\,\psi^{(j)}\left(\psi^{(j)}\wedge\psi^{(n)}\right)^{T}\right]\;\epsilon_{jnm}\;\;,\quad
j\neq m\,,$ (23)

with the side remark that this explicit expression of the eigenfunction
$F^{(jm)}$ cubically depends on just two vector solutions of the Lax pair.
Finally, the Fourier-like representation (16) can be more precisely written as

$\delta
Q(x,t)=\int_{\mathbb{S}}\mathrm{d}\lambda\sum_{j,m}\mu^{(jm)}(\lambda)F^{(jm)}(x,t,\lambda)\,,$
(24)

where the six functions $\mu^{(jm)}(\lambda)$ play the role of Fourier-like
transform of $\delta Q$.

##  3 Interaction of long and short continuous waves

In this section we consider reductions of the matrix $Q$ in (10) which lead,
via the compatibility condition (12), to a system of two evolution equations
able to model the long wave-short wave resonant interaction. To this purpose,
the matrix $Q$ should depend only on two wave fields: the amplitudes $L(x,t)$,
modulating a continuous wave with a (infinitely) long wave length, and
$S(x,t)$, which modulates the amplitude of a periodic short wave. As a
consequence, one should impose the condition that $L(x,t)$ be real and
dispersion-free, and that $S(x,t)$ be instead complex with a non-vanishing
dispersion coefficient (which, without any loss of generality, is rescaled to
unity). As we pointed out in Section 1, here we introduce the new integrable
model (5), which includes the YO and N systems (2) and, respectively (3), by
setting

$Q=\left(\begin{array}[]{ccc}0&S&iL\\\ \alpha S^{\ast}&0&S^{\ast}\\\
i\alpha^{2}L-i\beta&\alpha S&0\end{array}\right)\,.$ (25)

This system is integrable for _any_ real value of $\alpha$ and $\beta$; it
reduces to the YO equation (2) for $\alpha=0$, $\beta=1$ and to the N equation
(3) for $\alpha=\sigma$, $\beta=0$ and $L\mapsto\sigma L$ ($\sigma=\pm 1$).

From this choice of the matrix $Q$, and from the compatibility condition (12),
the coefficients $A$, $B$, $C$ of the matrix $T$ (see (10)) turn out to be:

$A=\frac{i}{3}\left(\begin{array}[]{ccc}1&0&0\\\ 0&-2&0\\\
0&0&1\end{array}\right)\,,\quad B=\left(\begin{array}[]{ccc}0&iS&0\\\ i\alpha
S^{\ast}&0&-iS^{\ast}\\\ 0&-i\alpha S&0\end{array}\right)\,,$ (26a)
$C=\left(\begin{array}[]{ccc}-i\alpha|S|^{2}&-\alpha LS+iS_{x}&i|S|^{2}\\\
-\alpha^{2}LS^{\ast}+\beta S^{\ast}-i\alpha
S^{\ast}_{x}&2i\alpha|S|^{2}&-\alpha LS^{\ast}-iS^{\ast}_{x}\\\
i\alpha^{2}|S|^{2}&-\alpha^{2}LS+\beta S+i\alpha
S_{x}&-i\alpha|S|^{2}\end{array}\right)\,.$ (26b)

In these formulae, the parameters $\alpha$, $\beta$ are arbitrary but
constrained to be real. These parameters may be considered as independent
constants which are responsible for the long-short wave cross-interaction.

Here we are concerned with the linear stability of a given solution $S(x,t)$,
$L(x,t)$ of the system (5), under the assumption that this solution is bounded
for all values of $x$. According to standard practice, we proceed by
investigating the time evolution of a small variation $\delta Q(x,t)$, namely
$\delta S(x,t)$, $\delta L(x,t)$, of this solution. At the early stage of the
evolution these variations are assumed to be sufficiently small so as to keep
only their linear contributions. Thus, they satisfy the linearised approximate
equation (14)

$\displaystyle\begin{array}[]{l}i\delta S_{t}+\\!\delta
S_{xx}+\\!\left(i\alpha L_{x}+\alpha^{2}L^{2}-\beta
L-4\alpha|S|^{2}\right)\\!\delta S+i\alpha S\delta
L_{x}+\\!\left(2\alpha^{2}L-\beta\right)\\!S\delta L-2\alpha S^{2}\delta
S^{\ast}\\!\\!=0\\\ \delta L_{t}=2\left(S\delta S^{\ast}+S^{\ast}\delta
S\right)_{x}\,.\end{array}$ (29)

Moreover, and for the sake of simplicity, we assume that the initial values
$\delta S(x,0)$ and $\delta L(x,0)$ are localised and bounded functions of
$x$. With these assumptions in mind, the solution $S(x,t)$, $L(x,t)$ is
linearly stable if $\delta S(x,t)$, $\delta L(x,t)$ remain small for all later
times $t>0$ or, equivalently because of the integral representation (24), if
the eigenfunctions $F^{(jm)}(x,t,\lambda)$, see (23), do not grow
exponentially in time.

As testing a recipe goes by tasting the cake, we apply our approach to the
simplest physically relevant solution of the long wave-short wave equation
(5), namely the continuous wave solution

$S(x,t)=ae^{i\theta},\quad L(x,t)=b,\quad\theta=qx-\nu
t,\quad\nu=q^{2}-\alpha^{2}b^{2}+\beta b+2\alpha a^{2}\,,$ (30)

which introduces three independent real parameters: the two amplitudes $a$ and
$b$, and the wave number $q$ of the short wave. The starting point is the
construction of a corresponding fundamental solution $\hat{\Psi}(x,t,\lambda)$
of the Lax equations (9). This is given by the expression

$\hat{\Psi}(x,t,\lambda)=e^{i\rho(\lambda)t}R(x,t)e^{i(xW(\lambda)-tW^{2}(\lambda))},\quad
R(x,t)=\text{diag}\\{1,e^{-i\theta},1\\}\,,$ (31)

where

$\rho(\lambda)=\frac{2}{3}\lambda^{2}+\alpha^{2}b^{2}-2\alpha a^{2}-\beta b,$
(32)

and where the $(x,t)$-independent matrix $W(\lambda)$ is

$W(\lambda)=\begin{pmatrix}\lambda&-ia&b\\\ -i\alpha a&q&-ia\\\
\alpha^{2}b-\beta&-i\alpha a&-\lambda\end{pmatrix}\,.$ (33)

