# Two-dimensional reductions of the Whitham modulation system for the
Kadomtsev-Petviashvili equation

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Optical Kerr spatiotemporal dark-lump dynamics of hydrodynamic origin

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Line soliton interactions of the Kadomtsev-Petviashvili equation

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Soliton solutions of the Kadomtsev-Petviashvili II equation

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Towards an inverse scattering theory for non-decaying potentials of the heat
equation

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Extended resolvent and inverse scattering with an application to KPI

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Building an extended resolvent of the heat operator via twisting
transformations

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# On the equivalence of different approaches for generating multisoliton
solutions of the KPII equations

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Thèorie Des Ondes et Des Remous Qui Se Propagent Le Long d’un Canal
Rectangulaire Horizontal, En Communiquant Au Liquide Contenu Dans Ce Canal Des
Vitesses Sensiblement Pareilles de La Surface Au Fond

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Dispersive shock waves and modulation theory

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# On the stability of solitary waves in weakly dispersive media

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Ion acoustic soliton experiments in a plasma

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Self-focusing of plane dark solitons in nonlinear defocusing media

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Stability of magnetoelastic solitons and self-focusing of sound in
antiferromagnet

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# The direct scattering problem for perturbed Kadomtsev-Petviashvili multi
line solitons

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

# Stability of Kadomtsev-Petviashvili multi-line solitons

Gino Biondini1, Alexander J. Bivolcic1, Mark A. Hoefer2 and Antonio Moro3 1
State University of New York, Department of Mathematics, Buffalo, NY 14260,
USA  2 University of Colorado, Department of Applied Mathematics, Boulder, CO
80303, USA  3 Northumbria University, Department of Mathematics, Physics and
Electrical Engineering, Newcastle, NE1 8ST, UK

###### Abstract

Two-dimensional reductions of the KP-Whitham system, namely the overdetermined
Whitham modulation system for five dependent variables that describe the
periodic solutions of the Kadomtsev-Petviashvili equation, are studied and
characterized. Three different reductions are considered corresponding to
modulations that are independent of $x$, independent of $y$, and of $t$ (i.e.,
stationary), respectively. Each of these reductions still describes dynamic,
two-dimensional spatial configurations since the modulated cnoidal wave
generically has a nonzero speed and a nonzero slope in the $xy$ plane. In all
three of these reductions, the properties of the resulting systems of
equations are studied. It is shown that the resulting reduced system is not
integrable unless one enforces the compatibility of the system with all
conservation of waves equations (or considers a reduction to the harmonic or
soliton limit). In all cases, compatibility with conservation of waves yields
a reduction in the number of dependent variables to two, three and four,
respectively. As a byproduct of the stationary case, the Whitham modulation
system for the Boussinesq equation is also explicitly obtained.

##  1 Introduction and background

The description of dispersive wave propagation has been a classical topic of
study dating back to the works of Boussinesq, Stokes, Rayleigh, Korteweg and
de Vries and others in the nineteenth century, and it continues to attract
significant attention. A scenario of both theoretical and applicative interest
is that in which dispersive effects are much smaller than nonlinear ones, a
regime that often leads to the generation of dispersive shock waves. Indeed, a
large number of works have been devoted to this subject (e.g., see [17] and
references therein). The mathematical framework for the description of small
dispersion problems in one spatial dimension and the formation of dispersive
shock waves in that context have been well characterized, beginning with the
seminal work of G. B. Whitham [36]. However, our understanding of dispersive
wave propagation and dispersive shock waves in more than one spatial dimension
is much less developed.

The purpose of this work is to study special solutions of the Kadomtsev-
Petviashvili (KP) equation [23],

$(u_{t}+6uu_{x}+\varepsilon^{2}u_{xxx})_{x}+\sigma u_{yy}=0\,,$ (1.1)

where $0<\varepsilon\ll 1$, subscripts $x$, $y$ and $t$ denote partial
differentiation and the values $\sigma=\mp 1$ distinguish between the KPI and
KPII variants of the KP equation, respectively. The KP equation, which is a
two-dimensional generalization of the celebrated Korteweg-de Vries (KdV)
equation, similarly arises in such diverse fields as plasma physics [22, 23,
26], fluid dynamics [6, 24], nonlinear optics [7, 29] and ferromagnetic media
[34]. The KP equation is also, like the KdV equation, a completely integrable
infinite-dimensional Hamiltonian system whose solutions possess a rich
mathematical structure [5, 8, 21, 22, 24, 25, 27]. The initial-value problem
for the KP equation is in principle amenable to exact solution via the inverse
scattering transform (IST) [5, 25, 27]. Yet, even though considerable work has
been devoted to the development of the IST for the KP equation throughout the
last twenty years [12, 13, 14, 15, 37], the IST has rarely been used to study
the dynamical behavior of solutions of the KP equation [38]. Conversely,
asymptotic methods such as Whitham modulation theory have recently been shown
to be quite effective in this regard [4, 11, 30, 32].

In this work, we derive and characterize several asymptotic reductions of the
KP equation, which we rewrite in evolution form as

$u_{t}+6uu_{x}+\varepsilon^{2}u_{xxx}+\sigma v_{y}=0\,,\qquad v_{x}=u_{y}\,.$
(1.2)

The linear dispersion relation of (1.2), obtained by looking for small-
amplitude plane-wave solutions $u(x,y,t)=u_{o}+A\,e^{i\theta(x,y,t)}$ with
$|A|\ll|u_{o}|$, $\theta(x,y,t)=(kx+ly-\omega t)/\varepsilon$, is
$\omega=(6u_{o}+\sigma q^{2})k-k^{3}$, with $q=l/k$. In addition, the KP
equation admits nonlinear, exact traveling wave solutions in the form of
“cnoidal waves”

$u(x,y,t)=r_{1}-r_{2}+r_{3}+2(r_{2}-r_{1})\mathop{\rm cn}\nolimits^{2}(2\theta
K_{m};m)\,,\qquad v(x,y,t)=qu+p\,,$ (1.3)

where $\mathop{\rm cn}\nolimits(\cdot)$ denotes the Jacobian elliptic cosine
[28], $K_{m}=K(m)$ and $E_{m}=E(m)$ are the complete elliptic integrals of the
first and second kind, respectively, and

$m=\frac{r_{2}-r_{1}}{r_{3}-r_{1}}$ (1.4)

is the elliptic parameter. The above solution is completely determined by five
parameters: $r_{1}$, $r_{2}$, $r_{3}$, $q$ and $p$. The local wavenumber $k$
and $l$ in the $x$ and $y$ directions and the frequency $\omega$ are then
obtained as

$\displaystyle k={\sqrt{r_{3}-r_{1}}}\,\big{/}{2K_{m}}\,,$ (1.0a)
$\displaystyle l=qk\,,$ (1.0b) $\displaystyle\omega=(V+\sigma q^{2})k\,,$
(1.0c)

with

$V=\frac{\omega}{k}-\sigma q^{2}=2(r_{1}+r_{2}+r_{3})\,.$ (1.1)

In [4], the method of multiple scales was used to derive the so-called KP-
Whitham system, i.e., a system of quasilinear first-order PDEs that describes
the slow modulation of the above periodic solutions of the KP equation. One
begins by seeking a solution of (1.2) in the form $u=u(\theta,x,y,t)$, with
rapidly varying variable $\theta(x,y,t)$ defined through its derivatives:

$\theta_{x}={k(x,y,t)}/{\varepsilon},\qquad\theta_{y}={l(x,y,t)}/{\varepsilon},\qquad\theta_{t}=-{\omega(x,y,t)}/{\varepsilon}\,.$
(1.2)

Here, $k(x,y,t)$ and $l(x,y,t)$ are the local wave numbers in the $x$ and $y$
directions, respectively, and $\omega(x,y,t)$ is the wave’s local frequency.
Imposing the equality of the mixed second derivatives of $\theta$ results in
the compatibility conditions

$\displaystyle k_{t}+\omega_{x}$ $\displaystyle=0\,,$ (1.0a) $\displaystyle
l_{t}+\omega_{y}$ $\displaystyle=0\,,$ (1.0b) $\displaystyle k_{y}-l_{x}$
$\displaystyle=0\,,$ (1.0c)

called the “conservation of waves” equations. One also introduces the
dependent variable

$q(x,y,t)=\frac{l}{k}\,$ (1.1)

consistent with the above periodic solutions, along with the slowly varying
variables $x$, $y$ and $t$. It was then shown in [4] that to leading order one
recovers the solution (1.3). When the parameters of the above periodic
solution are slowly modulated with respect to $x$, $y$ or $t$, they satisfy a
system of Whitham modulation equations. When writing down these equations, it
is convenient to define the “convective derivative”

$\frac{D}{Dy}{}=\frac{\partial}{\partial{y}}-q\frac{\partial}{\partial{x}}~{}.$
(1.2)

In component form, the KP-Whitham system (KPWS) is then comprised of the
following partial differential equations (PDEs)

