# Nonlinear spatial evolution of degenerate quartets of water waves

Conor Heffernan  School of Engineering, Computing & Mathematics,University of
Plymouth, Plymouth, PL4 8AA, United Kingdom  Amin Chabchoub  Hakubi Center for
Advanced Research, Kyoto University, Kyoto, 606-8501, Japan  Disaster
Prevention Research Insitute, Kyoto University, Uji, 611-0011, Japan  Raphael
Stuhlmeier  School of Engineering, Computing & Mathematics,University of
Plymouth, Plymouth, PL4 8AA, United Kingdom

###### Abstract

In this manuscript we investigate the Benjamin-Feir (or modulation)
instability for the spatial evolution of water waves from the perspective of
the discrete, spatial Zakharov equation, which captures cubically nonlinear
and resonant wave interactions in deep water without restrictions on spectral
bandwidth. Spatial evolution, with measurements at discrete locations, is
pertinent for laboratory hydrodynamic experiments, such as in wave flumes,
which rely on time-series measurements at a series of fixed gauges installed
along the facility. This setting is likewise appropriate for experiments in
electromagnetic and plasma waves. Through a reformulation of the problem for a
degenerate quartet, we bring to bear techniques of phase-plane analysis which
elucidate the full dynamics without recourse to linear stability analysis. In
particular we find hitherto unexplored breather solutions and discuss the
optimal transfer of energy from carrier to sidebands. Finally, we discuss the
observability of such discrete solutions in light of numerical simulations.

##  1 Introduction

The Benjamin-Feir (or modulation) instability of waves in deep water is one of
the most prominent discoveries of nonlinear science during the 20th century.
The fact that monochromatic waves distort while propagating in the laboratory,
and that these difficulties can be attributed to fundamental energy transfers
which impact our ability to forecast waves or understand extreme events
continues to be a source of fascination to this day.

The potential flow problem with free surface which describes water wave
propagation is nonlinear due to the surface boundary conditions. To deal with
this formidable problem, the investigation of the instability of a
monochromatic wave train by Benjamin & Feir [7] employed perturbation theory
in the spirit of G. G. Stokes, who pioneered its use in hydrodynamics more
than a century earlier. When the problem is linearised, periodic, traveling
waves consisting of a single Fourier harmonic are easily found. From the
second order in the perturbation expansion these waves change shape due to the
addition of bound harmonic terms, and at third order undergo a first
dispersion correction, such that their frequency depends on their amplitude.
These changes were known to Stokes by the mid 19th century.

What Stokes could not anticipate was how the dynamics would change if more
than one Fourier harmonic were present in the lowest order solution. Indeed,
the fact that the onerous perturbation expansions might yield something
worthwhile became apparent only with Phillips’ [34] discovery of a resonant
mechanism for energy exchange among water waves at third order. This gave the
impetus to explore the problem more deeply, introducing initially small
superharmonic and subharmonic perturbations into the water wave problem, and
culminating in the work of Benjamin & Feir. Simultaneously with these
developments, efforts were underway to develop compact model equations for the
nonlinear evolution of perturbed wave trains.

In water waves this resulted in the groundbreaking 1968 paper of Vladimir
Zakharov [44], wherein a Hamiltonian formulation of the water wave problem was
determined for the first time. In the same work this novel formulation was
used to derive a nonlinear Schrödinger equation (NLS) in the context of
hydrodynamics, which was then employed to analyse the linear stability of
uniform wave trains. The NLS is in fact a type of universal equation for the
slow variation of wave envelopes, as previously shown by Benney & Newell [8].
A linear stability analysis based on such a model equation had previously been
employed by Bogoliubov [10] in understanding the elementary excitations of a
Bose-Einstein condensate, and by Bespalov & Talanov [9] for nonlinear
electromagnetic waves.

These analyses, starting from the Gross-Pitaevski or nonlinear Schrödinger
equations, allow for the study of small perturbations – modulations – of a
plane wave solution. This is effectively a Fourier mode truncation, whereby
the partial differential equation is reduced to a system of coupled ordinary
differential equations which are linearised to obtain a threshold for
stability. The initial exponential growth soon renders the linearisation
invalid, and understanding the subsequent behaviour requires new methods. One
natural avenue of progress was numerical investigation, although this comes
with its own pitfalls, see, for example, Ablowitz & Herbst [1]. As numerical
solutions require the fine-tuning of initial conditions and inevitably involve
computations with finite accuracy, the complementary route of seeking exact
solutions came to play a significant role. From the point of view of the
nonlinear Schrödinger equation, the most remarkable exact solutions are the
breathers found by Kuznetsov [27] and Ma [30], Akhmediev et al [3] and
Peregrine [33]. These solutions represent the reversible amplification of
disturbances from a background state into a spatially, temporally or
spatiotemporally-localised coherent structure, and have been the subject of
intense interest since their discovery.

Another line of inquiry, led initially by the Fluid Mechanics Department at
TRW Defense and Space System and collaborators, and reviewed in Yuen & Lake
[43], sought to explore the modulation instability using relaxed assumptions.
In particular, an intermediate result in Zakharov’s seminal 1968 paper [44] –
namely a reduced form of the Hamiltonian equation which retains only resonant
contributions – provided a window into the instability without some of the
narrow-bandwidth restrictions imposed by the NLS formulation. Employing this
equation, which became known as the Zakharov equation, Crawford et al [19]
gave improved linear stability bounds, and contributed to a profusion of
interest in modulation instability with broader bandwidth [21, 41]. This
reduced Hamiltonian equation has, in fact, a rather generic form [45]: for a
dispersion law which permits four wave resonances but not three wave
resonances, the particular physics of the problem are contained only in the
corresponding integral kernel. In the context of water waves, the correct
Hamiltonian form of this kernel was given by Krasitskii [26]. Analogous
reduced Hamiltonians can be found, for example, for Langmuir waves in plasma
or optical waves described by the Maxwell equation [32].

Historically, most approaches to the problem of modulation instability and its
consequences focus on equations written in terms of temporal evolution. While
this is natural mathematically, as it is common to initiate the dynamics from
initial conditions, it does not correspond to the typical experimental set-up
in hydrodynamics, where times-series of surface wave evolution are measured
along the tank by means of wave gauges placed at fixed spatial locations,
i.e., the wave dynamics are initiated through boundary conditions. This calls
for a suitable spatial evolution equation, either in the form of the NLS [15]
or a spatial Zakharov equation [36]. The latter equation, derived by Shemer
and co-workers in the early 2000s, has been used for a small handful of
studies (e.g. [37, 23]), but its consequences for the Benjamin-Feir
instability have barely been explored (one notable exception is Shemer &
Chernyshova [35]).

Our objective is to employ the spatial Zakharov formulation, which arises from
the cubically nonlinear problem but otherwise makes no assumptions about
spectral bandwidth, to understand the entire spatial evolution of modulation
instability. Exactly as in studies of the instability threshold using NLS we
shall begin by truncating our system to three interacting Fourier modes. In
contrast to the classical approach, we find a subsequent linearisation to be
superfluous: the resulting system can be recast as a planar Hamiltonian
dynamical system, whose dynamics can be analysed by studying fixed points,
separatrices, and bifurcations. This dynamical system naturally encompasses
the bi-modal spectrum [35], which is in fact the natural counterpoint of the
classical Benjamin-Feir instability.

Our approach provides analytical insight into new solutions and associated
novel physics of this spatial equation, including an analogues of the
Kuznetsov-Ma breather solution. Moreover it shows a theoretical route towards
optimal conversion of energy from a monochromatic to a bichromatic sea, or
vice versa. We explore the stability of this optimal depletion solution to
higher harmonics both analytically and numerically, in a complement to the
phase plane analysis for the interacting degenerate quartet.

We now give an outline of the subsequent sections of this paper. In Section 2,
we provide a description of the fundamentals of the Zakharov equation. We show
how it is discretised to a finite number of interacting waves which lays the
groundwork for our analysis, before ending with a discussion of special cases,
which admit closed form solutions. In Section 3, we build on these
fundamentals discussed in Section 2 as we consider the degenerate quartet case
of three distinct frequencies and reduce the Zakharov equation to a discrete
set of ODEs for the amplitude and phase of the interacting waves. By
considering certain conserved quantities, we can reduce the dynamics to a two
dimensional dynamical system.