Because of the trace expressions

$\text{tr}(W)=q\,,\quad\text{tr}(W^{2})=\nu+3\rho\,,$ (34)

the matrix solution (31) has unit determinant,
det$(\hat{\Psi}(x,t,\lambda))=1$. To our purpose, and according to the
formalism detailed in the previous section, it is however more convenient to
choose the alternative solution $\Psi(x,t,\lambda)$ whose column vectors, see
(17), are

$\psi^{(j)}=\hat{\Psi}(x,t,\lambda)f^{(j)}(\lambda)\,,\quad j=1,\,2,\,3\,.$
(35)

Here the three constant vectors $f^{(j)}(\lambda)$ are the eigenvectors of the
matrix $W(\lambda)$ (33),

$Wf^{(j)}=w_{j}f^{(j)}\,,\quad j=1,\,2,\,3\,,$ (36)

which are generically linearly independent. Moreover, we normalise them so
that

$f^{(1)}\cdot\left(f^{(2)}\wedge f^{(3)}\right)=1\,,$ (37)

with the implication that det$(\Psi(x,t,\lambda))=1$. This construction
finally yields the expression of the column vectors of $\Psi(x,t,\lambda)$,
see (35), namely

$\psi^{(j)}=e^{i(\eta_{j}+\rho t)}Rf^{(j)}\,,$ (38)

with

$\eta_{j}=w_{j}x-w_{j}^{2}t\,,\quad j=1\,,2\,,3\,.$ (39)

We are now in the position to obtain the expression of the eigenfunctions
$F^{(jm)}(x,t,\lambda)$, which correspond to the continuous wave solution
(30). By inserting in (23) the vectors (38), and by taking into account the
relation

$\eta_{1}+\eta_{2}+\eta_{3}=\theta-3\rho t\,,$ (40)

and the matrix identity (for two arbitrary vectors $u$, $v$)

$e^{i\theta}(Ru)(Ru\wedge Rv)^{T}=R\,\left[u(u\wedge
v)^{T}\right]\,R^{-1}\,,\quad
R(x,t)=\text{diag}\left\\{1,e^{-i\theta},1\right\\}\,,$ (41)

we end up with the expression

$F^{(jm)}(x,t,\lambda)=e^{i(\eta_{j}-\eta_{m})}\,R\,\left[\Sigma\,,\,f^{(j)}(f^{(j)}\wedge
f^{(n)})^{T}\right]\,R^{-1}\,\epsilon_{jnm}\,,\quad j\neq m\,,$ (42)

where $\epsilon_{jnm}$ is the parity of the permutation $\\{j,n,m\\}$ of
$\\{1,2,3\\}$. This formula explicitly shows that, apart from the
$\lambda$-independent phase $\theta$, the eigenfunctions
$F^{(jm)}(x,t,\lambda)$ depend on $x$ and $t$ only via the exponentials
$e^{i(\eta_{j}-\eta_{m})}$. By taking into account the expression of
$\eta_{j}$, see (39), these exponentials for $j\neq m$ take the familiar
expression

$e^{\pm i(k_{n}x-\omega_{n}t)}\,,\quad
k_{n}=w_{n+1}-w_{n+2}\,,\quad\omega_{n}=w^{2}_{n+1}-w^{2}_{n+2}\;,\quad
n=1,2,3\;\;\text{mod}\,3\;\;,$ (43)

where the wave numbers $k_{j}(\lambda)$ and their corresponding frequencies
$\omega_{j}(\lambda)$ are explicitly defined as

$\begin{array}[]{ccc}k_{1}=w_{2}-w_{3}\;,&k_{2}=w_{3}-w_{1}\;,&k_{3}=w_{1}-w_{2}\;,\\\
\\\
\omega_{1}=w^{2}_{2}-w^{2}_{3}\;,&\omega_{2}=w^{2}_{3}-w^{2}_{1}\;,&\omega_{3}=w^{2}_{1}-w^{2}_{2}\,,\end{array}$
(44)

in terms of the eigenvalues $w_{j}$ of the matrix $W$, see (36), or,
equivalently, in terms of the roots of the characteristic polynomial

$P(w,\lambda)=\det[w\,\mathbf{1}-W(\lambda)]=(w-w_{1})(w-w_{2})(w-w_{3})=(w-q)(w^{2}-\lambda^{2}+p)+r\;\;,$
(45)

where the two parameters $p$ and $r$ have the expression

$p=2\alpha a^{2}-\alpha^{2}b^{2}+\beta b=\nu-q^{2}\,,\qquad
r=a^{2}[2\alpha(q+\alpha b)-\beta]\,.$ (46)

Because of the requirement that the basic solutions (42) be bounded functions
of $x$, we conclude that the spectral representation of $\delta L(x,t)$ and of
$\delta S(x,t)$ requires integrating with respect to the complex spectral
variable $\lambda$, see (24), over the subset of the complex $\lambda$-plane
where at least one of the wave numbers $k_{j}(\lambda)$ is real. These
considerations then lead to the following

###### Definition 3.1

The stability spectrum $\mathbb{S}$ is defined as the set of the complex
values of $\lambda$ such that at least one of the three wave number functions
$k_{1}(\lambda)$, $k_{2}(\lambda)$, $k_{3}(\lambda)$ is real.