$\displaystyle\frac{\partial r_{i}}{\partial{t}}+(V_{j}+\sigma
q^{2})\frac{\partial r_{j}}{\partial{x}}+2\sigma
q\frac{Dr_{j}}{Dy}+\sigma\nu_{j}\frac{Dq}{Dy}+\sigma\frac{Dp}{Dy}=0\,,\qquad
j=1,2,3,$ (1.0a) $\displaystyle\frac{\partial q}{\partial{t}}+(V_{2}+\sigma
q^{2})\frac{\partial q}{\partial{x}}+2\sigma
q\frac{Dq}{Dy}+(4-\nu_{4})\frac{Dr_{1}}{Dy}+(2+\nu_{4})\frac{Dr_{3}}{Dy}=0\,,$
(1.0b) $\displaystyle\frac{\partial
p}{\partial{x}}-(1-\alpha)\frac{Dr_{1}}{Dy}-\alpha\frac{Dr_{3}}{Dy}+\nu_{5}\frac{\partial
q}{\partial{x}}=0\,,$ (1.0c) $\displaystyle
b_{1}\frac{Dr_{1}}{Dy}+b_{2}\frac{Dr_{2}}{Dy}+b_{3}\frac{Dr_{3}}{Dy}+b_{4}\frac{\partial
q}{\partial{x}}=0\,.$ (1.0d)

[Note that (1.0a) is a consequence of the three equations (1.0a), while (1.0b)
and (1.0c) are equivalent to (1.0b) and (1.0d), respectively.] Here,
$V_{1},\dots,V_{3}$ are the characteristic speeds of the Whitham system for
the KdV equation, namely

$\displaystyle V_{1}=V-2b\frac{K_{m}}{K_{m}-E_{m}}\,,\quad
V_{2}=V-2b\frac{(1-m)K_{m}}{E_{m}-(1-m)K_{m}}\,,\quad
V_{3}=V+2b\frac{(1-m)K_{m}}{mE_{m}}\,,$

and $b=2(r_{2}-r_{1})$ is the amplitude of the cnoidal wave solution (1.3),
while the remaining coefficients are

$\displaystyle\nu_{1}=\frac{V}{6}+\frac{b}{3m}\frac{(1+m)E_{m}-K_{m}}{K_{m}-E_{m}}\,,\quad\nu_{2}=\frac{V}{6}+\frac{b}{3m}\frac{(1-m)^{2}K_{m}-(1-2m)E_{m}}{E_{m}-(1-m)K_{m}}\,,$
(1.0a)
$\displaystyle\nu_{3}=\frac{V}{6}+\frac{b}{3m}\frac{(2-m)E_{m}-(1-m)K_{m}}{E_{m}}\,,\quad\nu_{4}=\frac{2mE_{m}}{E_{m}-(1-m)K_{m}}\,,$
(1.0b)
$\displaystyle\nu_{5}=r_{1}-r_{2}+r_{3}\,,\quad\alpha=\frac{E_{m}}{K_{m}}\,,\quad
b_{1}=(1-m)(K_{m}-E_{m}),$ (1.0c) $\displaystyle b_{2}=E_{m}-(1-m)K_{m},\quad
b_{3}=-mE_{m},\quad b_{4}=2(r_{2}-r_{1})(1-m)K_{m}$ (1.0d)

Importantly, the modulation system (1.2) contains six PDEs for the five
dependent variables $r_{1}$, $r_{2}$, $r_{3}$, $q$ and $p$, and is therefore
overdetermined in general. In [4], the initial value problem for the system
(1.2) was shown to be compatible provided that (1.0c) and (1.0d) hold at
$t=0$, in which case it was shown that (1.0c) and (1.0d) remain satisfied for
all $t>0$. Consequently, in [4] a reduced system consisting of the five PDEs
(1.0a)–(1.0c) was considered, which is a minimal set of equations for the five
dependent variables ${\mathbf{r}}=(r_{1},r_{2},r_{3},q,p)^{T}$ that can be
written as

$I_{4}\,\frac{\partial{\mathbf{r}}}{\partial{t}}+A_{5}\,\frac{\partial{\mathbf{r}}}{\partial{x}}+B_{5}\,\frac{\partial{\mathbf{r}}}{\partial{y}}=0\,,$
(1.1)

where $I_{4}=\mathop{\rm diag}\nolimits(1,1,1,1,0)$ and $A_{5}$ and $B_{5}$
are $5\times 5$ matrices whose explicit form is given in (A.1). In [4] and
[11] the term “KP-Whitham system” was used to refer to the five equations
(1.1). However, in this work we will show that, in order for the modulation
system to inherit the integrability properties of the KP equation, it is
crucial to consider all six equations (1.2) on an equal footing. Accordingly,
we will henceforth refer to the system (1.1) as the “original KPWS”, and we
will refer to the six equations (1.2) as the “full KPWS”.

Generally, all the dependent variables in (1.2) depend on two spatial
dimensions ($x$ and $y$) and one temporal dimension ($t$), so we refer to the
KPWS (1.2) as $(2+1)$-dimensional, or equivalently 3-dimensional. A number of
asymptotic reductions of the system (1.2) and their properties were studied in
[11], and $(1+1)$-dimensional (i.e., 2-dimensional) reductions of the soliton
limit of (1.2) were used in [30, 31, 32] to study various concrete physical
problems. A number of important questions remain open, however. Among them is
the issue of integrability. Since the KP-Whitham system was derived as an
exact asymptotic reduction of the KP equation (1.1), which is integrable, one
would naturally expect that the modulation system is also integrable. On the
other hand, as was mentioned in [4], the system (1.2a–c) fails the Haantjes
tensor test for integrability [19]. At the same time, it was shown in [11]
that the harmonic and soliton limits of the system (1.2) are in fact
integrable. An obvious question is then whether there are other integrable
reductions of (1.2) and if so how one can identify them.

In this work, we begin to address this question by studying and characterizing
the two-dimensional ($1+1$ and $2+0$) reductions of the KPWS (1.2). We
demonstrate that these reductions of the original KPWS are integrable only if
the full KPWS is compatible. In section 2 we consider the situation in which
all fields are independent of $x$, and in section 3 the situation in which all
fields are independent of $y$. In section 4 we study the situation in which
all fields are stationary, i.e., independent of $t$. Finally, in section 5 we
discuss the compatbility of the full KPWS, and in section 6 we conclude this
work with some final remarks. We emphasize that, as in [30, 32, 31], even when
the solution of the KPWS is independent of one independent variable, the
reduced systems of equations still generically describe two-dimensional,
dynamical configurations of the KP equation, because nonzero $V$ (1.1) implies
propagation of the cnoidal wave and $q$ describes the orientation of the
periodic wave in the $xy$ plane. Variations of $q$ with respect to $x$ or $y$
correspond to curved wave profiles. Section 6 ends this work with some
concluding remarks.

To avoid confusion, we note that in this work we are using the normalization
of [4], not that of [11, 31, 30, 32]. In the latter works, the coefficient 6
in front of the term $uu_{x}$ in (1.2) was absent and the cnoidal wave’s
period was normalized to $2\pi$, whereas here it is normalized to unity. As a
result, several formulas are adjusted accordingly.

##  2 The YT system

In this section we consider solutions of the KPWS in which all fields are
independent of $x$. We begin by considering the original KPWS (1.1) [i.e., the
five-component system (1.2a–c)], neglecting the compatibility condition (1.0d)
at first. When solutions are independent of $y$, (1.1) reduces to a system of
five PDEs in the independent variables $y$ and $t$, which we refer to as the
“YT system”. In vector form, this YT system is

$I_{4}\frac{\partial{\mathbf{r}}}{\partial{t}}+B_{5}\frac{\partial{\mathbf{r}}}{\partial{y}}=0\,,$
(2.1)

where ${\mathbf{r}}=(r_{1},r_{2},r_{3},q,p)^{T}$ and $I_{4}=\mathop{\rm
diag}\nolimits(1,1,1,1,0)$ as before, and the coefficient matrix $B_{5}$ is
given in (A.0l).

###  2.1 Reduction of the YT system to a three-component system

The number of equations in the system (2.1) can be reduced through a suitable
change of variables. Explicitly, the last row of (2.1) is

$\frac{\partial
r_{1}}{\partial{y}}+\alpha\frac{\partial}{\partial{y}}(r_{3}-r_{1})=0\,.$
(2.2)

Then, we use the transformation

$s_{1}=r_{3}-r_{2}\,,\quad s_{2}=r_{3}-r_{1}\,,\quad s_{3}=r_{2}-r_{1}\,$
(2.3)

where we made the choice not to define these variables in cyclic fashion in
order to preserve the property that $s_{j}\geq 0$ $\forall j=1,2,3$ when the
Riemann-type variables $r_{1},\dots,r_{3}$ are well-ordered. This
transformation leads to the set of equations

$\displaystyle\frac{\partial s_{j}}{\partial{t}}+2\sigma q\frac{\partial
s_{j}}{\partial{y}}+\sigma\Delta\nu_{j}\frac{\partial
q}{\partial{y}}=0\,,\quad j=1,2,3,$ (2.0a) $\displaystyle\frac{\partial
q}{\partial{t}}+6\frac{\partial r_{1}}{\partial{y}}+(\nu_{4}+2)\frac{\partial
s_{2}}{\partial{y}}+2\sigma q\frac{\partial q}{\partial{y}}=0\,,$ (2.0b)

with $\Delta\nu_{1}=\nu_{3}-\nu_{2}$, $\Delta\nu_{2}=\nu_{3}-\nu_{1}$ and
$\Delta\nu_{3}=\nu_{2}-\nu_{1}$. The transformation (2.3) is not invertible.
However, (2.0a) with $j=1$ is decoupled from the rest of the system, since
$s_{1}$ does not appear in the remaining equations. Thus we can simply
disregard it moving forward, since the four dependent variables $r_{1}$,
$s_{2}$, $s_{3}$ and $q$, determined by the PDEs (2.0a) with $j=2,3$ plus
(2.0b), are a closed system. These dependent variables, together with (2.2),
are sufficient to recover the solution of the KP equation.