In Section 4 we discuss the various phenomena that result from three
interacting waves. We first discuss the periodic behaviour which represents
the generic, recurrent evolution of the Benjamin-Feir instability – also
called Fermi-Pasta-Ulam-Tsingou recurrence. We then analyse the breather
solutions present in our system, arising from both monochromatic and
bichromatic background states. We also discuss the issue of maximum energy
transfer away from the carrier, which is found in a particular breather
solution. In Section 5, we discuss the observability of these wave phenomena
when additional Fourier modes are present, and consider the implications for
wave flume experiments. Finally, in Section 6, we conclude with a summary of
our work as well as suggestions for some avenues of future study.

##  2 Fundamentals

Our starting point will be the spatial Zakharov equation developed in the
early 2000s by Shemer et al [36]. This is a modification of the temporal
evolution equation for nonlinear waves derived by Zakharov [44], and in
Hamiltonian form by Krasitskii [26]. The spatial Zakharov equation has the
form

$\displaystyle ic_{g}\frac{\partial B(x,\omega)}{\partial x}=$
$\displaystyle\iiint
T(k,k_{1},k_{2},k_{3})B^{*}(x,\omega_{1})B(x,\omega_{2})B(x,\omega_{3})$
$\displaystyle\cdot\exp(-i(k+k_{1}-k_{2}-k_{3})x)\delta(\omega+\omega_{1}-\omega_{2}-\omega_{3})d\omega_{1}d\omega_{2}d\omega_{3}.$
(1)

where $c_{g}$ denotes the deep-water linear group velocity, $k_{i}$ is a
wavenumber, $\omega_{i}=\omega(k_{i})=\sqrt{(}gk_{i})$ the linear dispersion
relation in deep water, and $*$ denotes the complex conjugate. By $\delta$ we
denote the Dirac delta distribution and each integral is taken over the real
line. The complex amplitudes $B$ are related to the Fourier transforms of the
free-surface elevation $\eta$ and the potential at the free surface, which may
be recovered from this formulation (see below). Full expressions for the
kernel $T(k,k_{1},k_{2},k_{3})$ are available in [26]. We note only that this
kernel has the following symmetries:

$T(i,j,k,l)=T(j,i,k,l)=T(i,j,l,k)=T(k,l,i,j).$

It is common to write the wavenumber-dependence of the kernels as a subscript,
and we shall employ the abbreviation $T_{jlmn}$ for the kernel
$T(k_{j},k_{l},k_{m},k_{n}).$ Owing to the symmetries we will also denote
$T_{jjjj}=T_{j}$ and $T_{ijij}=T_{ij}$ without risk of confusion.

This equation can be discretised as follows:

$ic_{g,j}\frac{dB_{j}(x)}{dx}=\sum_{l,m,n}T_{jlmn}B_{l}^{*}B_{m}B_{n}\exp(-i(k_{j}+k_{l}-k_{m}-k_{n})x)\delta(\omega_{j}+\omega_{l}-\omega_{m}-\omega_{n}),$
(2)

where $B_{i}=B(\omega_{i},x),$ and we denote by
$\Delta_{jl}^{mn}=k_{j}+k_{l}-k_{m}-k_{n}$ the wavenumber detuning, which is
of order $O(\epsilon^{2})$ for $\epsilon$ a characteristic wave steepness
[36].

The relationship between the complex amplitudes and the free surface elevation
is given (to lowest order) by

$\eta(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\frac{\omega}{2g}\right)^{1/2}\left[B(x,\omega)\exp(i(k(\omega)x-\omega
t))+\text{c.c.}\right]d\omega.$ (3)

Here “c.c.” stands for the complex conjugate of the preceding expression. More
detail on the background and derivation of this equation can be found in the
recent review by Stuhlmeier [39].

###  2.1 Simple solutions of the discrete ZE

The spatial Zakharov equation (1) can be explicitly solved in some special
cases, two of which are of particular interest for a study of the Benjamin-
Feir instability. We first consider the case of a single wave $\omega_{0},$
such that the spatial Zakharov equation becomes

$ic_{g,0}\frac{dB_{0}(x)}{dx}=T_{0}|B_{0}(x)|^{2}B_{0}(x),$ (4)

which admits the constant amplitude solution

$B_{0}(x)=A_{0}e^{-iA_{0}^{2}T_{0}x/c_{g,0}}.$ (5)

This is a monochromatic wave field with a nonlinear correction to the
wavenumber, corresponding to the free mode part of the well-known third-order
Stokes’ wave solution. Inserting into (3) and using
$T_{j}=\frac{k_{j}^{3}}{4\pi^{2}}$ it can be written as

$\eta(x,t)=a_{0}\cos(k_{0}[1-a_{0}^{2}k_{0}^{2}]x-\omega_{0}t).$ (6)

where we have normalised the constant amplitude via

$A_{0}=\pi a_{0}\left(\frac{2g}{\omega_{0}}\right)^{1/2}.$

The second simple case consists of two waves $\omega_{a}$ and $\omega_{b},$
resulting in a system of two equations

$\displaystyle ic_{g,a}\frac{dB_{a}}{dx}$
$\displaystyle=T_{a}|B_{a}|^{2}B_{a}+2T_{ab}|B_{b}|^{2}B_{a},$ (7a)
$\displaystyle ic_{g,b}\frac{dB_{b}}{dx}$
$\displaystyle=T_{b}|B_{b}|^{2}B_{b}+2T_{ab}|B_{a}|^{2}B_{b},$ (7b)

with solution

$\displaystyle B_{a}(x)$
$\displaystyle=A_{a}\exp(-i(T_{a}A_{a}^{2}+2T_{ab}A_{b}^{2})x/c_{g,a}),$ (8a)
$\displaystyle B_{b}(x)$
$\displaystyle=A_{b}\exp(-i(T_{b}A_{b}^{2}+2T_{ab}A_{a}^{2})x/c_{g,b}),$ (8b)

for $A_{a}$ and $A_{b}$ two constant amplitudes. The free surface is then a
bichromatic (sometimes called bimodal) sea-state, written

$\eta(x,t)=a_{a}\cos\left(k_{a}\left[1-a_{a}^{2}k_{a}^{2}-2a_{b}^{2}k_{a}^{3/2}k_{b}^{1/2}\right]x-\omega_{a}t\right)+a_{b}\cos\left(k_{b}\left[1-a_{b}^{2}k_{b}^{2}-2a_{a}^{2}k_{a}^{3/2}k_{b}^{1/2}\right]x-\omega_{b}t\right),$
(9)

where we take $k_{a}<k_{b}$ to resolve the two-wavenumber kernels as
$T_{ab}=\frac{k_{a}^{2}k_{b}}{4\pi^{2}}.$ This is the spatial counterpart of
the solution found using perturbation theory by Longuet-Higgins & Phillips
[29].

##  3 Reformulation of the discrete ZE

As soon as more than two Fourier modes are involved the equations become more
cumbersome. The principal reason is the appearance of nontrivial interactions
beyond the symmetric resonances encountered in Section 2.1. In particular,
three modes may interact to exchange energy if
$2\omega_{1}=\omega_{2}+\omega_{3}.$ A description of the resulting
interaction is significantly simplified by writing the complex amplitudes in
terms of magnitude and phase, as has been suggested and carried out by
Bretherton [11], Craik [18], Capellini & Trillo [13], Trillo & Wabnitz [40]
(in the context of the NLS), and Andrade & Stuhlmeier [4, 5] (for the temporal
Zakharov equation).