In this construction, we notice that the dispersion relation between the wave
number $k_{j}$ and its corresponding frequency $\omega_{j}$ is
_parametrically_ defined by the pair of functions $k_{j}(\lambda)$ and
$\omega_{j}(\lambda)$ by varying the parameter $\lambda$ only over the
stability spectrum $\mathbb{S}$. It follows that the continuous wave solution
(30) is linearly stable if $\omega_{j}(\lambda)$ is real for those values of
$j$ such that $k_{j}(\lambda)$ is real for any $\lambda$ over the entire
spectrum $\mathbb{S}$. On the contrary, if for some $j$ and
$\lambda\in\mathbb{S}$, $\omega_{j}(\lambda)$ is not real, then this solution
is linearly unstable. In this latter case, the relevant physical information
is provided by the computation of the gain function

$\Gamma_{j}(\lambda)=|\text{Im}(\omega_{j}(\lambda))|\;\;,\quad\lambda\in\mathbb{S}\;\;,\quad\text{Im}(k_{j}(\lambda))=0\,\quad.$
(47)

Before proceeding to classify the stability spectra $\mathbb{S}$ and gain
functions (47), it is worth drawing the reader’s attention to the following
fact. If the present approach to linear stability is applied to continuous
wave solutions of the NLS equation (1), then the stability spectrum introduced
above by the definition (3.1) coincides with the Lax spectrum (which follows
from the Lax differential operator in the space variable $x$ via standard
definition) [25, 19]. However, this coincidence of the two spectra is a
peculiar property of the $2\times 2$ matrix Lax equations. In the case of
$N\times N$ Lax equations with $N\geq 3$, the two spectra, i.e. the Lax
spectrum and the stability spectrum, as defined here by (3.1) for $N=3$, are
generically different from each other. For such larger matrices only the
spectrum $\mathbb{S}$ as defined above is relevant to stability.

We begin our analysis of the stability spectra $\mathbb{S}$ and of the
corresponding gains by observing that the expression of the coefficients of
the characteristic polynomial $P(w,\lambda)$ (45) depends on the wave number
$q$ of the short wave solution (30), on the parameter $r$ given in (46), and
on the parameter $p$ (see (46)) and the spectral variable $\lambda$ only via
the combination $\lambda^{2}-p$. This shows that it is sufficient to fix the
value of the parameters $q$ and $r$ in the polynomial (45) and to define the
stability spectrum as a curve in the complex plane of the variable
$\lambda^{2}-p$. This simplifies our task as the parameter space is reduced to
the $(q,r)$ plane. Thus, hereafter, we introduce the alternative, and more
convenient, complex variable $\Lambda$ defined as

$\Lambda=\lambda^{2}-p\,,$ (48)

with the implication that the parameter $p$ becomes irrelevant to our
characterisation and classification of spectra. In fact, by a minor abuse of
notation, we refer to the characteristic polynomial (45) as

$P(w,\Lambda)=\det[w\,\mathbf{1}-W]=(w-w_{1})(w-w_{2})(w-w_{3})=(w-q)(w^{2}-\Lambda)+r\,.$
(49)

In order to make this change of variable from $\lambda$ to $\Lambda$ explicit,
we denote as $\mathbb{S}^{\Lambda}$ the stability spectrum in the complex
$\Lambda$-plane by adopting the following definition

###### Definition 3.2

The stability spectrum $\mathbb{S}^{\Lambda}$ is defined as the set of all
complex values of $\Lambda$ such that at least one of the three wave number
functions $k_{1}(\Lambda)$, $k_{2}(\Lambda)$, $k_{3}(\Lambda)$, see (44), is
real.

It is obvious that this spectrum looks different from the spectrum
$\mathbb{S}$. Furthermore, we note that the parameter $q$, if non-zero, can be
rescaled to the value $q=1$ by rescaling $w$ by $q$, $\Lambda$ by $q^{2}$ and
$r$ by $q^{3}$, or, equivalently, by the change of variables

$w\rightarrow qw\,,\quad\Lambda\rightarrow q^{2}\Lambda\,,\quad r\rightarrow
q^{3}r\,.$ (50)

Nevertheless, we find it convenient to keep $q$ in our formulae, to
numerically set $q=1$ whenever $q\neq 0$, and to separately treat the case
$q=0$.

Let us consider now the part of the spectrum $\mathbb{S}^{\Lambda}$ which lies
on the _real_ axis Im$\,\Lambda=0$. In this case, all coefficients of the
characteristic polynomial $P(w,\Lambda)$ are real, and therefore either the
three zeros $w_{1}(\Lambda)$, $w_{2}(\Lambda)$, $w_{3}(\Lambda)$ are real, or
one is real and two are complex conjugate. In the first case the three wave
numbers $k_{j}$ are real, while in the second case none of them is real, which
leads to:

###### Proposition 3.1

If $\Lambda$ is real then it belongs to the spectrum $\mathbb{S}^{\Lambda}$ if
and only if the $w$-discriminant of the polynomial (49) is non-negative,
namely if $\Delta_{w}\,P(w,\Lambda)\geq 0$, where

$\Delta_{w}\,P(w,\Lambda)=k_{1}^{2}\,k_{2}^{2}\,k_{3}^{2}=4\Lambda^{3}-8q^{2}\Lambda^{2}+4q(q^{3}-9r)\Lambda-27r^{2}+4rq^{3}\;\;.$
(51)

Thus, as shown by this expression (51), the large and positive real values of
$\Lambda$ do belong to $\mathbb{S}^{\Lambda}$, while the large and negative
real values of $\Lambda$ do not. More explicitly, we approximately solve the
equation $P(w,\Lambda)=0$ around the point at infinity of the complex
$\Lambda$-plane and obtain the following asymptotic expressions of the three
roots $w_{j}$

$\begin{array}[]{l}w_{1}(\Lambda)=\sqrt{\Lambda}-\frac{r}{2\Lambda}+O(1/\Lambda^{3/2})\\\
w_{2}(\Lambda)=-\sqrt{\Lambda}-\frac{r}{2\Lambda}+O(1/\Lambda^{3/2})\\\
w_{3}(\Lambda)=q+\frac{r}{\Lambda}+O(1/\Lambda^{2})\,,\end{array}$ (52)