Next, one can use (2.2) to eliminate $r_{1}$ from (2.0b), obtaining the
following closed system of three PDEs for the three dependent variables
$s_{2}$, $s_{3}$ and $q$:

$\displaystyle\frac{\partial s_{2}}{\partial{t}}+2\sigma q\frac{\partial
s_{2}}{\partial{y}}+\sigma(\nu_{3}-\nu_{1})\frac{\partial
q}{\partial{y}}=0\,,$ (2.0a) $\displaystyle\frac{\partial
s_{3}}{\partial{t}}+2\sigma q\frac{\partial
s_{3}}{\partial{y}}+\sigma(\nu_{2}-\nu_{1})\frac{\partial
q}{\partial{y}}=0\,,$ (2.0b) $\displaystyle\frac{\partial
q}{\partial{t}}+(\nu_{4}-6\alpha+2)\frac{\partial s_{2}}{\partial{y}}+2\sigma
q\frac{\partial q}{\partial{y}}=0\,,$ (2.0c)

All the coefficients appearing in (2.1) are completely determined by $m$ and
$b$, which in turn are completely determined by $s_{2},s_{3}\&q$ as

$m=\frac{s_{3}}{s_{2}}\,,\qquad\frac{b}{m}=2s_{2}\,.$ (2.1)

Note that $r_{1}$ is also needed to recover the asymptotic solution of the KP
equation, but its value, up to an integration constant determined by the
initial conditions, can be obtained from $s_{2}$, $s_{3}$ by integrating
(2.2). Introducing the vector ${\mathbf{v}}=(s_{2},s_{3},q)^{T}$, we can write
the above system (2.1) in vector form as

$\frac{\partial{\mathbf{v}}}{\partial{t}}+B_{3}\frac{\partial{\mathbf{v}}}{\partial{y}}=0\,,$
(2.2)

with

$B_{3}=\left(\begin{array}[]{ccc}2\sigma q&0&\sigma(\nu_{3}-\nu_{1})\\\
0&2\sigma q&\sigma(\nu_{2}-\nu_{1})\\\ \nu_{4}-6\alpha+2&0&2\sigma
q\end{array}\right)\,.$ (2.3)

The eigenvalues of $B_{3}$ are

$\lambda_{1}=2\sigma q,\qquad\lambda_{2,3}=2\sigma q\pm\sqrt{\Delta}\,,$
(2.0a)

where

$\Delta=\sigma\left(\nu_{1}-\nu_{3}\right)\left(6\alpha-\nu_{4}-2\right)=4\sigma
s_{2}\frac{((1-m)K_{m}-2(2-m)E_{m}K_{m}+3E_{m}^{2})^{2}}{3E_{m}K_{m}(K_{m}-E_{m})(E_{m}-(1-m)K_{m})}\,.$
(2.0b)

By properties of $K_{m}$ and $E_{m}$, $\mathop{\rm
sgn}\nolimits\Delta=\mathop{\rm sgn}\nolimits\sigma$ since $s_{2}>0$. Hence,
if $\sigma=-1$, as for KPI, some of the eigenvalues are imaginary, implying
that the initial value problem for the above system is ill-posed, confirming
known results [4]. Incidentally, note that the PDEs for $s_{2}$ and $q$ do not
contain $s_{3}$ explicitly. However, the value of $s_{3}$ is required to
determine $m$.

###  2.2 Harmonic and soliton limits of the YT system

We now consider two distinguished limits of the system outlined in (2.0a),
which describe concrete physical scenarios: the harmonic limit and the soliton
limit. The limiting values of all coefficients of the KPWS in these limits are
found in the Appendix.

The harmonic limit, in which the elliptic parameter $m\to 0$, is obtained by
taking $r_{2}\to r_{1}^{+}$. In this limit, the cnoidal wave becomes a
vanishing-amplitude trigonometric wave (cf. section 1 and the well-known
properties of the elliptic functions [28]). Note $s_{3}\to 0$ in this limit,
consistent with the fact that $m$ is obtained from $s_{2}$ and $s_{3}$ via
(2.1). The harmonic limit of system (2.0a) results in the partially decoupled
system

$\displaystyle\frac{\partial s_{2}}{\partial{t}}+2\sigma q\frac{\partial
s_{2}}{\partial{y}}=0\,,$ (2.0a) $\displaystyle\frac{\partial
q}{\partial{t}}+2\sigma q\frac{\partial q}{\partial{y}}=0\,,$ (2.0b)

with identical equations for the variables $s_{2}\&q$. In the limit $m\to 0$
of (1.3), the mean flow is simply $\bar{u}=r_{3}$. Moreover, (2.2) implies
that, in the harmonic limit, $\partial{r_{3}}/\partial{y}\to 0$. Thus the mean
flow is constant with respect to all spatial variables, but it does not play
any role in the system. Moreover, taking into account the reduction of (2.2),
we see that the system (2.2) coincides with the harmonic limits in [11] and
[4] when derivatives with respect to $x$ are neglected.

The soliton limit, i.e., $m\to 1$, is obtained when $r_{2}\to r_{3}^{-}$,
which amounts to $s_{3}\to s_{2}$, and the cnoidal wave solution (1.3)
limiting to a line soliton. The first two equations of (2.2) are identical in
this limit, consistent with $s_{3}=s_{2}$, while the remaining equations are

$\displaystyle\frac{\partial s_{2}}{\partial{t}}+2\sigma q\frac{\partial
s_{2}}{\partial{y}}+\frac{4}{3}\sigma s_{2}\frac{\partial q}{\partial{y}}=0$
(2.0a) $\displaystyle\frac{\partial q}{\partial{t}}+4\frac{\partial
s_{2}}{\partial{y}}+2\sigma q\frac{\partial q}{\partial{y}}=0\,.$ (2.0b)

As before, (2.2) coincides with the reduction of the soliton limit in [4] and
[11] when derivatives with respect to $x$ are neglected. In this case, since
$\alpha\to 0$ as $m\to 1$, (2.2) implies $\partial r_{1}/\partial y=0$ and
$\partial{s_{2}}/\partial{y}=\partial{r_{3}}/\partial{y}$. Unlike the harmonic
limit, the two-component system (2.2) in the soliton limit is coupled. On the
other hand, the system can be diagonalized in a straightforward way. The
eigenvalues of the system and the associated left eigenvectors are,
respectively [11],

$\lambda_{\pm}=2\sigma q\pm 4\sqrt{{\sigma
s_{2}}/{3}}\,,\qquad{\mathbf{w}}_{\pm}=(\pm\sqrt{3},\sqrt{\sigma s_{2}})\,.$
(2.1)

Using these, we obtain the characteristic differential forms $\sqrt{3}{\rm
d}s_{2}\pm\sqrt{\sigma s_{2}}{\rm d}q=\sqrt{3\sigma/s_{2}}{\rm d}s_{2}\pm{\rm
d}q=0$, which yields the Riemann invariants

$R_{\pm}=q\pm 2\sqrt{3\sigma s_{2}}\,.$ (2.2)

In turn, the change of variable from $s_{2}$ and $q$ to $R_{\pm}$ transforms
the system (2.2) into diagonal form:

$\frac{\partial R_{\pm}}{\partial{t}}+\lambda_{\pm}\frac{\partial
R_{\pm}}{\partial{x}}=0\,.$ (2.3)

###  2.3 Integrability and Riemann invariant of the YT system

We now return to the three-component YT system (2.2). The Haantjes tensor test
for integrability of a system of hydrodynamic equations [19] (see also the
Appendix) is a relatively simple way to determine whether a strictly
hyperbolic system is diagonalizable. It is generally believed that asymptotic
reductions of an integrable system preserve integrability. While the KP
equation (1.2) is integrable, the five-component original KP Whitham system
(1.2a–c) fails the Haantjies tensor test, a necessary condition for the
integrability of three-dimensional quasi-linear systems [18]. At the same
time, it was shown in [11] that the three-dimensional harmonic and soliton
limits of the full system are indeed integrable, which then raises the natural
question of whether two-dimensional reductions of the KPWS are integrable.

The three-component YT system (2.2) fails the Haantjes tensor test, since 19
out of the 27 components of the Haantjes tensor vanish in general. The
Haantjes tensor does vanish in the limit $m\to 0$, but its entries diverge in
the limit $m\to 1$, even though, as we showed above, the limiting system can
be reduced to an integrable two-component system for $s_{2}$ and $q$.

The reason for this discrepancy, and the reason why the original KPWS and the
reduction (2.2) fails the Haantjes tensor test is that, in order for (1.2) or
(1.1) to be compatible, $q$ and $k$ must be related by condition (1.0c), i.e.,
$k_{y}=l_{x}$, or, equivalently, (1.2d). When the initial conditions do not
satisfy the compatibility condition (1.2d), the original KPWS does not
describe actual solutions of the KP equation, and that explains why the
original KPWS system may not be integrable despite the integrability of the KP
equation. Indeed, we show below that, once the compatibility condition
$k_{y}=l_{x}$ is enforced, the $x$-independent reduction of the full KPWS (the
YT system) is in fact integrable. Later, in sections 3 and 4, we will see how
similar considerations apply to the $y$-independent and $t$-independent
reductions of the KPWS.