In the discrete spatial Zakharov equation (2) we write the complex amplitude
$B_{j}(x)$ as $|B_{j}|\exp(i\phi_{j}),$ where both magnitude and phase may
depend on $x.$ Separating into real and imaginary parts leads to:

$\displaystyle
c_{g,j}\frac{d|B_{j}|}{dx}=-\sum_{l,m,n}T_{jlmn}\delta_{jl}^{mn}|B_{l}||B_{m}||B_{n}|\sin(\Delta_{jl}^{mn}x+\theta_{jlmn}),$
(10) $\displaystyle-
c_{g,j}|B_{j}|\frac{d\phi_{j}}{dx}=\sum_{l,m,n}T_{jlmn}\delta_{jl}^{mn}|B_{l}||B_{m}||B_{n}|\cos(\Delta_{jl}^{mn}x+\theta_{jlmn}),$
(11)

with

$\displaystyle{\theta_{jlmn}=\phi_{j}+\phi_{l}-\phi_{m}-\phi_{n},}$

We identify $\Delta_{jl}^{mn}$ as the wavenumber detuning parameter.

###  3.1 Reduction to a degenerate quartet

If we assume that the indices in equations (10)–(11) take on values in the set
$\\{a,b,c\\}$ only, with the proviso that

$2\omega_{a}=\omega_{b}+\omega_{c},$

so that the resonance condition imposed by the Kronecker delta
$\delta_{jl}^{mn}$ is fulfilled, we obtain a set of six ODEs:

$\displaystyle c_{g,a}|B_{a}|^{\prime}$
$\displaystyle=-2T_{aabc}|B_{a}||B_{b}||B_{c}|\sin(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12a) $\displaystyle c_{g,c}|B_{c}|^{\prime}$
$\displaystyle=T_{aabc}|B_{a}|^{2}|B_{b}|\sin(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12b) $\displaystyle c_{g,b}|B_{b}|^{\prime}$
$\displaystyle=T_{aabc}|B_{a}|^{2}|B_{c}|\sin(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12c) $\displaystyle-c_{g,a}|B_{a}|\phi_{a}^{\prime}$
$\displaystyle=\Gamma_{a}+2T_{aabc}|B_{a}||B_{b}||B_{c}|\cos(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12d) $\displaystyle-c_{g,b}|B_{b}|\phi_{b}^{\prime}$
$\displaystyle=\Gamma_{b}+T_{aabc}|B_{a}|^{2}|B_{c}|\cos(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12e) $\displaystyle-c_{g,c}|B_{c}|\phi_{c}^{\prime}$
$\displaystyle=\Gamma_{c}+T_{aabc}|B_{a}|^{2}|B_{b}|\cos(\Delta_{aa}^{bc}x+\theta_{aabc})$
(12f)

where

$\Gamma_{i}=|B_{i}|^{3}T_{i}+2\sum_{j\neq i}|B_{i}||B_{j}|^{2}T_{ij}.$ (13)

As we are restricted to this so-called degenerate quartet we shall henceforth
drop the sub and superscripts on the detuning parameter $\Delta_{aa}^{bc}$
where there is no risk of confusion.

A key observation is that the phases $\phi_{i}$ of the individual modes appear
only in the single combination $\Theta=\Delta x+\theta_{aabc},$ which we
identify as the combined (or dynamic) phase variable of the problem. We write
an evolution equation for this dynamic phase variable [Bustamante2009, 12] as

$\displaystyle\frac{d\Theta}{dx}$ $\displaystyle=\frac{d}{dx}\left(\Delta
x+\theta_{aabc}\right)=\Delta+2\phi_{a}^{\prime}-\phi_{b}^{\prime}-\phi_{c}^{\prime}$
$\displaystyle=\Delta-\left(\frac{2\Omega_{a}}{c_{g,a}}-\frac{\Omega_{b}}{c_{g,b}}-\frac{\Omega_{c}}{c_{g,c}}\right)-T_{aabc}\cos(\Theta)\left(\frac{4|B_{b}||B_{c}|}{c_{g,a}}-\frac{|B_{c}||B_{a}|^{2}}{c_{g,b}|B_{b}|}-\frac{|B_{b}||B_{a}|^{2}}{c_{g,c}|B_{c}|}\right),$
(14)

where we use ′ to denote the derivative in $x$ and define

$\Omega_{i}=\Gamma_{i}/|B_{i}|=|B_{i}|^{2}T_{i}+2\sum_{j\neq
i}|B_{j}|^{2}T_{ij}.$ (15)

Note that this differs from [5, Eq. (2.5)] in the signs of all terms except
the first. This is due to the temporal Zakharov equation [5, Eq. (2.1)]
containing a term $\exp(i\Delta_{np}^{qr}t)$ rather than the term
$\exp(-i\Delta_{jl}^{mn}x)$ found in (2).

We find that a quantity akin to wave action

$c_{g,a}|B_{a}|^{2}+c_{g,b}|B_{b}|^{2}+c_{g,c}|B_{c}|^{2}=A$ (16)

is conserved, as is the difference in side-band magnitudes

${c_{g,b}|B_{b}|^{2}}-{c_{g,c}|B_{c}|^{2}}=A\alpha.$ (17)

In light of these conserved quantities, it is useful to reformulate our
equations in terms of the squared magnitudes $I_{i}:=|B_{i}|^{2},$ whereupon

$\frac{d}{dx}I_{i}=2\frac{d|B_{i}|}{dx}|B_{i}|$

and we obtain the three equations

$\displaystyle I_{a}^{\prime}$
$\displaystyle=\frac{-4T_{aabc}}{c_{g,a}}I_{a}\sqrt{I_{b}I_{c}}\sin(\Theta),$
(18) $\displaystyle I_{b}^{\prime}$
$\displaystyle=\frac{2T_{aabc}}{c_{g,b}}I_{a}\sqrt{I_{b}I_{c}}\sin(\Theta),$
(19) $\displaystyle I_{c}^{\prime}$
$\displaystyle=\frac{2T_{aabc}}{c_{g,c}}I_{a}\sqrt{I_{b}I_{c}}\sin(\Theta).$
(20)

Making the substitution

$\displaystyle I_{a}$ $\displaystyle=\frac{A}{c_{g,a}}\eta,$ (21)
$\displaystyle I_{b}$ $\displaystyle=\frac{A}{2c_{g,b}}(1-\eta+\alpha),$ (22)
$\displaystyle I_{c}$ $\displaystyle=\frac{A}{2c_{g,c}}(1-\eta-\alpha).$ (23)

finally reduces the six coupled equations for the magnitudes and phases to a
dynamical system involving two parameters $A$ and $\alpha:$

$\displaystyle\frac{d\eta}{dx}$ $\displaystyle=-\frac{2T_{aabc}\eta
A\sqrt{(1-\eta)^{2}-\alpha^{2}}\sin(\Theta)}{c_{g,a}\sqrt{c_{g,b}c_{g,c}}}$
(24) $\displaystyle\frac{d\Theta}{dx}$
$\displaystyle=\frac{2AT_{aabc}(\alpha^{2}-2\eta^{2}+3\eta-1)\cos(\Theta)}{c_{g,a}\sqrt{c_{g,b}c_{g,c}}\sqrt{(1-\eta)^{2}-\alpha^{2}}}+A\eta\Xi_{1}+A\Xi_{0}+\Delta$
(25)

where

$\displaystyle\Xi_{1}=-2\left(\left({\tilde{T}_{a}}-2{\tilde{T}_{ab}}+\frac{1}{4}{\tilde{T}_{b}}\right)+\left(-2{\tilde{T}_{ac}}+{\tilde{T}_{bc}}\right)+\frac{{\tilde{T}_{c}}}{4}\right)$
(26)
$\displaystyle\Xi_{0}=\frac{\left(\left(1+\alpha\right)\left(-4{\tilde{T}_{ab}}+{\tilde{T}_{b}}\right)+4\left({\tilde{T}_{ac}}\left(-1+\alpha\right)+{\tilde{T}_{bc}}\right)-{\tilde{T}_{c}}\left(-1+\alpha\right)\right)}{2}$
(27)

and for compactness we write $\tilde{T}_{ij}=T_{ij}/(c_{g,i}c_{g,j})$ and
$\tilde{T}_{i}=T_{i}/c_{g,i}^{2}.$

When the side-band energy is equally distributed and $\alpha=0$ this reduces
to the simpler system