where the labelling index $j$ is arbitrary. Consequently, if $\Lambda$ is
real, large and positive then also the three wave numbers $k_{j}(\Lambda)$,
see (44), are real and large. If instead $\Lambda$ is real and negative, and
its modulus is large enough, no real wave number $k_{j}$ exists and $\Lambda$
does not belong to the spectrum. If $\Lambda$ is real, large and positive, the
$w$-discriminant (51) is positive, and it remains so while moving $\Lambda$
along the real axis towards the origin, until it reaches its first zero, say
$\Lambda_{+}$, of the discriminant $\Delta_{w}P(w,\Lambda)$, see (51). At this
zero, this discriminant generically changes its sign and one of the three wave
numbers ${k_{j}(\Lambda)}$ vanishes. In turn, the $w$-discriminant (51) is a
cubic polynomial of the variable $\Lambda$ with real coefficients. Therefore,
it has either one or three real $\Lambda$-zeros. In the former case, the
discriminant $\Delta_{w}P(w,\Lambda)$ is negative for all values
$\Lambda<\Lambda_{+}$ and hence, according to our Proposition 3.1, the real
part of the spectrum $\mathbb{S}^{\Lambda}$ is the semi-axis
$\Lambda_{+}\leq\Lambda<+\infty$. If instead the discriminant
$\Delta_{w}P(w,\Lambda)$ has three real zeros, say
$\Lambda_{0}<\Lambda_{-}<\Lambda_{+}$, then the interval
$\Lambda_{-}<\Lambda<\Lambda_{+}$ cannot belong to the spectrum since in this
interval the discriminant $\Delta_{w}P(w,\Lambda)$ is negative (see
Proposition 3.1). Thus, this interval is a finite gap of the spectrum. Indeed,
in this case the real part of the spectrum consists on the finite interval
$\Lambda_{0}<\Lambda<\Lambda_{-}$ where the discriminant
$\Delta_{w}P(w,\Lambda)$ is positive, and the semi-axis
$\Lambda_{+}\leq\Lambda<+\infty$. This finite gap should be considered as a
distinctive feature of the spectra in our classification. Indeed, the
$\Lambda$-discriminant of the discriminant $\Delta_{w}P(w,\Lambda)$, that is
$\Delta_{\Lambda}\Delta_{w}P(w,\Lambda)$, depends only on the two parameters
$q$, $r$ and serves our classification purpose, as summarised by the following

###### Proposition 3.2

Let $\Delta_{\Lambda}\Delta_{w}P(w,\Lambda)$ be the $\Lambda$-discriminant of
the discriminant (51), that is

$\Delta_{\Lambda}\Delta_{w}P(w,\Lambda)=16r(8q^{3}-27r)^{3}\,.$ (53)

The spectrum $\mathbb{S}^{\Lambda}$ has one, and only one, finite gap (G) on
the real axis if and only if $\Delta_{\Lambda}\Delta_{w}P(w,\Lambda)>0$,
namely, if and only if $r(8q^{3}-27r)>0$, and it has no gap if
$r(8q^{3}-27r)<0$. The gap opening and closing threshold values of the
parameters are $r=0$ and $r=(8/27)\,q^{3}$.

Let us now turn our attention to the complex values of $\Lambda$ that are not
on the real axis and yet belong to the spectrum,
$\Lambda\in\mathbb{S}^{\Lambda}$, Im$(\Lambda)\neq 0$. To this purpose, it is
far more convenient to introduce the new polynomial

$\mathcal{P}(\zeta,\Lambda)=(\zeta-k^{2}_{1})(\zeta-k^{2}_{2})(\zeta-k^{2}_{3})=\zeta^{3}+\gamma_{2}\,\zeta^{2}+\gamma_{1}\,\zeta+\gamma_{0}\,,$
(54a) whose roots $\zeta_{j}(\Lambda)$, $\,j=1,2,3$, are the squares of the
differences of the $\Lambda$-dependent roots $w_{j}(\Lambda)$ of the
polynomial $P(w,\Lambda)$, see (44),
$\zeta_{j}(\Lambda)=k^{2}_{j}(\Lambda)=(w_{j+1}-w_{j+2})^{2}\,,\quad
j=1,\,2,\,3\,\;\textrm{mod}\,3\,.$ (54b)

The coefficients of the polynomial $\mathcal{P}(\zeta,\Lambda)$ (54a) can be
computed explicitly in terms of the coefficients of the polynomial
$P(w,\Lambda)$ (49) (see [21]), and read

$\begin{array}[]{l}\gamma_{2}=-2(3\Lambda+q^{2})\,,\\\
\gamma_{1}={(3\Lambda+q^{2})}^{2}\,,\\\
\gamma_{0}=-4\Lambda^{3}+8q^{2}\Lambda^{2}-4q(q^{3}-9r)\Lambda+27r^{2}-4rq^{3}\,.\end{array}$
(54c)

The definition (54a) shows that the $w$-discriminant (51) of the polynomial
$P(w,\Lambda)$ is simply related to this new polynomial (54a) as it reads

$\Delta_{w}P(w,\Lambda)=-\mathcal{P}(0,\Lambda)\,.$ (55)

This property allows one to rewrite our Proposition (3.2) on gap
characterisation in terms of $\mathcal{P}(0,\Lambda)$. The spectrum
$\mathbb{S}^{\Lambda}$ itself can be redefined as the locus of the
$\Lambda$-zeros of $\mathcal{P}(\zeta,\Lambda)$ which correspond to non-
negative values of the variable $\zeta$, say $\zeta\geq 0$. Thus, to each
$\zeta\geq 0$ there correspond three values of $\Lambda$ which belong to
$\mathbb{S}^{\Lambda}$. In particular, since this polynomial
$\mathcal{P}(\zeta,\Lambda)$ also takes the expression

$\begin{array}[]{l}\mathcal{P}(\zeta,\Lambda)=-4\left[\Lambda-\Lambda_{1}(\zeta)\right]\left[\Lambda-\Lambda_{2}(\zeta)\right]\left[\Lambda-\Lambda_{3}(\zeta)\right]=\\\
\\\
=-4\Lambda^{3}+\Lambda^{2}(9\zeta+8q^{2})-2\Lambda(3\zeta^{2}-3q^{2}\zeta+2q^{4}-18qr)+\zeta^{3}-2q^{2}\zeta^{2}+q^{4}\zeta+27r^{2}-4q^{3}r,\end{array}$
(56)

with real coefficients for real $\zeta$, we conclude that

###### Proposition 3.3

The spectrum $\mathbb{S}^{\Lambda}$ is symmetric with respect to the the real
axis.