The coefficient matrix $A$ in (A.0l) has the eigenvalue $2\sigma q$, which is
inherited by the matrix $B$ in (2.3). Using the corresponding left eigenvector
in (2.3), we have the following characteristic relation

$(\nu_{2}-\nu_{1}){\rm d}s_{2}-(\nu_{3}-\nu_{1}){\rm d}s_{3}=0\,,$ (2.4)

along ${\rm d}y/{\rm d}t=2\sigma q$. To integrate this differential form, we
eliminate $s_{3}$ in favor of $m$ using (2.1), to obtain

$(\nu_{2}-\nu_{1}-m(\nu_{3}-\nu_{1}))\,{\rm
d}s_{2}-s_{2}(\nu_{3}-\nu_{1})\,{\rm d}m=0\,.$ (2.5)

Multiplying by the integrating factor
$1/[s_{2}(\nu_{2}-\nu_{1}-m(\nu_{3}-\nu_{1}))]$ yields

$\frac{1}{s_{2}}{\rm
d}s_{2}+\bigg{(}\frac{1}{m}-\frac{E_{m}}{m(1-m)K_{m}}\bigg{)}{\rm d}m=0\,,$
(2.6)

and recalling the derivative of $K_{m}$ [cf. (A.0a)], we express the above
characteristic relation as

${\rm d}\big{[}\frac{1}{2}\log s_{2}-\log K_{m}\big{]}=0,$ (2.7)

which yields the Riemann invariant $R_{o}={\sqrt{s_{2}}}\,\big{/}{2K_{m}}$
that satisfies the PDE

$\frac{\partial R_{o}}{\partial{t}}+2\sigma q\frac{\partial
R_{o}}{\partial{y}}=0\,.$ (2.8)

In light of (1.0a) and (2.3), however, we see that $R_{o}\equiv k$! That is,
the Riemann invariant $R_{o}$ is just the local wavenumber $k$ in the $x$
direction. Moreover, the fact that $k$ is a Riemann invariant is deeply
connected with the integrability of the full KPWS. This is because, when all
fields are independent of $x$, the conservation of waves and compatibility
condition (1.2) immediately yield $k_{t}=k_{y}=0$, i.e., $k={}$const.
Enforcing the constancy of the local wavenumber $k$ (as needed to ensure the
compatibility of the full KPWS with the KP equation, as per the above
discussion) reduces the three-component YT system (2.2) to a two-component
system that is locally integrable. Any two-component system can always be
reduced to Riemann invariant form and is therefore always locally integrable
via the classical hodograph transform. We will see in sections 3 and 4 that a
similar phenomenon will also arise for the $y$-independent and $t$-independent
reductions of the KPWS.

###  2.4 Further reduction of the YT system and its diagonalization

Since the wavenumber $k$ is simultaneously a Riemann invariant of the three-
component YT system (2.2) and a conserved quantity for $x$-independent one-
phase solutions of the KP equation, we now derive a reduction of the YT system
in which $k$ is constant. We choose two different sets of dependent variables:
a reduced system for the dependent variables $(s_{2},q)^{T}$ and the reduced
system for the dependent variables $(m,q)^{T}$.

We begin by performing a change of variable from
${\mathbf{v}}=(s_{2},s_{3},q)^{T}$ to ${\tilde{\mathbf{v}}}=(s_{2},q,k)^{T}$.
We choose to keep $s_{2}$ as opposed to $s_{3}$ because $s_{2}$ never
vanishes, whereas $s_{3}\to 0$ in the harmonic limit. Then
${\tilde{\mathbf{v}}}$ satisfies the system
${\tilde{\mathbf{v}}}_{t}+{\tilde{B}}_{3}\,{\tilde{\mathbf{v}}}_{y}=0$, with
${\tilde{B}}_{3}=T^{-1}B_{3}T$ and $T=\big{(}{\partial
v_{i}}/{\partial{\tilde{v}}_{j}}\big{)}$ . One can verify that the new
coefficient matrix is block-diagonal, ${\tilde{B}}_{3}=\mathop{\rm
diag}\nolimits(B_{2},2\sigma q)$, with the $2\times 2$ matrix $B_{2}$ given by

$\displaystyle B_{2}=\left(\\!\begin{array}[]{cc}2\sigma q&2\sigma
s_{2}\displaystyle\frac{(1-m)K_{m}^{2}-2(2-m)K_{m}E_{m}+3E_{m}^{2}}{3E_{m}(E_{m}-K_{m})}\\\
\displaystyle
2+2\left(\frac{m}{E_{m}-(1-m)K_{m}}-\frac{3}{K_{m}}\right)E_{m}&2\sigma
q\end{array}\\!\right)\,.$ (2.11) (2.12)

Since compatibility of the KPWS requires $k_{t}=k_{y}=0$, we therefore
consider the compatible solution $k\equiv{}$const, which simplifies the
$3\times 3$ system
${\tilde{\mathbf{v}}}_{t}+{\tilde{B}}_{3}\,{\tilde{\mathbf{v}}}_{y}=0$ to the
following $2\times 2$ system for the dependent variable
${\mathbf{u}}=(s_{2},q)^{T}$:

${\mathbf{u}}_{t}+B_{2}\,{\mathbf{u}}_{x}=0\,.$ (2.13)

The coefficient matrix $B_{2}$ in (2.12) contains the elliptic parameter $m$,
which is defined implicitly in terms of $s_{2}$ and $k$ by the relation
(1.0a), implying

$\sqrt{s_{2}}=2k_{0}K_{m}\,,$ (2.14)

where $k_{0}$ is the positive constant value of $k$. We can solve for $m$ by
inverting $K_{m}$ via
$m=K_{m}^{-1}(\sqrt{s_{2}}/(2k_{0}),m)=1-\mathrm{dn}^{2}(\sqrt{s_{2}}/(2k_{0}),m)$,
where $\mathrm{dn}$ is a Jacobi elliptic function.

In light of (2.14), we see that (2.13) is actually a one-parameter family of
hydrodynamic type systems, parametrized by the constant value of $k_{0}$ via
(2.14). Equivalently, one can use (2.14) to express $s_{2}$ as a function of
$m$ and $k_{0}$. Note however that $q$ does not enter in the relation between
$s_{2}$ and $m$. One can therefore easily replace (2.13) with the equivalent
hydrodynamic system
${\tilde{\mathbf{u}}}_{t}+{\tilde{B}}_{2}\,{\tilde{\mathbf{u}}}_{x}=0$ for the
modified dependent variable ${\tilde{\mathbf{u}}}=(m,q)^{T}$.

We use the eigenvalues and eigenvectors of $B_{2}$ to complete the
diagonalization of the YT system. The eigenvalues $\lambda_{\pm}$ coincide
with $\lambda_{2,3}$ in (2.3), while the associated left eigenvectors are

${\mathbf{w}}_{\pm}=\big{(}\pm\sqrt{3\sigma
E_{m}(K_{m}-E_{m})/[s_{2}K_{m}(E_{m}-(1-m)K_{m}))]}\,,\,1\,\big{)}\,,$ (2.15)

leading to the characteristic relations

${\rm d}q\pm\sqrt{\frac{3\sigma
E_{m}(K_{m}-E_{m})}{s_{2}K_{m}(E_{m}-(1-m)K_{m})}}{\rm d}s_{2}=0$ (2.0a)

along the characteristic curves

${\rm d}y/{\rm d}t=2\sigma q\mp\sqrt{\Delta}\,,$ (2.0b)

with $\Delta$ as in (2.3). We now differentiate (2.14) and use the known
differential equations for $K_{m}$, (A.0a), to express ${\rm
d}s_{2}=s_{2}[(E_{m}-(1-m)K_{m})/(m(1-m)K_{m})]\,{\rm d}m$, thereby
simplifying (2.0a) to

${\rm d}q\pm\sqrt{\sigma}k_{0}f(m)\,{\rm d}m=0\,,$ (2.0a)

where

$f(m)=\frac{2\sqrt{3E_{m}(K_{m}-E_{m})(E_{m}-(1-m)K_{m})}}{m(1-m)\sqrt{K_{m}}}\,.$
(2.0b)

Therefore, Riemann invariants for the two-component hydrodynamic system for
${\tilde{\mathbf{u}}}$ are

$R_{\pm}=q\pm\sqrt{\sigma}k_{0}\,F(m)\,,$ (2.0a)

with

$F(m)=\int_{0}^{m}f(\mu)\,{\rm d}\mu\,.$ (2.0b)

Although we were unable to compute this integral in closed form, the above
expression of the Riemann invariants is the same as those for the p-system
modeling isentropic gas dynamics and nonlinear elasticity [33] where $f(m)$ is
related to the sound speed of the medium. A plot of $f(m)$ is shown in Fig. 1.
Note that $f(m)>0$ for all $m\in[0,1)$, and $\lim_{m\to 0^{+}}f(m)=\pi/2$.
However, $f(m)=2[1+2/(\log(1-m)-4\log 2)]^{1/2}/(1-m)+O(1)$ as $m\to 1^{-}$.
As a result, we also have $F(m)\to+\infty$ and $R_{\pm}\to\pm\infty$
logarithmically in that limit, implying that the limit $m\to 1^{-}$ of the
two-component system for $m$ and $q$ is singular. This is expected because
$k_{0}\equiv{}$ const $\neq 0$ is incompatible with the soliton limit, for
which $k\to 0$ [cf. (1.0a)]. Note that $k_{0}=0$, $m\to 1$ is a valid
reduction that corresponds to the zero amplitude limit $s_{2}\to 0$ for the
soliton modulation equations (2.0b). In Fig. 1 we also plot $F(m)$ as a
function of $m$, which can be used to obtain the relationship between $m$ and
$q$ satisfied by simple wave solutions of (2.13) with $R_{\pm}\equiv{}$const.

Figure 1: The quantities $\Delta(m)/(\sigma k_{0}^{2})$ (left), $f(m)$
(center) and $F(m)$ (right) as functions of $m$.