$\displaystyle\frac{d\eta}{dx}$
$\displaystyle=\beta_{2}\eta(\eta-1)\sin(\Theta)$ (28)
$\displaystyle\frac{d\Theta}{dx}$
$\displaystyle=\beta_{2}(2\eta-1)\cos(\Theta)+\beta_{1}\eta+\beta_{0}$ (29)

where

$\displaystyle\beta_{2}$
$\displaystyle=A\frac{2T_{aabc}}{c_{g,a}\sqrt{c_{g,b}c_{g,c}}},$ (30)
$\displaystyle\beta_{1}$
$\displaystyle=A\frac{\left(-{\tilde{T}_{b}}-4{\tilde{T}_{bc}}-{\tilde{T}_{c}}\right)+\left(8{\tilde{T}_{ab}}+8{\tilde{T}_{ac}}\right)-4{\tilde{T}_{a}}}{2},$
(31) $\displaystyle\beta_{0}$
$\displaystyle=\Delta+A\frac{\left(-4{\tilde{T}_{ab}}+{\tilde{T}_{b}}\right)+\left(-4{\tilde{T}_{ac}}+4{\tilde{T}_{bc}}\right)+{\tilde{T}_{c}}}{2}.$
(32)

It is possible to integrate the evolution equation for $\eta$ with respect to
$\Theta$ and the evolution equation for $\Theta$ with respect to $\eta$ and so
obtain a Hamiltonian for the planar system being analysed. We write the
Hamiltonian for $\alpha=0$ as

$H(\eta,\Theta)=-\beta_{2}\eta(\eta-1)\cos\left(\Theta\right)-\frac{\eta^{2}}{2}\beta_{1}-\beta_{0}\eta.$
(33)

Despite the significant differences in the spatial and temporal formulations,
this transformed, planar Hamiltonian is essentially analogous to that found in
the temporal case by Andrade & Stuhlmeier [5].

###  3.2 Phase-plane dynamics and fixed points

The fact that we have a planar Hamiltonian dynamical system means it is a
simple matter to compute the trajectories, which are simply the level lines of
the Hamiltonian. The phase space is the truncated cylinder $\mathbf{p}\in
C=\\{(\Theta,\eta)\in\mathbb{T}\times\mathbb{R}:-\pi\leq\Theta\leq\pi,0\leq\eta\leq
1\\},$ whose top $\eta=1$ corresponds to monochromatic waves and whose bottom
$\eta=0$ corresponds to bichromatic waves (see Section 2.1) as can be seen
immediately from (21)–(23). Points in the interior correspond to some mixing
of three modes.

The specification of a trajectory requires that we fix the Fourier modes under
consideration, which is done by selecting a central (or carrier) frequency $f$
(1/s) or $\omega$ (rad/s) and a mode separation parameter $p$ such that
$\omega_{a}=\omega=2\pi f,$ $\omega_{b}=\omega-p$ and $\omega_{c}=\omega+p.$
In addition we must specify $A$ in some physically meaningful sense. This is
simplest if we employ the relations between the complex amplitudes $B$, the
energy scale parameter $\eta$, and the free surface elevation $\eta$ in (3).

We write this correspondence for a single wave, which we can think of
classifying a phase portrait based on the monochromatic carrier (6) which
forms the top of the phase space. Substituting $\eta=1$ into (21)–(23) with
$\alpha=0$ shows $A=c_{g,a}|B_{a}|^{2}$ and yields the relation

$A=\frac{a_{a}^{2}\pi^{2}g}{k_{a}}=\frac{\epsilon_{a}^{2}\pi^{2}g}{k_{a}^{3}},$
(34)

where $a_{a}$ is the physical amplitude of the carrier wave, and
$\epsilon_{a}=a_{a}k_{a}$ is the wave steepness. The dynamics in the phase
space are governed by fixed points and associated separatrices, which are
depicted as black circles and dashed curves in Figure 1. By setting the right-
hand side of (28)–(29) equal to zero we can find expressions for the fixed
points of the system.

$\displaystyle\mathbf{p}_{1}$
$\displaystyle=\left(0,\frac{\beta_{2}-\beta_{0}}{2\beta_{2}+\beta_{1}}\right),$
(35) $\displaystyle\mathbf{p}_{2}$
$\displaystyle=\left(\pm\pi,\frac{\beta_{0}+\beta_{2}}{2\beta_{2}-\beta_{1}}\right),$
(36) $\displaystyle\mathbf{p}_{3,4}$
$\displaystyle=\left(\Theta_{1},0\right),$ (37)
$\displaystyle\mathbf{p}_{5,6}$ $\displaystyle=\left(\Theta_{2},1\right),$
(38)

with $\Theta_{1}$ and $\Theta_{2}$ defined as the solutions to the
trigonometric equations

$\displaystyle\cos(\Theta_{1})$ $\displaystyle=\frac{\beta_{0}}{\beta_{2}},$
(39) $\displaystyle\cos(\Theta_{2})$
$\displaystyle=-\frac{\beta_{0}+\beta_{1}}{\beta_{2}},$ (40)

Figure 1: Phase portraits for $(\Theta,\eta)$ for various values of the mode
separation parameter $p$ with $f=1$ and $\epsilon=0.2$. Panels (a)–(f) show
$p=0,\,0.6p_{c},\,p_{c},\,1.1p_{c},\,2p_{c}$ and $3p_{c}$ respectively for
$p_{c}\approx 0.5791.$ Fixed points of the dynamical system are denoted by
black circles, separatrices are denoted by dashed curves connecting pairs of
fixed points.

We note that these fixed points must be either centres or saddle points. This
follows immediately from considering the Jacobian of the system (28)–(29):

$J=\begin{pmatrix}H_{\eta\Theta}&H_{\Theta\Theta}\\\
-H_{\eta\eta}&-H_{\eta\Theta}\end{pmatrix}=\begin{pmatrix}\beta_{2}(2\eta-1)\sin(\Theta)&\beta_{2}\eta(1-\eta)\cos(\Theta)\\\
2\beta_{2}\cos(\Theta_{1})+\beta_{1}&\beta_{2}(1-2\eta)\sin(\Theta)\
\end{pmatrix}$

which has vanishing trace and determinant

$\det(J)=\beta_{2}\eta(1-\eta)\cos(\Theta)\left(2\beta_{2}\cos(\Theta)+\beta_{1}\right)-(1-2\eta)^{2}\beta_{2}^{2}\sin^{2}(\Theta).$

We find that the existence of the $\eta=1$ fixed points depends on the
condition

$\displaystyle-1\leq-\frac{\beta_{0}+\beta_{1}}{\beta_{2}}\leq 1,$ (41)

which we may square to give

$\displaystyle D$
$\displaystyle=\left({\Delta}+2|B_{a}|^{2}\left(\frac{T_{ab}}{c_{g,b}}+\frac{T_{ac}}{c_{g,c}}-\frac{T_{a}}{c_{g,a}}\right)\right)^{2}-\frac{4T_{aabc}^{2}}{c_{g,b}c_{g,c}}|B_{a}|^{4},$
(42)

such that fixed points exist when $D\geq 0.$ This is exactly the growth rate
found from linear stability analysis of the spatial Benjamin-Feir instability
(see also A). Indeed, if fixed points exist at $\eta=1$ the eigenvalues of the
Jacobi matrix – which satisfy $\lambda_{1,2}=\pm\sqrt{-\det(J)}$ – are
positive exactly when $\det(J)$ is negative. It can be readily verified by
substitution that at $\eta=1$

$\det(J)=\left(\Delta+2A\left(\tilde{T}_{ab}+\tilde{T}_{ac}-\tilde{T}_{a}\right)\right)^{2}-4A^{2}\tilde{T}_{aabc}^{2}.$

Existence of $\eta=0$ fixed points (and so the instability of the underlying
bichromatic sea) depends on the condition

$-1\leq\frac{\beta_{0}}{\beta_{2}}\leq 1,$

which may be resolved (recalling that here
$A=c_{g,b}|B_{b}|^{2}+c_{g,c}|B_{c}|^{2}$), after squaring, as

$\left(\Delta+\frac{c_{g,b}|B_{b}|^{2}+c_{g,c}|B_{c}|^{2}}{2}\left[\tilde{T}_{c}+\tilde{T}_{b}-4\tilde{T}_{ab}+4\tilde{T}_{bc}-4\tilde{T}_{ac}\right]\right)^{2}\leq\frac{4T_{aabc}^{2}(c_{g,b}|B_{b}|^{2}+c_{g,c}|B_{c}|^{2})^{2}}{c_{g,a}^{2}c_{g,b}c_{g,c}}.$

The regions of instability of the monochromatic waves (i.e. the spatial
Benjamin-Feir instability) and bichromatic waves are shown in Figure 2 for
fixed carrier frequency $f=1$ Hz. Fixed points appear on the top nullcline
$\eta=1$ for values of carrier steepness $\epsilon$ and frequency separation
$p$ within the coloured region shown in the left panel, and on the bottom
nullcline $\eta=0$ as shown in the right panel. Colour denotes the linear
growth rate of the unstable modes, with lighter yellow denoting higher growth
rate.