Indeed, if $\Lambda$ and $\Lambda^{\ast}$ are roots of
$\mathcal{P}(\zeta,\Lambda)$ for a given non-negative $\zeta$, then both
$\Lambda$ and $\Lambda^{\ast}$ belong to the spectrum. Thus the spectrum
$\mathbb{S}^{\Lambda}$ is a symmetric piecewise smooth curve in the complex
$\Lambda$-plane. The transition from a triplet of real $\Lambda$-roots of
$\mathcal{P}(\zeta,\Lambda)$ to a pair of complex conjugate $\Lambda$-roots
and one real $\Lambda$-root (or viceversa) originates from a collision of two
real (or two complex conjugate) $\Lambda$-roots, while changing the value of
the variable $\zeta$. This happens at a zero $\zeta_{j}$ of the discriminant

$\ Q(\zeta)=\Delta_{\Lambda}\mathcal{P}(\zeta,\Lambda)\,,$ (57a) namely, if
$Q(\zeta_{j})=0$, provided this discriminant changes its sign. This
discriminant $Q(\zeta)$ turns out to be of fifth degree. However, it
factorises as $\begin{array}[]{c}Q(\zeta)=4\,Q_{1}^{2}(\zeta)\,Q_{2}(\zeta)\\\
\\\ Q_{1}(\zeta)=18q\zeta+27r-8q^{3},\quad
Q_{2}(\zeta)=\zeta^{3}-8q^{2}\zeta^{2}+8q(2q^{3}-9r)\zeta+4r(8q^{3}-27r)\,.\end{array}$
(57b)

After the collision where the discriminant $Q(\zeta)$ changes its sign, the
two $\Lambda$-roots scatter off the real axis. The change of sign makes the
factor $Q_{1}^{2}(\zeta)$, see (57b), irrelevant to this analysis, so that
hereafter we focus our attention on the factor $Q_{2}(\zeta)$ only. This
polynomial has generically three simple roots,

$Q_{2}(\zeta)=(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})\,,$ (58)

which depend only on the parameters $q$, $r$, and may be either all three
real, if the discriminant of $Q_{2}(\zeta)$ is positive,

$\Delta_{\zeta}Q_{2}(\zeta)=16r{\left(16q^{3}-27r\right)}^{3}>0\,,\quad\zeta_{1}<\zeta_{2}<\zeta_{3}\,,$
(59a) or two of them are complex conjugate and one is real, if this
discriminant is instead negative,
$\Delta_{\zeta}Q_{2}(\zeta)=16r{\left(16q^{3}-27r\right)}^{3}<0\,,\quad\zeta_{1}=\mu+i\rho\,,\quad\zeta_{2}=\mu-i\rho\,,\quad\text{Im}(\zeta_{3})=0\,,$
(59b)

where $\mu$, $\rho$ and $\zeta_{3}$ are real. Thus, our classification is
mainly based on the function $Q_{2}(\zeta)$ and on its discriminant
$\Delta_{\zeta}Q_{2}(\zeta)$. In this context, the necessary existence
condition of a gap, which follows from our Proposition (3.2), can also be
established as $Q_{2}(0)=-\zeta_{1}\zeta_{2}\zeta_{3}=4r(8q^{3}-27r)>0$. In
the following classification, we find it more convenient to play with the sign
of $qr$ rather than with the sign of $q$ and of $r$ separately. Thus, for
instance, the existence condition of a gap of the spectrum reads
$(qr)\left[8q^{4}-27(qr)\right]>0$ which is never satisfied if $qr<0$ and it
is satisfied only for $0<qr<\left(8/27\right)q^{4}$.

Consider first a spectrum $\mathbb{S}^{\Lambda}$ whose parameters $q$, $r$
satisfy the positive discriminant inequality (59a), namely
$(qr)\left[16q^{4}-27(qr)\right]>0$. In this case, for large enough values of
positive $\zeta$, namely for $\zeta_{3}<\zeta<+\infty$, as well as for
$\zeta_{1}<\zeta<\zeta_{2}$, the function $Q_{2}(\zeta)$ is positive,
$Q_{2}(\zeta)>0$, while $Q_{2}(\zeta)<0$ for $\zeta_{2}<\zeta<\zeta_{3}$ and
for $-\infty<\zeta<\zeta_{1}$. In order to look for distinctive features of
the spectrum, we have to analyse the following four different cases separately
according to the positiveness of the three $\zeta$-roots of $Q_{2}(\zeta)$.

  1. 1.

$\zeta_{1}<\zeta_{2}<\zeta_{3}<0$: this case (all roots are negative) is
excluded by the Vieta relation $\zeta_{1}+\zeta_{2}+\zeta_{3}=8q^{2}$.

  2. 2.

$\zeta_{1}<\zeta_{2}<0<\zeta_{3}$: the Vieta relation
$\zeta_{1}\zeta_{2}\zeta_{3}=4r(27r-8q^{3})$, implies
$(qr)\left[27(qr)-8q^{4}\right]>0$, while the positive discriminant condition
(59a), $(qr)\left[16q^{4}-27(qr)\right]>0$, does not allow $qr$ to be
negative. We conclude that only the interval $(8/27)q^{4}<qr<(16/27)q^{4}$ is
allowed. In this case, two $\Lambda$-roots collide for $\zeta=\zeta_{3}$ and,
for $0\leq\zeta<\zeta_{3}$, the spectrum acquires two complex conjugate curves
in the $\Lambda$-plane. We refer to this complex part of the spectrum as
branch (B), since the two end-points at $\zeta=0$ of the two $\Lambda$-roots
trajectories do not generically coincide with each other (see Figure 1(a)
below). No gap is possible and we term this spectrum type 0G 1B 0L, or of
B-type.

  3. 3.

$\zeta_{1}<0<\zeta_{2}<\zeta_{3}$: the same Vieta relation used above
necessarily requires $0<qr<(8/27)q^{4}$, which is also compatible with the
positive discriminant condition. Two $\Lambda$-roots collide for
$\zeta=\zeta_{3}$, get off the real axis and collide again for
$\zeta=\zeta_{2}$ on the real $\Lambda$-axis thereby forming one complex
closed curve, which we term loop (L) (see Figure 2(a) below). In this case
there is no branch. However, a gap exists because its existence condition
$(qr)\left[8q^{4}-27(qr)\right]>0$, see our Proposition (3.2), is satisfied.
Thus, we call this spectrum of type 1G 0B 1L, or of LG-type.

  4. 4.

$0<\zeta_{1}<\zeta_{2}<\zeta_{3}$: this case is not allowed by two Vieta
relations, which lead to the inequalities $6q^{4}-27(qr)>0$ and
$(qr)\left[27(qr)-8q^{4}\right]>0$, and by the positive discriminant condition
(59a).