In closing, we return our attention to the eigenvalues $\lambda_{\pm}$ of the
system, given by $\lambda_{\pm}=2\sigma q\pm\sqrt{\Delta(m)}$ [cf. (2.3)],
where in light of (2.14), we can now express $\Delta$ as

$\Delta(m)=16\sigma
k_{0}^{2}K_{m}\frac{((1-m)K_{m}-2(2-m)E_{m}K_{m}+3E_{m}^{2})^{2}}{3E_{m}(K_{m}-E_{m})(E_{m}-(1-m)K_{m})}\,.$
(2.1)

Note that $\Delta(m)/(\sigma k_{0}^{2})\to 0$ as $m\to 0$ and
$\Delta(m)/(\sigma k_{0}^{2})\to+\infty$ as $m\to 1$ (cf. Fig. 1). When $m=0$,
$\lambda_{\pm}=2\sigma q$ and $R_{\pm}=q$ so that (2.13) exhibits a one-
component reduction in the harmonic limit to the inviscid Burgers equation. By
monotonicity of $\Delta(m)$, the reduced $2\times 2$ system is strictly
hyperbolic according to the following definition of strict hyperbolicity:
$\lambda_{+}=\lambda_{-}$ if and only if $R_{+}=R_{-}$ [17]. It can also be
shown that $({\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{m,q}\lambda_{\pm})\cdot{\mathbf{w}}_{\pm}\neq
0$, which implies that the system is genuinely nonlinear.

##  3 The XT system

Next we consider the reduction of the KPWS in which all fields are independent
of $y$. Similarly to section 2, at first we consider the five-component
original KPWS (1.1), ignoring the compatibility condition (1.0d). When
solutions are independent of $y$, this system reduces to what we call the “XT
system”, which in vector form is

$I_{4}\frac{\partial{\mathbf{r}}}{\partial{t}}+A_{5}\frac{\partial{\mathbf{r}}}{\partial{x}}=0\,,$
(3.1)

with ${\mathbf{r}}=(r_{1},r_{2},r_{3},q,p)^{T}$ as before, and $A_{5}$ given
in (A.0f). We will obtain analogous results to section 2 even though the
analysis of the systems and the physics they describe are significantly
different.

###  3.1 Reduction to a four-component system, harmonic and soliton limits

In this case, reducing the size of the system is much easier than in section
2, because when all derivatives in $y$ vanish, the last equation in (3.1)
determines $p$ in terms of $r_{1},r_{2},r_{3}$ by direct integration.
Substituting the resulting expression into the PDEs for $r_{1},r_{2},r_{3}$
leads to the reduced system

$\displaystyle\frac{\partial r_{j}}{\partial{t}}+(V_{j}-\sigma
q^{2})\frac{\partial r_{j}}{\partial{x}}+\sigma q^{2}(1-\alpha)\frac{\partial
r_{1}}{\partial{x}}+\sigma q^{2}\alpha\frac{\partial
r_{3}}{\partial{x}}+\sigma q(\nu_{5}-\nu_{j})\frac{\partial
q}{\partial{x}}=0\,,\quad j=1,2,3\,,$ (3.0a) $\displaystyle\frac{\partial
q}{\partial{t}}-q(4-\nu_{4})\frac{\partial
r_{1}}{\partial{x}}+q(2+\nu_{4})\frac{\partial
r_{3}}{\partial{x}}+(V_{2}-\sigma q^{2})\frac{\partial q}{\partial{x}}=0\,,$
(3.0b)

or equivalently, in vector form,

$\frac{\partial{\mathbf{r}}_{4}}{\partial{t}}+A_{4}\frac{\partial{\mathbf{r}}_{4}}{\partial{x}}=0\,,$
(3.1)

where now ${\mathbf{r}}_{4}=(r_{1},r_{2},r_{3},q)^{T}$ and

$A_{4}=\left(\begin{array}[]{cccc}V_{1}-\sigma q^{2}\alpha&0&\sigma
q^{2}\alpha&\sigma q(\nu_{5}-\nu_{1})\\\ \sigma q^{2}(1-\alpha)&V_{2}-\sigma
q^{2}&\sigma q^{2}\alpha&\sigma q(\nu_{5}-\nu_{2})\\\ \sigma
q^{2}(1-\alpha)&0&V_{3}+\sigma q^{2}(\alpha-1)&\sigma q(\nu_{5}-\nu_{3})\\\
-q(4-\nu_{4})&0&-q(2+\nu_{4})&V_{2}-\sigma q^{2}\end{array}\right)\,.$ (3.2)

The above system simplifies considerably in the harmonic and soliton limits.
The limiting values of all coefficients are found in the Appendix. In the
harmonic limit, $m\to 0$, the PDEs for $r_{1}$ and $r_{2}$ coincide, and we
can therefore choose the reduced set of dependent variables
${\mathbf{r}}_{3}=(r_{1},r_{3},q)^{T}$, obtaining

$\frac{\partial{\mathbf{r}}_{3}}{\partial{t}}+A_{3}^{(0)}\frac{\partial{\mathbf{r}}_{3}}{\partial{x}}=0\,,$
(3.3)

with

$A_{3}^{(0)}=\left(\begin{array}[]{ccc}12r_{1}-6r_{3}-\sigma q^{2}&\sigma
q^{2}&0\\\ 0&6r_{3}&0\\\ 0&6&12r_{1}-6r_{3}-\sigma q^{2}\end{array}\right)\,.$
(3.4)

Equation (3.3) coincides with the $y$-independent reduction of the systems in
[11], which were shown to be integrable. Conversely, in the soliton limit,
$m\to 1$, the PDEs for $r_{2}$ and $r_{3}$ coincide. Choosing again
${\mathbf{r}}_{3}=(r_{1},r_{3},q)^{T}$, we obtain (3.3), but with
$A_{3}^{(0)}$ replaced by

$A_{3}^{(1)}=\left(\begin{array}[]{ccc}6r_{1}&0&0\\\ \sigma
q^{2}&2r_{1}+4r_{3}-\sigma q^{2}&\frac{4\sigma q}{3}(r_{1}-r_{3})\\\
-2q&-4q&2r_{1}+4r_{3}-\sigma q^{2}\end{array}\right)\,.$ (3.5)

This system also coincides with the $y$-independent reduction of the systems
in [11], and is therefore integrable.

###  3.2 Riemann invariant, integrability, further reduction and
diagonalization of the XT system

The matrix $A_{4}$ has the eigenvalue

$\lambda_{o}=V_{2}-\sigma q^{2}\,,$ (3.6)

with associated left eigenvector

$\displaystyle{\mathbf{w}}_{o}=\big{(}q(1-m)(K_{m}-E_{m})\,,\,q(E_{m}-(1-m)K_{m})\,,\,qmE_{m}\,,\,-2m(1-m)(r_{3}-r_{1})K_{m}\,\big{)}\,.$
(3.7)

These expressions allow us to find a Riemann invariant and, in turn, to
partially diagonalize the XT system (3.1). In this case however, the
calculations are more complicated than those of section 2. We begin by
applying ${\mathbf{w}}_{o}^{T}$ to (3.1), obtaining the characteristic
relation

$\frac{1}{(r_{3}-r_{1})K_{m}}\bigg{(}\frac{K_{m}-E_{m}}{m}{\rm
d}r_{1}+\frac{E_{m}-(1-m)K_{m}}{(1-m)m}{\rm d}r_{2}-\frac{E_{m}}{1-m}{\rm
d}r_{3}\bigg{)}-2\frac{{\rm d}q}{q}=0\,$ (3.8)

along ${\rm d}{x}/{\rm d}{t}=V_{2}-\sigma q^{2}$ . To integrate this
differential form and find the Riemann invariant, we first eliminate $r_{2}$
in favor of $m$ using (1.4), implying ${\rm d}r_{2}=(1-m)\,{\rm
d}r_{1}+m\,{\rm d}r_{3}+(r_{3}-r_{1})\,{\rm d}m$, which yields

$\frac{{\rm
d}(r_{3}-r_{1})}{r_{3}-r_{1}}+\frac{E_{m}-(1-m)K_{m}}{m(1-m)K_{m}}\,{\rm
d}m-2\frac{{\rm d}q}{q}=0\,.$ (3.9)

Next, we eliminate $q$ in favor of $l$ using (1.1), implying ${\rm d}q={\rm
d}l/k-l\,{\rm d}k/k^{2}$, and we use (1.0a) to replace $k$ and express ${\rm
d}k={\rm d}(r_{3}-r_{1})/(4K_{m}\sqrt{r_{3}-r_{1}})-\sqrt{r_{3}-r_{1}}{\rm
d}K_{m}/(2K_{m}^{2})$. Using equation (A.0a), substituting and simplifying,
the resulting expression finally yields ${\rm d}l=0$. The Riemann invariant
$R_{o}$ in this case is nothing else but the wavenumber in the $y$ direction,
$R_{o}=l$, which satisfies the PDE

$\frac{\partial R_{o}}{\partial{t}}+(V_{2}-\sigma q^{2})\frac{\partial
R_{o}}{\partial{x}}=0\,.$ (3.10)

The fact that $l$ is a Riemann invariant for the XT system is not an accident,
and is related to its compatibility with the full KPWS and (as we will see
below) with its integrability. This is because, when all fields are
independent of $y$, the conservation of waves equations (1.2) yield
$l_{x}=l_{t}=0$, implying that, for one-phase solutions of the KP equation,
$l$ must be constant. We can use this relation to reduce the XT system (3.1)
to a three-component system. To do this, we perform a change of dependent
variable from ${\mathbf{r}}_{4}=(r_{1},r_{2},r_{3},q)^{T}$ to
${\mathbf{v}}=(r_{1},r_{2},r_{3},l)^{T}$, which results in the partially
decoupled system