Figure 2: Plot in $\epsilon,\,p$ parameter space of the existence of fixed
points (i.e. instability) at $\eta=1$ (monochromatic waves, left panel) and
$\eta=0$ (bichromatic waves, right panel). Colours denote the growth rate,
calculated from the eigenvalues of the Jacobi matrix. The red curve shows the
location in $\epsilon,\,p$ space of the maximum depletion of the carrier,
associated with the vertical separatrix shown in panel (c) of Figure 1.

##  4 Dynamics of interacting waves

Even the relatively simple situation of three interacting waves contains a
rich diversity of phenomena. These include periodic trajectories as well as
special cases such as the separatrices and interior fixed points.

###  4.1 Periodic solutions

It is well known that the generic, long-time behaviour found when Benjamin-
Feir instability is triggered is a periodic recurrence, known as the Fermi-
Pasta-Ulam-Tsingou recurrence. Indeed, this periodic behaviour can be readily
observed in our solutions. Figure 3 shows such a solution from three vantage
points. We consider a wave configuration with carrier frequency $f=1$ Hz and
carrier steepness $\epsilon=0.2$. This fixes the total energy of the system,
but any of the dynamics found in Figure 1 are possible. We can fix a
particular phase portrait by selecting the mode separation $p$, which is
chosen to be unity to yield a configuration where the carrier is unstable to
disturbances.

We can select a particular trajectory in phase space by computing the
Hamiltonian for given values of $\eta$ and $\Theta.$ In the lower left panel
of Figure 3 the red curve is the contour for $\eta=0.95$ when $\Theta=0.$ We
see that this periodic energy exchange is characterised by a being confined
within the separatrix surrounding the centre point $\mathbf{p}_{1}$ (see
(35)), and the dynamic phase $\Theta$ takes on values between approximately
$-1.5$ and $1.5$ only. These trajectories are “unwrapped” in the lower right
panel, which shows the individual Fourier amplitudes $|B_{i}|$ as well as the
dynamic phase as functions of $x.$ The interplay between dynamic phase and
Fourier amplitudes can be clearly observed: $\Theta(x)$ is at a maximum or
minimum when the energy exchange is greatest. Finally the free surface
envelope $|A(x,t)|$ is shown for one recurrence period ($x\approx 40$ m) in
the top panel of the same figure.

Figure 3: Three views of a period solution: (top panel) free-surface envelope
in space and time. (Bottom left) solution depicted in phase space (red curve).
(Bottom right) Fourier amplitudes $|B_{i}(x)|$ and dynamic phase $\Theta(x)$
plotted with distance $x$.

###  4.2 Breather solutions

In addition to the recurrent solutions, which are those usually encountered in
numerical simulations, there are also solutions with asymptotic behaviour.
These solutions tend asymptotically to either the bichromatic or monochromatic
wave field, and are therefore termed breathers (in that context we usually
speak of a mono/bichromatic background). We can look for breather solutions of
our equations by considering the orbits written in the form

$\frac{d\eta}{d\Theta}=\frac{\beta_{2}\eta(\eta-1)\sin(\Theta)}{\beta_{0}+\beta_{1}\eta+\beta_{2}(2\eta-1)\cos(\Theta)}.$
(43)

This can be integrated explicitly, although it is advisable to simplify first.
Solutions of interest are those for which $\eta\rightarrow 1,0$ for
$\Theta\rightarrow\Theta^{*},$ where $\Theta^{*}$ is the dynamic phase
corresponding to a fixed point on the boundary of the phase plane.

####  4.2.1 Breather solutions with monochromatic background

When we are looking for explicit solutions with monochromatic background, i.e.
separatrices which connect two fixed points at $\eta=1,$ we find the explicit
expression

$\eta(\Theta)=\frac{\beta_{2}\cos(\Theta)-(\beta_{2}\cos(\Theta)+\beta_{0}+\beta_{1})-\beta_{0}}{\beta_{1}+2\beta_{2}\cos(\Theta)},$
(44)

which tends towards 1 as $\Theta\longrightarrow\Theta_{0},$ for

$\Theta_{0}=\arccos\left(\frac{-\beta_{0}-\beta_{1}}{\beta_{2}}\right)$

the dynamic phase of the fixed point. The differential equation governing
$\Theta$ can also be integrated, and the solution written as

$\tan\left(\frac{\Theta}{2}\right)=\textrm{sgn}(\sin(\Theta_{0}))\tan\left(\frac{\Theta_{0}}{2}\right)\tanh\left(\frac{\beta_{2}}{2}x\sin(\Theta_{0})\right).$
(45)

This breather solution corresponds to the class of separatrix found in panel
(e) of Figure 1. A depiction of the free surface envelope in space and time is
shown in Figure 4. It should be noted that the envelope of a monochromatic
wave field is a constant, as observed for large values of $|x|$. At the
focusing location $x=0$ the breather is periodic in time, analogous to the
well-known Kuznetsov-Ma breather solution of the nonlinear Schrödinger
equation.

Figure 4: The free surface envelope of a breather with monochromatic
background, corresponding to $f=1$ Hz, $\epsilon=0.2,$ and $p=1.4.$ The
focusing occurs at $x=0,$ which is the minimum of the energy scale parameter
$\eta$ along the separatrix. As $x$ tends to $\pm\infty,$ $\eta$ tends toward
1 and the wave field asymptotically becomes monochromatic.

####  4.2.2 Breather solutions with bichromatic background

We may apply the same process to the $\eta=0$ fixed point, which has breathers
associated with the separatrices shown in panels (a) and (b) of Figure 1.
Using $H=0$ and simplifying equation (28), we have an exact solution
$\eta(\Theta)$ given by

$\displaystyle\eta(\Theta)$
$\displaystyle=\frac{\beta_{0}-\beta_{2}\cos(\Theta)}{\beta_{2}(1-\cos(\Theta))-\beta_{2}-\beta_{1}/2},$
(46)

where it can be seen that $\eta\rightarrow 0$ as $\Theta\rightarrow\Theta^{*}$
where

$\displaystyle\Theta^{*}$
$\displaystyle=\cos^{-1}\left(\frac{\beta_{0}}{\beta_{2}}\right).$ (47)

We can also solve explicitly the differential equation for $\Theta$ which
yields

$\displaystyle\tan\left(\frac{\Theta}{2}\right)$
$\displaystyle=\frac{\beta_{2}}{\beta_{2}+\beta_{0}}\sin(\Theta^{*})\tanh\left(\frac{\beta_{2}}{2}\sin(\Theta^{*})x\right),$
(48)

As above, the corresponding free surface envelope is shown in Figure 5.

Figure 5: The free surface envelope of a breather with bichromatic background,
corresponding to $f=1$ Hz, $\epsilon=0.2,$ and $p=0.4.$ The focusing (in the
form of a “demodulation”) occurs at $x=0,$ which is the maximum of the energy
scale parameter $\eta$ along the separatrix. At this location $x=0$ the wave
field is close to monochromatic, as components $b$ and $c$ are very small. As
$x$ tends to $\pm\infty$, $\eta$ tends toward 0 and the wave field
asymptotically becomes bichromatic.