Let us consider now those spectra $\mathbb{S}^{\Lambda}$ whose parameters $q$,
$r$ satisfy the negative discriminant inequality (59b) instead, namely
$(qr)\left[16q^{4}-27(qr)\right]<0$. Again we distinguish all possible cases
according to the $\zeta$-roots (59b).

  1. 1.

$\zeta_{3}<0$: no spectrum exists because the Vieta relations for the
polynomial $Q_{2}(\zeta)$, see (57b), and the sign of the discriminant (59b)
are never compatible with each other.

  2. 2.

$\zeta_{3}>0$, $\mu<0$: the Vieta relations and the discriminant sign imply
that the spectrum exists if either $qr<0$ or $(16/27)q^{4}<qr$. This spectrum
is of type 0G 1B 0L.

  3. 3.

$\zeta_{3}>0$, $\mu>0$: again, combining Vieta relations and the discriminant
negative sign shows that the spectrum exists only if $qr<0$, and it is of type
0G 1B 0L.

These observations above conclude our characterisation and classification of
spectra, which are summarised in the following Table 1.

Table 1: Stability spectra $qr<0\qquad$ | $0<qr<\frac{8}{27}q^{4}\qquad$ | $\frac{8}{27}q^{4}<qr$
---|---|---
0G 1B 0L | 1G 0B 1L | 0G 1B 0L

As stated after our Definition 3.2 of the spectrum $\mathbb{S}^{\Lambda}$, see
also (50), the classification as in Table 1 has been derived with the
assumption that $q\neq 0$. For the sake of completeness, we now remove this
limitation by computing the spectrum for $q=0$ with $r\neq 0$ (see below for
$r=0$). We first observe that in this special case the polynomial
$\mathcal{P}(\zeta,\Lambda)$ reads

$\mathcal{P}(\zeta,\Lambda)=-\left(4\Lambda-\zeta\right)\left(\Lambda-\zeta\right)^{2}+27r^{2}\,,$
(60)

and therefore $\mathcal{P}(0,\Lambda)=-4\Lambda^{3}+27r^{2}$. According to our
Proposition (3.1) and to the relation (55), the real part of the spectrum is
the semi-axis $\Lambda_{0}\leq\Lambda<+\infty$ with
$\Lambda_{0}=\left[(27/4)r^{2}\right]^{1/3}$. As for the complex part of the
spectrum, we note that $Q_{2}(\zeta)=\zeta^{3}-108r^{2}$, see (57b), and
therefore the only real zero of $Q_{2}(\zeta)$ is the positive number
$\zeta_{3}=(108r^{2})^{1/3}$. For this value of $\zeta$, the polynomial
$\mathcal{P}(\zeta,\Lambda)$ must have a double real $\Lambda$-root
$\Lambda_{B}$, which is found to be
$\Lambda_{B}=(1/2)\zeta_{3}=(1/2)\left(108r^{2}\right)^{1/3}=2^{1/3}\Lambda_{0}$.
As $\zeta$ is moved from $\infty$ towards the origin, two real $\Lambda$-roots
collide with each other at $\Lambda_{B}$ for $\zeta=\zeta_{3}$ to give rise to
a branch, whose off real parts terminate for $\zeta=0$ at the complex
conjugated points $\Lambda_{0}\exp(2i\pi/3)$, $\Lambda_{0}\exp(-2i\pi/3)$.
Thus, for $q=0$, $r\neq 0$ the spectrum is of type 0G 1B 0L.

A similar analysis can be carried out at any zero of discriminants of cubic
polynomials, in particular to discuss threshold phenomena. Indeed, the
inequalities shown in Table 1 divide the plane $(q,r)$ into regions with
different types of spectra. At the boundaries of these regions, the
inequalities in Table 1 turn into equalities which characterise thresholds.
One simple and interesting instance of such thresholds occurs at the special
value $r=0$. In this special case, the spectrum $\mathbb{S}^{\Lambda}$ is
readily found because the roots of the characteristic polynomial (49)
$P(w,\Lambda)=(w-q)(w^{2}-\Lambda)$ are explicit. The corresponding spectrum
is entirely real and we will call it of type 0G 0B 0L. Indeed, the wave number
functions $k_{j}(\Lambda)$ have the simple expression

$k_{1}(\Lambda)=-q-\sqrt{\Lambda}\,,\quad
k_{2}(\Lambda)=q-\sqrt{\Lambda}\,,\quad k_{3}(\Lambda)=2\sqrt{\Lambda}\,,\quad
0\leq\Lambda<+\infty\,.$ (61)

A second threshold occurs for $27r-8q^{3}=0$, where a gap disappears. At these
threshold points of the $(q,r)$-plane the discriminant $Q(0)$, see (57b),
vanishes with the implication that the polynomial
$\mathcal{P}(0,\Lambda)=-4\left[\Lambda+(1/3)q^{2}\right]^{2}\left[\Lambda-(8/3)q^{2}\right]$
has a double $\Lambda$-root at $\Lambda=-(1/3)q^{2}$, where a branch closes up
and becomes a loop. The corresponding spectrum can be classified as 1G 0B 1L
where however the gap is reduced to just one point.

Let us now turn our attention to the time variable $t$ and to the stability of
the plane wave solution (30). This point requires investigating whether the
frequencies $\omega_{1}(\Lambda)$, $\omega_{2}(\Lambda)$,
$\omega_{3}(\Lambda)$, see (44), are real numbers for
$\Lambda\in\mathbb{S}^{\Lambda}$. Our starting point is their relation to the
wave numbers $k_{j}(\Lambda)$

$\omega_{j}=\frac{1}{3}k_{j}\left(2q+k_{j}+2k_{j+1}\right),\quad
j=1,2,3\;\text{mod}\;3\,,$ (62)

which can be derived from their definition (44) by first inverting the map
$\\{w_{j}\\}\rightarrow\\{k_{j}\\}$, that is (see (44))

$w_{j}=\frac{1}{3}\left(q+k_{j}+2k_{j+2}\right),\quad
j=1,2,3\;\text{mod}\;3\,.$ (63)