${\mathbf{v}}_{t}+A_{4}^{\prime}\,{\mathbf{v}}_{x}=0\,,$ (3.11)

with

$A_{4}^{\prime}=\left(\begin{array}[]{cc}A_{3}&{\mathbf{a}}_{3}\\\
{\mathbf{0}}_{3}^{T}&V_{2}-\sigma q^{2}\end{array}\right)\,,\qquad$ (3.12)

where the three-component vector ${\mathbf{a}}_{3}$ is immaterial for our
purposes, ${\mathbf{0}}_{3}=(0,0,0)^{T}$ and

$\displaystyle A_{3}=A_{\mathrm{KdV}}+\frac{4\sigma
K_{m}l^{2}}{3(r_{2}-r_{1})}A_{3}^{(2)}\,,$ (3.0a) $\displaystyle
A_{\mathrm{KdV}}=\mathop{\rm diag}\nolimits(V_{1},V_{2},V_{3})\,,$ (3.0b)
$\displaystyle
A_{3}^{(2)}=\left(\begin{array}[]{ccc}2a_{3}&\frac{a_{1}a_{4}}{(1-m)(K_{m}-E_{m})}&\frac{mE_{m}a_{2}}{(1-m)(K_{m}-E_{m})}\\\\[1.72218pt]
\frac{-(K_{m}-E_{m})a_{4}}{a_{1}}&-\frac{2a_{2}}{1-m}&-\frac{E_{m}ma_{3}}{(1-m)a_{1}}\\\\[1.72218pt]
\frac{(K_{m}-E_{m})a_{2}}{E_{m}}&\frac{a_{1}a_{3}}{(1-m)E_{m}}&-\frac{2ma_{4}}{1-m}\end{array}\right)\,,$
(3.0f)

with $V_{1},\dots,V_{3}$ as in (1) and

$\displaystyle a_{1}=E_{m}-(1-m)K_{m}\,,\quad a_{2}=(1-m)K_{m}+(2m-1)E_{m}\,,$
(3.0g) $\displaystyle a_{3}=(1-m)K_{m}-(1+m)E_{m}\,,\quad
a_{4}=2(1-m)K_{m}-(2-m)E_{m}\,.$ (3.0h)

Since $l$ is constant for compatible solutions of the full KPWS, we can solve
the fourth equation in (3.11) by taking $l\equiv l_{0}$, thereby arriving at a
three-component system of the same form as (3.3)

$\frac{\partial{\mathbf{r}}_{3}}{\partial{t}}+A_{3}\,\frac{\partial{\mathbf{r}}_{3}}{\partial{x}}=0\,,$
(3.1)

except that now ${\mathbf{r}}_{3}=(r_{1},r_{2},r_{3})^{T}$ and the coefficient
matrix $A_{3}$ is given by (3.12). The case $l_{0}=0$ yields
$A_{3}=A_{\mathrm{KdV}}$, so the system (3.1) reduces exactly to the KdV-
Whitham system of Whitham modulation equations for the KdV equation [35]. We
have therefore showed that, once compatibility is enforced, the XT system
reduces to a one-parameter deformation of the KdV-Whitham system, parametrized
by the value of the wavenumber $l_{0}$ along the transverse dimension.

We now turn to the issue of the integrability of the XT system. We apply the
Haantjes tensor test to the four-component XT system (3.1) and find that it
fails. At the same time, the harmonic and soliton limits of the XT system
discussed earlier are clearly integrable, since they are exact one-dimensional
reductions of the the harmonic and soliton limits of the full KPWS, which were
shown to be integrable in [11]. On the other hand, the three-component reduced
system (3.1) does pass the Haantjes test, in that all the terms of its
Haantjes tensor associated with $A_{3}$ in (3.12) vanish identically. Thus,
while the system (3.1) is not integrable, the system (3.1) is an integrable,
one-parameter family of deformations of the KdV-Whitham system.

The harmonic limit (i.e., $r_{2}\to r_{1}$, implying $m\to 0$) of the deformed
three-component system (3.1) yields the coefficient matrix

$\lim_{r_{2}\to r_{1}}A_{3}=\left(\begin{array}[]{ccc}12r_{1}-6r_{3}-\sigma
q^{2}&0&\sigma q^{2}\\\ 0&12r_{1}-6r_{3}-\sigma q^{2}&\sigma q^{2}\\\
0&0&6r_{3}\end{array}\right)$ (3.2)

with $q^{2}=\pi^{2}l_{0}^{2}/(r_{3}-r_{1})$, which is consistent with the
system (3.3). On the other hand, like with the reduced YT system, the soliton
limit (i.e., $r_{2}\to r_{3}$, implying $m\to 1$) of the reduced XT system
(3.1) is singular for $l_{0}\neq 0$, since some of the entries of
$A_{3}^{(2)}$ diverge in that limit. Again, this is to be expected, because in
the soliton limit one has $l_{0}=0$ [cf. (1.0a)].

###  3.3 Deformed Riemann invariants

Since the one-parameter deformation (3.1) of the KdV-Whitham system is
integrable, it can be written in diagonal form. We have not been able to
determine the deformed Riemann invariants for all parameter values $l$ but we
can obtain approximate Riemann invariants for small $l$ following standard
methodology (e.g., see [20]). It is convenient to define the coefficient

$c_{2}=\frac{4K_{m}}{3m(r_{2}-r_{1})}\,,$ (3.3)

so that (3.12) reads $A_{3}=A_{\mathrm{KdV}}+A_{\mathrm{def}}$ and the
“deformation matrix” is simply $A_{\mathrm{def}}=\ell c_{2}A_{3}^{(2)}$ with
the (signed) deformation parameter

$\displaystyle\ell\equiv\sigma l_{0}^{2}.$ (3.4)

We seek an expansion of the deformed eigenvalues ${\tilde{V}}_{j}$ and
corresponding right eigenvectors ${\tilde{\mathbf{v}}}_{j}$ in powers of
$\ell$ as

$\displaystyle{\tilde{V}}_{j}$ $\displaystyle=V_{j}+\ell
V_{j}^{(2)}({\mathbf{r}}_{3})+O(\ell^{2}),\qquad{\tilde{\mathbf{v}}}_{j}$
$\displaystyle={\mathbf{e}}_{j}+\ell{\mathbf{v}}_{j}^{(2)}({\mathbf{r}}_{3})+O(\ell^{2}),\qquad
j=1,2,3,$ (3.5)

where ${\mathbf{r}}_{3}=(r_{1},r_{2},r_{3})^{T}$ are the unperturbed KdV-
Whitham Riemann invariants, the unperturbed speeds $V_{j}$ are given in (1),
and the unperturbed eigenvectors
${\mathbf{e}}_{1},{\mathbf{e}}_{2},{\mathbf{e}}_{3}$ are simply the canonical
basis in ${\mathbb{R}}^{3}$, i.e.,
$({\mathbf{e}}_{1},{\mathbf{e}}_{2},{\mathbf{e}}_{3})={\mathbb{I}}_{3}$, where
${\mathbb{I}}_{n}$ is the $n\times n$ identity matrix. We begin by computing
the perturbation to the characteristic speeds. Since
$A_{\mathrm{def}}=O(\ell)$, the first correction terms appear at $O(\ell)$.
The deformed eigenvalue problem is

$(A_{\mathrm{KdV}}+A_{\mathrm{def}})\,{\tilde{\mathbf{v}}}_{j}={\tilde{V}}_{j}{\tilde{\mathbf{v}}}_{j}\,,\qquad
j=1,2,3.$ (3.6)

The unperturbed eigenvalue problem, obtained at $O(1)$, is simply
$A_{\mathrm{KdV}}{\mathbf{e}}_{i}=V_{i}{\mathbf{e}}_{i}$, which is satisfied
because the KdV-Whitham system is the $\ell=0=l_{0}$ reduction of the XT
system (3.1). Collecting terms $O(\ell)$ in (3.6) yields

$(A_{\mathrm{KdV}}-V_{j}{\mathbb{I}}_{3}){\mathbf{v}}_{j}^{(2)}=(V_{j}^{(2)}-c_{2}A_{3}^{(2)}){\mathbf{e}}_{j}\,,$
(3.7)

and multiplying from the left by ${\mathbf{e}}_{j}^{T}$ yields the first-order
correction to the characteristic velocities as

$V_{j}^{(2)}={\mathbf{e}}_{j}^{T}c_{2}A_{3}^{(2)}{\mathbf{e}}_{j}\,,\qquad
j=1,2,3\,,$ (3.8)

since the ${\mathbf{e}}_{j}$ are orthonormal. Explicitly,

$\displaystyle{\tilde{V}}_{1}=V_{1}^{(0)}-2c_{2}\ell\left((1+m)E_{m}-(1-m)K_{m}\right)+O(\ell^{2}),$
(3.0a)
$\displaystyle{\tilde{V}}_{2}=V_{2}^{(0)}-2c_{2}\ell\frac{(1-m)K_{m}-(1-2m)E_{m}}{1-m}+O(\ell^{2}),$
(3.0b) $\displaystyle{\tilde{V}}_{3}=V_{3}^{(0)}+2c_{2}\ell
m\frac{(2-m)E_{m}-2(1-m)K_{m}}{1-m}+O(\ell^{2})\,.$ (3.0c)