####  4.2.3 Breather solutions with maximum depletion

The most striking breather solution occurs when a separatrix connects a fixed
point at the top of the phase space $\eta=1$ with a fixed point at the bottom
of the phase space $\eta=0$. For this breather solution to exist we require
that the fixed points at $\eta=0$ and $\eta=1$ have the same phase, which
requires

$\beta_{0}=-\frac{\beta_{1}}{2}.$ (49)

Such separatrices are vertical orbits $\Theta=\Theta_{c}$, as shown in Panel
(c) of Figure 1. Consequently they are simple to obtain from the dynamical
system, since (28) decouples from (29), and can be immediately integrated as a
separable ODE for $\eta.$ The solution is

$\eta(x)=\frac{1}{1+C\exp(-\beta_{2}x\sin(\Theta_{c}))}.$ (50)

For unidirectional waves as considered herein the coefficient $\beta_{2}$ is
strictly non-negative, so that the behaviour of the solution as
$x\rightarrow\pm\infty$ depends only on the sign of $\sin(\Theta_{c}).$ For
symmetry, the constant of integration $C$ can be chosen as unity, such that
$\eta(0)=1/2$ is centred in the phase space. The explicit formula for the
optimal conversion breather (50) also allows us to calculate analytically the
evolution length.

The free surface envelope of this breather is shown in Figure 6. This solution
is particularly interesting, since it represents the optimal conversion of
energy (or transformation) from a monochromatic wave train to a bichromatic
wave train (or vice versa). Thus, while any instability of the monochromatic
carrier (shown in the left panel of Figure 2) gives rise to a breather of the
type discussed in Section 4.2.1, the optimal energy transfer is obtained for a
unique value of mode separation for a given carrier steepness – shown as the
red curve in both panels of Figure 2. In addition, the phases of the waves
must be tuned in order to obtain this solution, see panel (c) of Figure 1. A
key observation is that, for a given carrier, the maximal energy transfer
occurs at for much closer side-bands (smaller $p$) than the fastest linear
growth rate.

Figure 6: The free surface envelope of a breather with maximal depletion,
corresponding to $f=1$ Hz, $\epsilon=0.2,$ and $p\approx 0.5791.$ For
$x\rightarrow\infty$ the wave field tends toward a monochromatic state, while
for $x\rightarrow-\infty$ the wave tends towards the bichromatic state.

##  5 Observability of discrete wave interactions

A principal advantage of the spatial Zakharov equation over its better-known
temporal sister equation is that it relates directly to properties that can be
measured in wave flume experiments. Analogous interaction equations can be
used to describe other nonlinear dispersive media, such as electromagnetic
Kerr media [31, 42]. A natural question therefore concerns the significance of
the three-mode truncation results for experimental work, in particular for
optimising energy transfer between a carrier wave and its side bands.

The two interactions that occur in our dynamical systems description of the
Benjamin-Feir instability are transfers of energy from one mode $\omega$ to
two modes $\omega_{\pm 1}$ and vice versa. These are depicted schematically in
Figure 7. The principal energy exchange is indicated by the blue arrows,
moving either from one mode to two (left panel) or two modes to one (right
panel). What occurs when additional Fourier modes are incorporated? If these
are equidistant modes $\omega_{\pm 2},\,\omega_{\pm 3},\ldots$ new energy
exchanges become available, and the possibility of spectral broadening
appears. The most important interaction is with the superharmonics
$\omega_{+2}$ and $\omega_{-2},$ which is easily subsumed into the foregoing
theory.

We will limit our discussion to two cases: the periodic recurrence discussed
in Section 4.1 and the optimal energy conversion from Section 4.2.3, and
investigate whether these are robust from the expanded point of view which
allows for higher harmonics. We shall begin with the case of optimal energy
conversion, which demonstrates a particular instability whereby small
perturbations – with specially tuned mode separation and phases – are able to
asymptotically convert a monochromatic wave train into a bichromatic state, or
vice versa.

Figure 7: Instabilities of monochromatic and bichromatic waves shown embedded
in a discrete spectrum, illustrating the possibilities for spectral broadening
through energy transfer. (Left panel) The Benjamin-Feir instability of a
monochromatic wave $f$ (black arrow) transfers energy to $f_{\pm 1}$ (blue
arrows), but may continue to transfer energy to outlying modes $f_{\pm 2}.$
(Right panel) The instability of a bichromatic wave train $f_{\pm 1}$ (black
arrows) to monochromatic disturbances transfers energy to $f$ (blue arrows),
but may also transfer energy to outlying modes $f_{\pm 2}$ (red arrows).

One problem is immediately evident from Figure 1: given a carrier steepness
and frequency, we find the optimal conversion for a particular value of $p$
which we shall call $p_{c}$ (panel (c) Figure 1). However the same carrier is
also unstable to superharmonic perturbations with $2p_{c}$ (panel (e) Figure
1). Moreover, the growth rate of the superharmonics $2p_{c}$ is larger than
that of the fundamental $p_{c},$ as can be seen in Figure 8. In fact,
perturbations with $2p_{c}$ (dashed red curve) are very close to the curve of
maximum growth rate (black curve), and so will be preferentially amplified.
This is precisely the scenario shown in the top panel of Figure 9, which
compares the same initial conditions for a three-mode and a five-mode
system.111Tackling the same problem from the other side of the phase plane,
and attempting to make a bichromatic into a monochromatic wave suffers from
similar problems. While $\omega_{-1}$ may transfer energy to mode $\omega_{0}$
in the desired interaction $\omega_{-1}+\omega_{1}=2\omega_{0}$, it much
prefers to transfer energy to both $\omega_{0}$ and $\omega_{-2}$ via the
equally accessible interaction $2\omega_{-1}=\omega_{0}+\omega_{-2}.$

Figure 8: Instability domain and linear growth rates (colours) for a
monochromatic wave field (as in Figure 2), showing the optimal conversion
breather (solid red curve), largest growth rate (black curve), and twice the
mode separation of both of the aforementioned (dashed red curve – breather
superharmonic, dashed black curve – largest growth rate superharmonic).

The foregoing discussion highlights some of the differences between a strict
truncation and the subsequent evolution when more modes are allowed into the
interaction. The analogous arguments can be made for periodic recurrences,
which exist alongside the asymptotic solutions and fixed points. In Figure 8
we note that the most unstable perturbation $p_{\text{max}}$ for a given
carrier steepness $\epsilon$ (shown in the solid black curve) is not
susceptible to superharmonic instabilities since the curve of
$2p_{\text{max}}$ lies outside the instability domain. Again, this fact is
borne out by numerical simulations comparing the three mode and five mode
system, as shown in the bottom panel of Figure 9.

Figure 9: Numerical integration with 3 and 5 Fourier modes, with carrier wave
$f_{a}=1$ Hz, $\epsilon_{a}=0.2.$ (Top panel) Three mode initial conditions
corresponding to the maximal energy transfer breather solution found in
Section 4.2.3, with $p\approx 0.5798.$ The three-mode system (solid curves)
behaves as expected, with energy transferring asymptotically from $B_{a}$ to
$B_{a+p}$ and $B_{a-p}.$ The five-mode system (dashed curves) behaves
analogously for very short times, but the fact that modes $f_{a+\pm 2p}$ are
unstable gives rise to energy exchange and chaotisation of the trajectories.
(Bottom panel) Three mode initial conditions corresponding to the largest
linear growth rate with $p\approx 1.1532.$ The three mode system (solid
curves) undergoes periodic energy exchange. As the higher harmonics are
linearly stable these do not participate in any significant energy exchange,
but are entrained in the dynamics of the principal triad $a,a+p,a-p.$

It may seem initially surprising that the superharmonics $\omega_{\pm 2p}$
oscillate despite being linearly stable. This can be seen from linear
stability analysis (see A), which shows that higher harmonics are entrained
and grow linearly in concert with the dominant unstable triad
$\omega,\,\omega_{\pm 1}$. While the qualitative behaviour, and in particular
the recurrent nature of the solution, is preserved some quantitative
differences do manifest, among these the aperiodicity of the five-mode
solution. Another significant difference is the extent to which the carrier
(denoted $|B_{a}|$) can deplete: in a case with more modes the cascade of
energy from the carrier into the sidebands and superharmonics leads to a
greater depletion of the carrier amplitude itself – a process that is brought
to its apotheosis by Akhmediev or Kuznetsov-Ma breathers [27, 2] (see Chin et
al [17]).