This relation (62) is clearly not a dispersion relation since each frequency
$\omega_{j}$ is written in terms of two wave numbers, $k_{j}$ and $k_{j+1}$.
In order to compute the dispersion relation, we need to eliminate the variable
$\Lambda$ among the algebraic relations $P(w,\Lambda)=0$,
$\mathcal{P}(\zeta,\Lambda)=0$ and $\omega_{j}=k_{j}(q-w_{j})$. This requires
a tedious but straightforward computation aimed to factorise high degree
polynomials. This is done with the help of the additional algebraic relation
$\mathcal{R}(\xi,\Lambda)=0$ with the following specifications

$\mathcal{R}(\xi,\Lambda)=(\xi-\omega^{2}_{1})(\xi-\omega^{2}_{2})(\xi-\omega^{2}_{3})=\xi^{3}+\delta_{2}\xi^{2}+\delta_{1}\xi+\delta_{0}\;\,,$
(64a) where the coefficients $\delta_{n}(\Lambda)$ can be explicitly computed
as in [16], and have the following expression
$\begin{array}[]{l}\delta_{2}=-2\left(q^{4}-6qr-2q^{2}\Lambda+\Lambda^{2}\right),\\\
\delta_{1}=\left(q^{4}-6qr-2q^{2}\Lambda+\Lambda^{2}\right)^{2},\\\
\delta_{0}=r^{2}\left(-4q^{3}r+27r^{2}-4q^{4}\Lambda+36qr\Lambda+8q^{2}\Lambda^{2}-4\Lambda^{3}\right)\,.\end{array}$
(64b)

This additional polynomial $\mathcal{R}(\xi,\Lambda)$, as shown in (64a), is
defined in the same way as $\mathcal{P}(\zeta,\Lambda)$ by substituting the
wave numbers $k_{j}$ with the frequencies $\omega_{j}$, and by replacing the
polynomial $P(w,\Lambda)$ with the characteristic polynomial of the matrix
$W^{2}$, that is

$\det\left[w^{2}\,\mathbf{1}-W^{2}\right]=-P(w,\Lambda)P(-w,\Lambda)=\left(w^{2}-q^{2}\right)\left(w^{2}-\Lambda\right)^{2}+2qr\left(w^{2}-\Lambda\right)-r^{2}\,.$
(65)

The upshot of this calculation is the three-branch dispersion relation $H(\pm
k_{j},\pm\omega_{j})=0$, where $H(k,\omega)$ is the polynomial

$H(k,\omega)=\omega^{3}-4qk\omega^{2}+k^{2}\left(4q^{2}-k^{2}\right)\omega-4rk^{3}\,.$
(66)

It is worth noticing that this dispersion relation $H(k,\omega)=0$ can be
obtained by means of a standard Fourier approach to the linearised equations.
In other words, the continuous wave solution (30) is so special that its
stability properties can be investigated by both Fourier expansion and our Lax
pair method. However, since the Fourier expansion approach works only for this
particular solution, we have turned our focus on the integrability methods in
order to explicitly apply the mathematical formalism (see Section 2) in a
simple context. The next, and harder, task will be to extend the present
analysis to other known solutions, such as solitary waves, say solitons,
periodic and rogue waves.

Let us now consider the linear stability of the plane wave solution (30) for
_generic_ values of $q$ and $r$. Our analysis has shown that the
$\Lambda$-spectra have a real component and a complex, off real one. The
former lies on the real axis, and at each point all three wave numbers
$k_{1}(\Lambda)$, $k_{2}(\Lambda)$, $k_{3}(\Lambda)$ are real. Since the
corresponding frequencies $\omega_{j}(\Lambda)$, according to their expression
(62), are also real, the eigenfunctions (42) on this component of the spectrum
are bounded in $t$, and do not cause instabilities. On the contrary, if
$\Lambda$ is off the real axis, namely on a branch or on a loop, which are the
complex components of $\mathbb{S}^{\Lambda}$, only one of the three wave
numbers, for instance $k_{3}(\Lambda)$, is real, while the other two
$k_{1}(\Lambda)$, $k_{2}(\Lambda)$ are non-real. On branches and loops, while
$k_{3}(\Lambda)$ is real, the corresponding frequency $\omega_{3}(\Lambda)$ is
necessarily complex with a non-vanishing imaginary part. This follows from the
expression (62). Precisely, since
$\omega_{3}=\frac{1}{3}k_{3}\,\left(2q+k_{3}+2k_{1}\right)$, the real
frequency is

$\text{Re}(\omega_{3})=\frac{1}{3}k_{3}\left(2q+k_{3}\right)+\frac{2}{3}k_{3}\,\text{Re}(k_{1})\,.$
(67)

The corresponding instability is characterised by the gain function (47)

$\Gamma_{3}(\lambda)=\left|\text{Im}(\omega_{3})\right|=\frac{2}{3}\,\left|k_{3}\,\text{Im}(k_{1})\right|\,,$
(68)

which is non-vanishing as far as $\Lambda$ remains on the non-real component
of the spectrum. We conclude this analysis with the following

###### Proposition 3.4

All stability spectra $\mathbb{S}^{\Lambda}$ are classified with respect to
the parameters $q$ and $r\neq 0$ in two types: the B-type (having a real part
and one branch), and the LG-type (having a real part with one finite gap, and
one loop). Only for $r=0$ the spectrum is totally real with no complex part.
The continuous wave solution is linearly stable if and only if $r=0$, with
$\omega_{1}=k_{1}^{2}+2qk_{1}$, $\omega_{2}=-k_{2}^{2}+2qk_{2}$,
$\omega_{3}=0$, and is unstable for all values of $q$ and all non-vanishing
values of $r$.

Examples of these two types of spectra have been numerically computed, by
evaluating the $\Lambda$-roots of $\mathcal{P}(\zeta,\Lambda)$ in (60), as
$\zeta$ varies from 0 to the largest, positive, real $\zeta$-root of
$Q_{2}(\zeta)$ in (57b), as detailed in Appendix C of [16]. In Figures 1, we
show the B-type spectrum (Fig. 1(a)); the corresponding gain function $\Gamma$
on the branch (Fig. 1(b)), which proves that this instability is of baseband
type, namely waves are unstable $|k|=0$; the functions $k_{j}(\Lambda)$, for
$j=1,2,3$, if $\Lambda$ is real, together with the function
$k_{3}(\text{Re}(\Lambda))$ on the branch (Fig. 1(c)); the real frequency on
the branch as function of $k=k_{3}$ (Fig. 1(d)). The same functions are
plotted in Figures. 2(b), 2(c) and 2(d), for an LG-type spectrum, which is
shown in Figure 2(a). In particular, Figure 2(b) shows that in this case the
instability is of passband type, namely, waves are stable for sufficiently
small values of $|k|$.