Note that if $\sigma=1$, the corrections to the first and second
characteristic velocities are negative, while the correction to the third
characteristic velocity is positive (opposite if $\sigma=-1$). Note also that
all three corrections remain finite in the limit $m\to 0$, but diverge as
$m\to 1$, indicating that the harmonic limit of the deformed system is finite,
but its soliton limit is singular for $\ell\neq 0$. This is expected because
the soliton limit requires $l_{0}=0$, which is only true when the XT system
(3.1) reduces to the KdV-Whitham system. Next, we use (3.7) and (3.8) to
compute the correction to the eigenvectors. The matrix
$A_{\mathrm{KdV}}-V_{j}{\mathbb{I}}_{3}$ is singular for $j=1,2,3$.
Nonetheless, the inhomogeneous linear system admits a solution, which yields
the deformed eigenvectors in the form

$\displaystyle{\mathbf{{\tilde{v}}_{1}}}=\left(\\!\\!\begin{array}[]{c}1\\\
0\\\ 0\end{array}\\!\\!\right)+c_{2}\ell\left(\\!\\!\begin{array}[]{c}0\\\
1\\\
m\end{array}\\!\\!\right),\quad{\mathbf{{\tilde{v}}_{2}}}=\left(\\!\\!\begin{array}[]{c}0\\\
1\\\ 0\end{array}\\!\\!\right)+c_{2}\ell\left(\\!\\!\begin{array}[]{c}m-1\\\
0\\\
m\end{array}\\!\\!\right),\quad{\mathbf{{\tilde{v}}_{3}}}=\left(\\!\\!\begin{array}[]{c}0\\\
0\\\ 1\end{array}\\!\\!\right)+c_{2}\ell\left(\\!\\!\begin{array}[]{c}m-1\\\
1\\\ 0\end{array}\\!\\!\right),$ (3.19)

up to $O(\ell^{2})$ terms.

Next we compute the deformed Riemann invariants. It is non-trivial to find the
correct integrating factor for the deformed Riemann invariants. To circumvent
this issue, we take advantage of the fact that the total differential of each
Riemann invariant is zero along the associated characteristic curve. We expand
the deformed Riemann invariants
${\tilde{R}}_{1},{\tilde{R}}_{2},{\tilde{R}}_{3}$ as

${\tilde{R}}_{j}=r_{j}+\ell R^{(2)}_{j}({\mathbf{r}}_{3})+O(\ell^{2})\,,\qquad
j=1,2,3\,.$ (3.21)

We have

${\rm d}{\tilde{R}}_{j}=0$ (3.22)

along the characteristic curve ${\rm d}x/{\rm d}t={\tilde{V}}_{j}$, for
$j=1,2,3$. Expanding (3.22) yields

${\rm d}{\tilde{R}}_{j}={\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}{\tilde{R}}_{j}\cdot{\rm
d}{\mathbf{r}}={\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}{\tilde{R}}_{j}\cdot\bigg{(}\frac{\partial{\mathbf{r}}_{3}}{\partial{t}}{\rm
d}t+\frac{\partial{\mathbf{r}}_{3}}{\partial{x}}{\rm d}x\bigg{)}=0\,,\qquad
j=1,2,3\,,$ (3.23)

where ${\kern-0.83337pt\lower 0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower
0.20004pt\hbox{\kern 0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}=(\partial_{r_{1}},\partial_{r_{2}},\partial_{r_{3}})^{T}$.
Next we use (3.1) and (3.6) to rewrite (3.23) as

$({\kern-0.83337pt\lower 0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower
0.20004pt\hbox{\kern 0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}{\tilde{R}}_{j})^{T}({\tilde{V}}_{j}{\mathbb{I}}_{3}-A_{3})\frac{\partial{\mathbf{r}}_{3}}{\partial
x}\,{\rm d}t=0\,,\qquad j=1,2,3\,,$ (3.24)

along the curve ${\rm d}x/{\rm d}t={\tilde{V}}_{j}$. If the above differential
must be zero for all $\frac{\partial{\mathbf{r}}_{3}}{\partial x}$, one can
constrain each component to be zero, i.e.,

$({\kern-0.83337pt\lower 0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower
0.20004pt\hbox{\kern 0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}{\tilde{R}}_{j})^{T}({\tilde{V}}_{j}{\mathbb{I}}_{3}-A_{3})={\mathbf{0}}\,,\qquad
j=1,2,3\,.$ (3.25)

One can check that $\det({\tilde{V}}_{j}{\mathbb{I}}_{3}-A_{3})=O(\ell^{2})$
for $j=1,2,3$, which allows for a nontrivial solution at $O(\ell)$. For each
$j=1,2,3$, (3.25) yields a system of three differential equations for
$R^{(2)}_{j}$. Note that, even though it might not seem obvious a priori,
these differential equations must necessarily be compatible since we know that
the system is integrable and therefore admits Riemann invariants.

We present the calculations for $R^{(2)}_{1}$ in detail. Keeping terms up to
$O(\ell)$, the first equation in (3.25) is trivially satisfied, while the
remaining two equations are

$\displaystyle\frac{\partial R^{(2)}_{1}}{\partial{r_{2}}}$
$\displaystyle=\frac{(E_{m}-(1-m)K_{m})^{2}}{3(1-m)(r_{2}-r_{1})^{2}}\,,$
(3.0a) $\displaystyle\frac{\partial R^{(2)}_{1}}{\partial{r_{3}}}$
$\displaystyle=-\frac{m^{2}E_{m}^{2}}{3(1-m)(r_{2}-r_{1})^{2}}\,,$ (3.0b)

and one can check that the equality of the mixed second derivatives, namely
$\partial^{2}R^{(2)}_{1}/\partial{r_{2}}\partial{r_{3}}=\partial^{2}{R^{(2)}_{1}}\partial{r_{3}}\partial{r_{2}}$,
is indeed satisfied. Next, we need to integrate (3.3) to find $R^{(2)}_{1}$.
We can integrate the equations manually, employing a process akin to that of
finding a potential for a conservative vector field. We begin with (3.0b)
since it is simpler. Because of the presence of elliptic integrals, it is
convenient to perform a change of variables from $r_{1},r_{2},r_{3}$ to
$r_{1},r_{2}$ and $m$. Solving (1.4) for $r_{3}$ as a function of $m$, we have

$\frac{\partial R^{(2)}_{1}}{\partial{m}}=\frac{\partial
R^{(2)}_{1}}{\partial{r_{3}}}\frac{\partial
r_{3}}{\partial{m}}=\frac{E_{m}^{2}}{3(1-m)(r_{2}-r_{1})}\,.$ (3.1)

Integrating this equation (with $r_{1}$ and $r_{2}$ held constant) then yields
$R^{(2)}_{1}$ as

$\displaystyle R^{(2)}_{1}=\frac{g(m)}{3(r_{2}-r_{1})}\,,$ (3.0a)
$\displaystyle g^{\prime}(m)=\frac{E_{m}^{2}}{1-m}\,.$ (3.0b)

Note that we have taken the arbitrary function of $r_{1}$ and $r_{2}$ in
(3.0a) to be zero. By substituting (3.0a) into (3.0a) yields

$g(m)=-(1-m)K_{m}^{2}+2E_{m}K_{m}-E_{m}^{2}\,,$ (3.0c)

and one can confirm that (3.0c) is indeed compatible with (3.0b), which means
we have successfully integrated (3.3), obtaining the first approximate
deformed Riemann invariant of the XT system as

${\tilde{R}}_{1}=r_{1}+\frac{\ell}{3(r_{2}-r_{1})}(2K_{m}E_{m}-E_{m}^{2}-(1-m)K_{m}^{2})+O(\ell^{2})\,.$
(3.0a)

One can apply an identical process to find the remaining deformed Riemann
invariants as

$\displaystyle{\tilde{R}}_{2}=r_{2}+\frac{\ell}{4K_{m}}\bigg{(}2K_{m}E_{m}+\frac{E_{m}^{2}}{1-m}-(1-m)K_{m}^{2}\bigg{)}+O(\ell^{2})\,,$
(3.0b)
$\displaystyle{\tilde{R}}_{3}=r_{3}-\frac{\ell}{3(r_{2}-r_{1})}\frac{m((1-m)K_{m}^{2}-E_{m}^{2})}{1-m}+O(\ell^{2})\,.$
(3.0c)

The expressions of the deformed speeds and deformed Riemann invariants may
prove to be useful when investigating the dynamics of weakly slanted wave
fronts in the KP equation.

Figure 2: The functions $L_{\pm}(m)$ that determine the critical values
$m=m_{\pm}$ at which two characteristic velocities of the XT system coalesce.