##  6 Discussion

We have set out to examine the spatial Benjamin-Feir instability from the
perspective of the spatial Zakharov equation. While the temporal case has been
studied very extensively, this physically important case has received much
less attention. In order to employ techniques of phase plane analysis we
restrict ourselves to the three Fourier modes forming a degenerate quartet,
the germ of the Benjamin-Feir instability. With this restriction it is
possible to describe the entire subsequent behaviour simply and analytically –
classifying the phase portraits, identifying separatrices with breathers and
fixed points with steady-state solutions, see [28].

Two particularly important insights are obvious from our approach, but
obscured by the classical treatment of linear stability analysis and
subsequent numerical integration. The first is that the largest (linear)
growth rate does not correspond to the largest energy exchange among the
modes. This is clear from Figure 2, where the latter is shown in the red
curves, and is observed to occur consistently for a smaller value of mode
separation than the highest growth rate at equal carrier steepness. Just
because the a disturbance grows the fastest does not mean that it grows the
most.

A second key insight concerns the stability thresholds. Monochromatic and
bichromatic waves are identified with the top and bottom nullclines of our
dynamical system, respectively, and their orbital stability is determined by
the existence of fixed points thereon. Indeed the existence criteria for such
fixed points are identical to the linear stability thresholds. However, the
phase portraits (see Figure 1, panel (f)) make clear that energy exchange
persists outside this instability threshold, as $\eta$ changes along the
trajectories. Contrary to what one might expect from the linear analysis
alone, even stable configurations generally exhibit oscillations between the
Fourier amplitudes.

Breathers are famed exact solutions of the nonlinear Schrödinger equation [3],
and can be observed experimentally [16, 14, 25], which raises the question
whether they can appear in more general equations governing inviscid
propagation of water waves. Early numerical simulations affirming this using
the equations for potential flow were performed by Dyachenko & Zakharov [20].
A more detailed study was undertaken by Slunyaev & Shrira [38] using numerical
solutions of the Euler equation, who found that breathers carefully
initialised with NLS initial conditions could be propagated numerically
without significant change. This is quite surprising given the simple nature
of the NLS and the numerous restrictions made in its derivation.

A natural question is whether breather solutions exist within the framework of
the reduced Zakharov equation, which – while limited to third order in
nonlinearity – is at least free of the bandwidth restrictions which appear in
the NLS. The answer to this question is positive, although no explicit
expressions comparable to those found by Akhmediev et al. [3] have been found.
Using the Petviashvili method, pioneered in investigations of the so-called
compact Dyachenko-Zakharov equation by Fedele & Dutykh [22], Kachulin et al
[24] have successfully obtained a numerical breather solution of the spatial
Zakharov equation.

As is the case for the Akhmediev breather solution of the NLS [17], this
solution manifests the Benjamin-Feir instability and so must occur within the
linear instability domain (see Figure 2). However, its higher harmonics must
be linearly stable, in order that the energy transfer is reversible and the
monochromatic background is obtained as $x\rightarrow\pm\infty.$ In analogy
with the situation found in the NLS by Bendahmane et al. [6], it is likely
that the largest realisable depletion of the carrier wave is to be found along
this breather trajectory.

Finding an explicit Kuznetsov-Ma type breather solution of the Zakharov
equation is not so straightforward due to the high-dimensional phase-space
involved. Such a solution, tending asymptotically to a fixed point on the
submanifold consisting of monochromatic waves would be a kind of coda to
Section 5, presenting an exactly reversible cascade of energy to the
superharmonics. Evidently, finding an experimental setting where the distinct
features of such a breather could be realised would also be of great interest.

##  Appendix A Linear stability analysis for the spatial Zakharov equation

It may be surprising to find that, even when the modes $\omega_{\pm 2}$ are
linearly stable, they nevertheless grow from zero. We can appreciate this fact
by a closer look at the linear stability analysis of the discretised spatial
Zakharov equation (2)

$ic_{g,j}b_{j}^{\prime}(x)=-\frac{\omega_{j}}{2}b_{j}+\sum
T_{jlmn}b_{l}^{*}b_{m}b_{n}\delta_{jlmn}.$ (51)

Note that $k_{j}c_{g,j}=\omega_{j}/2.$

If we initially assume that mode $b_{0}\gg b_{i}$ for all $i\neq 0,$ and we
neglect products of small terms, we obtain the single equation

$ic_{g,0}b_{0}^{\prime}=-\frac{\omega_{0}}{2}b_{0}+T_{0,0}|b_{0}|^{2}b_{0}=\Omega_{0}b_{0}.$
(52)

where we write $\Omega_{i}=-\frac{\omega_{i}}{2}+T_{0,i}|b_{0}|^{2}.$ The
solution to this equation is the Stokes’ wave (see Section 2.1)

$b_{0}=A_{0}\exp(-i\Omega_{0}/c_{g,0}).$

Neglecting the second harmonics $b_{\pm 2}$ as small, and retaining only terms
linear in $b_{\pm 1}$ we obtain the system of equations

$\displaystyle ic_{g,1}b_{1}^{\prime}$
$\displaystyle=\Omega_{1}b_{1}+T_{-1,1,0,0}b_{-1}^{*}b_{0}^{2},$ (53)
$\displaystyle ic_{g,-1}b_{-1}^{\prime}$
$\displaystyle=\Omega_{-1}b_{-1}+T_{-1,1,0,0}b_{1}^{*}b_{0}^{2}.$ (54)

This is a linear system in the side-bands $b_{\pm 1}$, and can be solved by
substituting the Ansatz

$\displaystyle
b_{1}=A_{1}\exp\left(x\left(\sigma-i\frac{\Omega_{1}}{c_{g,1}}-i\frac{\beta}{2}\right)\right),$
$\displaystyle
b_{-1}=A_{-1}\exp\left(x\left(\sigma^{*}-i\frac{\Omega_{-1}}{c_{g,{-1}}}-i\frac{\beta}{2}\right)\right),$

where
$\beta=-\beta_{0}-\beta_{1}=2\Omega_{0}/c_{g,0}-\Omega_{1}/c_{g,1}-\Omega_{-1}/c_{g,-1}.$
Substitution shows that this linear system has a solution when the determinant
of the coefficient matrix vanishes, precisely the condition (42) previously
obtained from phase-plane analysis.

At the next order, assuming $b_{\pm 2}$ are small and retaining terms
containing $b_{0},\,b_{\pm 1},$ one obtains the linear system

$\displaystyle ic_{g,2}\frac{db_{2}}{dt}$
$\displaystyle=2T_{-1,2,0,1}b_{-1}^{*}b_{0}b_{1}+T_{0,2,1,1}b_{0}^{*}b_{1}^{2},$
(55) $\displaystyle ic_{g,-2}\frac{db_{-2}}{dt}$
$\displaystyle=2T_{-2,1,-1,0}b_{1}^{*}b_{-1}b_{0}+T_{-2,0,-1,-1}b_{0}^{*}b_{-1}^{2}.$
(56)

These are forced equations for modes $\pm 2,$ which show that these modes grow
linearly due to the dominant interaction between the carrier and the side-
bands $\pm 1.$

## Funding sources

RS and CH acknowledge the support of EPSRC Grant EP/V012770/1.