(a) Spectrum $\mathbb{S}^{\Lambda}$.

(b) Gain function $\Gamma$ versus $k_{3}$.

(c) Wave numbers versus real (blue) and non-real (red) values of $\Lambda$.

(d) Real frequency versus $k_{3}$ (see (67)).

Figure 1: B-type spectrum, $q=1$, $r=-4$.

(a) Spectrum $\mathbb{S}^{\Lambda}$.

(b) Gain function $\Gamma$ versus $k_{3}$.

(c) Wavenumbers versus real (blue) and non-real (red) values of $\Lambda$.

(d) Real frequency versus $k_{3}$ (see (67)).

Figure 2: LG-type spectrum, $q=1$, $r=0.1$.

##  4 Summary and conclusions

Research on instabilities of nonlinear waves has witnessed a renewed interest
in recent years, following the study of the modulational instability of
continuous wave solutions of the focusing NLS equation. We exploited
integrability to construct a method to predict whether a nonlinear wave,
described by an integrable, nonlinear system, is linearly stable against small
perturbations. Here, we displayed how this approach works in the resonant
coupling of long and short waves. As a side result, we first introduce a new
model equation which combines two very well known integrable models, namely
the Yajima-Oikawa equation (2) and the Newell equation (3), into the more
general YON equation (5), which features two arbitrary, real parameters and
thus, it is likely to fit a broader range of physical contexts. This outcome
is similar to the one that proves that the Korteweg-de Vries (KdV) and the
modified-KdV equations are just two particular cases of the Gardner equation
[26, 27].

A common feature of all these long wave-short wave equations is that a long
wave always arises as generated by short waves. This process, which is guessed
by observing the evolution equations (5) themselves, is revealed by the first
two conservation laws $\rho_{nt}=g_{nx}$, $n=1,2$, where $\rho_{n}$ is the
density and $g_{n}$ is the current. The first conservation law is the long
wave equation itself, with $\rho_{1}=L$ and $g_{1}=2|S|^{2}$. The second
conservation law turns out to have the conserved density
$\rho_{2}=|S|^{2}-(\alpha/2)L^{2}$ and the current
$g_{2}=2\,\text{Im}(SS^{\ast}_{x})-2\alpha L|S|^{2}$.

In this work, we give the $3\times 3$ Lax pair associated to the new YON model
and display how this Lax pair allows us to construct the basic solutions, or
eigenfunctions, of the linearised model equations. These computations are
specialised to the continuous wave (or plane-wave) solutions (30) of the YON
equations. In this case, the explicit construction of the eigenfunctions of
the linearised wave equations allows us to define, and compute, their
associated wave numbers $k$ and frequencies $\omega$. Not only are these
physical quantities related to each other by an explicit or implicit
dispersion relation (66), as in standard Fourier analysis, but they are also
parametrically related as $k=k(\Lambda)$, $\omega=\omega(\Lambda)$, where the
variable $\Lambda$ lives in the complex plane. In turn, the complex variable
$\Lambda$ is simply connected to the spectral variable $\lambda$, which
appears in the Lax pair. The key point is the definition of the _stability
spectrum_ , an algebraic curve whose geometrical properties convey relevant
physical information, and which generically differs from the Lax spectrum.
Eigenfunctions, wave numbers and frequencies are defined on the stability
spectrum. In particular, on the real $\Lambda$ part of the spectrum the
eigenfunctions are bounded functions of time (stability), while on the non-
real part of the spectrum the eigenfunctions exponentially grow with time
(instability). This strictly complex part of the spectrum brings its own
information: it displays one open branch, if the instability is of baseband
type, or a closed loop, if the instability is instead of passband type.

All the spectra are classified by their dependence on essentially one real
parameter $r$, see (46). This is a simple function of the amplitudes $a$ and
$b$ of the short and, respectively, long waves, of the wave number $q$ of the
short wave, and of the self- and cross-coupling constants $\alpha$ and $\beta$
of the model. For a generic choice of all these parameters, only two types of
spectra, and therefore of perturbed wave behaviours, emerge from our analysis.
Plots illustrate these two typical stability spectra. Their properties show
that all values of the parameters lead to instabilities, with the only
exception of the case $r=0$, which instead guarantees stability since the
associated spectrum has neither a branch nor a loop. This special case
corresponds to more than one choice of the physical parameters, according to
the expression (46) $r=a^{2}[2\alpha(q+\alpha b)-\beta]=0$. Interesting cases
depend on the particular physical setting and context. For instance, if the
short wave amplitude $a$, as well as the coupling constants $\alpha$ and
$\beta$ are not vanishing, the plane wave solution is stable if the wave
number $q$ is so chosen to take the value $q=(\beta/2\alpha)-\alpha b$. It is
plain that other choices are possible to satisfy the stability condition
$r=0$.

The algebraic construction of the eigenfunctions of the linearised long wave-
short wave equation (29), according to the method that we have displayed in
this work, requires not only the computation of a solution of the YON system
(5), but also the computation of the corresponding solution
$\Psi(x,t,\lambda)$ of the Lax pair associated to that solution. Generically
this task may turn out to be quite difficult; however, it may be feasible for
particular solitary wave solutions, either periodic (breathers) or localised
solitons.

Few integrable models have been systematically investigated so far by the
present method [16, 21, 28, 29]. Subsequent research should be devoted to
investigate linear stability of solutions of other integrable wave equations
of applicative relevance. In particular, this linear stability approach needs
to be formulated so as to deal with solutions which are different from just
continuous waves, whereas results obtained in the scalar cases should be
extended to multicomponent systems. Our present results are likely to be
relevant to the investigation of solutions which are homoclinic to continuous
waves, such as solitons propagating on a background, and rogue waves.

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