## References

  * [1] M. J. Ablowitz and B. M. Herbst.  On homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation.  SIAM J. Appl. Math., 50(2):339–351, 1990.
  * [2] N. Akhmediev, V. Eleonskii, and N. Kulagin.  Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions.  Sov. Phys. JETP, 62(5):894–899, 1985.
  * [3] N. N. Akhmediev, V. M. Eleonskiĭ, and N. E. Kulagin.  First-order exact solutions of the nonlinear Schrödinger equation.  Teoret. Mat. Fiz., 72(2):183–196, 1987.
  * [4] D. Andrade and R. Stuhlmeier.  Instability of waves in deep water—a discrete hamiltonian approach.  European Journal of Mechanics-B/Fluids, 101:320–336, 2023.
  * [5] D. Andrade and R. Stuhlmeier.  The nonlinear Benjamin-Feir instability - Hamiltonian dynamics, discrete breathers, and steady solutions.  Journal of Fluid Mechanics, 958:A17, 2023.  doi:10.1017/jfm.2023.96.
  * [6] A. Bendahmane, A. Mussot, A. Kudlinski, P. Szriftgiser, M. Conforti, S. Wabnitz, and S. Trillo.  Optimal frequency conversion in the nonlinear stage of modulation instability.  Opt. Express, 23(24):30861, 2015.
  * [7] T. B. Benjamin and J. E. Feir.  The disintegration of wave trains on deep water Part 1. Theory.  J. Fluid Mech., 27(03):417–430, 1967.
  * [8] D. Benney and A. Newell.  The propagation of nonlinear wave envelopes.  Journal of mathematics and Physics, 46(1-4):133–139, 1967.
  * [9] V. I. Bespalov and V. I. Talanov.  Filamentary structure of light beams in nonlinear liquids.  Soviet Journal of Experimental and Theoretical Physics Letters, 3:307, 1966.
  * [10] N. Bogoliubov.  On the theory of superfluidity.  J. Phys, 11(1):23, 1947.
  * [11] F. Bretherton.  Low frequency oscillations trapped near the equator.  Tellus, 16(2):181–185, 1964.
  * [12] M. D. Bustamante and E. Kartashova.  Effect of the dynamical phases on the nonlinear amplitudes’ evolution.  Europhys. Lett., 85(3), 2009.
  * [13] G. Cappellini and S. Trillo.  Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects.  J. Opt. Soc. Am. B, 8(4):824, 1991.
  * [14] A. Chabchoub, N. Akhmediev, and N. P. Hoffmann.  Experimental study of spatiotemporally localized surface gravity water waves.  Phys. Rev. E, 86(1):016311, jul 2012.
  * [15] A. Chabchoub and R. H. Grimshaw.  The hydrodynamic nonlinear Schrödinger equation: Space and time.  Fluids, 1(3):1–10, 2016.
  * [16] A. Chabchoub, N. Hoffmann, and N. Akhmediev.  Rogue wave observation in a water wave tank.  Physical Review Letters, 106(20):204502, 2011.
  * [17] S. A. Chin, O. A. Ashour, and M. R. Belić.  Anatomy of the Akhmediev breather: Cascading instability, first formation time, and Fermi-Pasta-Ulam recurrence.  Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys., 92(6):1–9, 2015\.
  * [18] A. D. D. Craik.  Wave Interactions and Fluid Flows.  Cambridge University Press, jan 1986.
  * [19] D. R. Crawford, B. M. Lake, P. G. Saffman, and H. C. Yuen.  Stability of weakly nonlinear deep-water waves in two and three dimensions.  J. Fluid Mech., 105:177–191, 1981.
  * [20] A. I. Dyachenko and V. E. Zakharov.  On the formation of freak waves on the surface of deep water.  JETP letters, 88:307–311, 2008.
  * [21] K. B. Dysthe.  Note on a modification to the nonlinear Schrodinger equation for application to deep water waves.  Proc. R. Soc. A Math. Phys. Eng. Sci., 369:105–114, aug 1979.
  * [22] F. Fedele and D. Dutykh.  Special solutions to a compact equation for deep-water gravity waves.  Journal of Fluid Mechanics, 712:646–660, 2012.
  * [23] M. Galvagno, D. Eeltink, and R. Stuhlmeier.  Spatial deterministic wave forecasting for nonlinear sea-states.  Phys. Fluids, 33(10), 2021.
  * [24] D. Kachulin, A. Dyachenko, and A. Gelash.  Interactions of coherent structures on the surface of deep water.  Fluids, 4(2):1–21, 2019.
  * [25] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley.  The Peregrine soliton in nonlinear fibre optics.  Nat. Phys., 6(10):790–795, aug 2010.
  * [26] V. P. Krasitskii.  On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves.  J. Fluid Mech., 272:1–20, 1994.
  * [27] E. A. Kuznetsov.  Solitons in a parametrically unstable plasma.  236:575–577, 1977.
  * [28] S. Liao, D. Xu, and M. Stiassnie.  On the steady-state nearly resonant waves.  J. Fluid Mech., 794:175–199, 2016.
  * [29] M. S. Longuet-Higgins and O. M. Phillips.  Phase velocity effects in tertiary wave interactions.  J. Fluid Mech., 12(3):333–336, 1962.
  * [30] Y.-C. Ma.  The perturbed plane-wave solutions of the cubic schrödinger equation.  Studies in Applied Mathematics, 60(1):43–58, 1979.
  * [31] M. Onorato, F. Baronio, M. Conforti, A. Chabchoub, P. Suret, and S. Randoux.  Hydrodynamic and optical waves: A common approach for unidimensional propagation.  Rogue and Shock Waves in Nonlinear Dispersive Media, pages 1–22, 2016.
  * [32] M. Onorato, S. Residori, and F. Baronio, editors.  Rogue and Shock Waves in Nonlinear Dispersive Media.  Springer, 2016.
  * [33] D. H. Peregrine.  Water waves, nonlinear Schrödinger equations and their solutions.  J. Aust. Math. Soc. Ser. B. Appl. Math., 25(1):16–43, 1983.
  * [34] O. M. Phillips.  On the dynamics of unsteady gravity waves of finite amplitude Part 1\. The elementary interactions.  J. Fluid Mech., 9(2):193–217, oct 1960.
  * [35] L. Shemer and A. Chernyshova.  Spatial evolution of an initially narrow-banded wave train.  J. Ocean Eng. Mar. Energy, 3(4):333–351, 2017.
  * [36] L. Shemer, H. Jiao, E. Kit, and Y. Agnon.  Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation.  J. Fluid Mech., 427:107–129, 2001.
  * [37] L. Shemer, E. Kit, and H. Jiao.  An experimental and numerical study of the spatial evolution of unidirectional nonlinear water-wave groups.  Phys. Fluids, 14(10):3380–3390, oct 2002.
  * [38] A. V. Slunyaev and V. I. Shrira.  On the highest non-breaking wave in a group: Fully nonlinear water wave breathers versus weakly nonlinear theory.  J. Fluid Mech., 735:203–248, 2013.
  * [39] R. Stuhlmeier.  An introduction to the Zakharov equation for modelling deep water waves.  In D. Henry, editor, Nonlinear Dispersive Waves. Springer, to appear.
  * [40] S. Trillo and S. Wabnitz.  Dynamics of the nonlinear modulational instability in optical fibers.  Optics letters, 16(13):986–988, 1991.
  * [41] K. Trulsen and K. B. Dysthe.  A modified nonlinear schrödinger equation for broader bandwidth gravity waves on deep water.  Wave Motion, 24(3):281–289, 1996.
  * [42] G. Xu, A. Chabchoub, D. E. Pelinovsky, and B. Kibler.  Observation of modulation instability and rogue breathers on stationary periodic waves.  Physical Review Research, 2(3):033528, 2020.
  * [43] H. C. Yuen and B. M. Lake.  Nonlinear Dynamics of Deep-Water Gravity Waves.  In Adv. Appl. Mech., pages 68–229. Academic Press, 1982.
  * [44] V. Zakharov.  Stability of periodic waves of finite amplitude on the surface of a deep fluid.  J. Appl. Mech. Tech. Phys., 9(2):190–194, 1968.
  * [45] V. E. Zakharov, V. S. L’vov, and G. Falkovich.  Kolmogorov Spectra of Turbulence I.  Springer-Verlag Berlin Heidelberg, 1992.