###  3.4 Hyperbolicity

The hyperbolicity of the three-component reduction (3.1) of the full KPWS can
be determined by analyzing the eigenvalues of the coefficient matrix
$A_{3}({\mathbf{r}}_{3})$, given in (3.12), which are the characteristic
velocities $\tilde{V}_{j}$, $j=1,2,3$. Because the characteristic polynomial

$p(\lambda)=\det(A_{3}-\lambda{\mathbb{I}}_{3})=-\lambda^{3}+b_{2}\lambda^{2}+b_{1}\lambda+b_{0},$
(3.1)

is a cubic with real coefficients, it has either three real roots or one real
root and a complex conjugate pair. Equation (3.8) demonstrates that the
$\tilde{V}_{j}$ are real for all $r_{j}\in{\mathbb{R}}$ with $r_{1}\leq
r_{2}\leq r_{3}$ when the deformation parameter $\ell=\sigma l_{0}^{2}$ is
sufficiently small in magnitude. Because the coefficients $b_{0},b_{1},b_{2}$
in (3.1) are smooth functions of ${\mathbf{r}}_{3}$, a bifurcation from all
real roots to a complex conjugate pair can only occur if the discriminant of
$p(\lambda)$,

$D({\mathbf{r}}_{3})=4b_{1}^{3}+b_{2}^{2}b_{1}^{2}-18b_{0}b_{2}b_{1}-4b_{0}b_{2}^{3}-27b_{0}^{2}\,,$
(3.2)

is zero. To evaluate $D({\mathbf{r}}_{3})$, we first simplify the calculation
by restricting ourselves to the set

$S=\left\\{{\mathbf{r}}_{3}=(0,m,1)^{T}~{}|~{}0\leq m\leq 1\right\\}\,,$ (3.3)

and we simply write $D=D(m)$, which can be shown to be a quintic polynomial in
$\ell$ with coefficients depending on $m$. Setting $D(m)=0$ and solving for
$\ell$, one finds two complex conjugate solutions, which are not of interest,
plus two real solutions $\ell=L_{-}(m)$ and $\ell=L_{+}(m)$, the latter of
which is a double root. Explicitly,

$\displaystyle
L_{+}(m)=\frac{((2-m)E_{m}-2(1-m)K_{m})((1+m)E_{m}-(1-m)K_{m})((1-m)K_{m}-(1-2m)E_{m})}{3E_{m}(K_{m}-E_{m})(E_{m}-(1-m)K_{m})\left(2(2-m)K_{m}E_{m}-3E_{m}^{2}-(1-m)K_{m}^{2}\right)},$

while the expression for $L_{-}(m)$ is more complicated, so it is omitted for
brevity. The expansions of $L_{\pm}(m)$ for small $m$ are

$\displaystyle L_{+}(m)$ $\displaystyle=\frac{12}{\pi^{2}}(1-m)+O(m^{2}),$
(3.5) $\displaystyle L_{-}(m)$
$\displaystyle=-\frac{12}{\pi^{2}}\left(1-\frac{4}{3}(2m)^{2/3}-m\right)+O(m^{4/3}).$
(3.6)

Figure 2 shows that $L_{+}(m)>0$ and $L_{-}(m)<0$ for $m\in[0,1)$. Therefore,
there are two critical values of $m$: $\ell=\pm l_{0}^{2}=L_{\pm}(m_{\pm})$.
The fact that $\tilde{V}_{j}\in{\mathbb{R}}$ for $|\ell|$ sufficiently small
implies that $D(m)>0$ when $0<m<m_{\pm}$, namely the XT system is (strictly)
hyperbolic when $m\in(0,m_{\pm})$. When $m=m_{\pm}$, two characteristic
velocities coalesce.

In the case of the plus sign, the fact that $\ell=L_{+}(m_{+})$ is a double
root of $D(m)=0$ implies that ${\rm d}D/{\rm d}m\,|_{m_{+}}=0$. Then, in a
neighborhood of $m=m_{+}$, the discriminant (3.2) exhibits parabolic behavior

$\displaystyle
D(m)=\frac{1}{2}\left.\frac{\mathrm{d}^{2}D}{\mathrm{d}m^{2}}\right|_{m_{+}}(m-m_{+})^{2}+O(m-m_{+})^{3}\,,$
(3.7)

and it must be the case that $D(m)\geq 0$, i.e., ${\rm d}^{2}D/{\rm
d}m^{2}|_{m_{+}}>0$, because $D(m)>0$ for $0<m<m_{+}$. Since $m_{+}$ is the
only point at which $D(m)=0$, this implies $D(m)>0$ for
$m\in[0,m_{+})\cup(m_{+},1)$ and that the characteristic speeds are always
real. Indeed, a direct calculation shows that, when $m=m_{+}$,

$\displaystyle\tilde{V}_{1}$
$\displaystyle=\tilde{V}_{2}=\frac{6\left((1+m)E(m)^{2}-(1-m)^{2}K(m)^{2}-2(1-m)mK(m)E(m)\right)}{(1-m)K(m)^{2}-2(2-m)K(m)E(m)+3E(m)^{2}},$
(3.0a) $\displaystyle\tilde{V}_{3}$
$\displaystyle=\frac{2}{3}\left(3(m+1)+\frac{2mE(m)}{(m-1)K(m)+E(m)}+\frac{2mE(m)}{K(m)-E(m)}-\frac{2(1-m)K(m)}{E(m)}\right)$
(3.0b)

and there are three corresponding linearly independent eigenvectors
${\bf\tilde{v}_{j}}$, $j=1,2,3$. Consequently, we conclude that the XT system
is hyperbolic for $m\in(0,1)$ and strictly hyperbolic when $m\neq m_{+}$.

In the case of the minus sign, the critical point $m_{-}$ satisfying
$\ell=-l_{0}^{2}=L_{-}(m_{-})$ is a simple root of $D(m)$. Since $D(m)>0$ for
$0<m<m_{-}$ and $D$ depends upon $m$ smoothly, it necessarily is the case that
${\rm d}D/{\rm d}m\,|_{m_{-}}<0$, so that the discriminant (3.2) becomes
negative in a right neigborhood of $m=m_{-}$. This implies that, for
$m_{-}<m<1$, the XT system exhibits a complex conjugate pair of characteristic
speeds and is not hyperbolic.

The above discussion of hyperbolicity of the XT system was limited to the set
$S$ defined in (3.3), where it was observed that a bifurcation occurs at the
point $m=m_{\sigma}$. However, using the scaling symmetry $r_{j}(x,t)\to
a^{2}r_{j}(ax,a^{3}t)$, $q(x,t)\to aq(ax,a^{3}t)$ with
$a=(r_{3}-r_{1})^{-1/2}$ and the Galilean symmetry $r_{j}(x,t)\to
b+r_{j}(x-6bt,t)$, $q(x,t)\to q(x-6bt,t)$ with $b=-r_{1}$ [4], we can map any
vector ${\mathbf{r}}_{3}=(r_{1},r_{2},r_{3})^{T}\in{\mathbb{R}}^{3}$ with
$r_{1}\leq r_{2}\leq r_{3}$ to a vector
${\tilde{\mathbf{r}}_{3}}=(0,m,1)^{T}\in S$. For ${\mathbf{r}}_{3}$, the
bifurcation occurs on the surface

$\displaystyle\Sigma_{\pm}=\left\\{{\mathbf{r}}_{3}=(r_{1},r_{2},r_{3})^{T}~{}\Big{|}~{}\pm
l_{0}^{2}=L_{\pm}(m_{\pm})(r_{3}-r_{1})^{2},~{}~{}m_{\pm}=\frac{r_{2}-r_{1}}{r_{3}-r_{1}}\right\\}.$
(3.1)

In the case of the plus sign, the XT system is hyperbolic, and strictly so for
${\mathbf{r}}_{3}\notin\Sigma_{+}$. In the case of the minus sign, the XT
system is hyperbolic so long as $(r_{2}-r_{1})/(r_{3}-r_{1})<m_{-}$ where,
$L_{-}(m_{-})=-l_{0}^{2}(r_{3}-r_{1})^{2}$. When
$(r_{2}-r_{1})/(r_{3}-r_{1})>m_{-}$, the XT system loses hyperbolicity.

##  4 The XY system

###  4.1 KPWS in a comoving frame and the XY system

The third and final class of reductions of the KPWS we consider is that of
time-independent solutions, to be defined precisely below. While in Sections 2
and 3 we considered reductions that are evolutionary, exhibiting well-posed
initial value problems (at least when $\sigma=1$), here we are considering a
spatial problem, independent of $t$, for the modulations. As we will see, this
does not preclude dynamics in the full solution to KP itself. In the previous
sections we saw that, in order to ensure the compatibility of the XT and YT
reductions with the KP equation, one must make sure that all three
conservation of waves equations (1.2) are satisfied. We will see that this is
also the case with stationary reductions of the KPWS.

Note that, even though one may think that a more general scenario is obtained
by looking for traveling wave solutions, i.e., solutions that are stationary
in a traveling frame of reference $({\tilde{x}},{\tilde{y}},{\tilde{t}})$,
with ${\tilde{x}}=x-ct$, ${\tilde{y}}=y-dt$ and ${\tilde{t}}=t$, this is not
the case in practice. This is because the Galilean and pseudo-rotation
invariance of the KP equation allow one to perform appropriate transformations
of the dependent and independent variables to rewrite any traveling wave
solution of the KP equation as stationary in a suitable reference frame. Since
the KPWS preserves these invariances, the same transformations will also work
for the KPWS, see Appendix A.2 for details.

Based on the above discussion, consider situations in which the temporal
derivatives in the original KPWS (1.1) can be neglected, which then yields

$A_{5}\,\frac{\partial{\mathbf{r}}}{\partial{x}}+B_{5}\,\frac{\partial{\mathbf{r}}}{\partial{y}}=0\,.$
(4.1)

Contrary to the reductions discussed in 2 and 3, here the independence from
one of the coordinates does not automatically result in a reduction in the
number of degrees of freedom. That is, all five dependent variables appear in
(4.1). Assuming invertibility of $A_{5}$ and $B_{5}$, one could equivalently
write (4.1) as an evolutionary system with respect to either $x$ or $y$, e.g.,
as ${\mathbf{r}}_{x}+C\,{\mathbf{r}}_{y}=0$. However, the resulting
coefficient matrix $C=(A_{5})^{-1}B_{5}$ is quite complicated, and therefore
the resulting system is difficult to analyze.

###  4.2 Harmonic and soliton limits of the XY system

Similar to the XT and YT systems, the XY system admits finite harmonic and
soliton limits, in which case the system simplifies considerably.
Specifically, in the harmonic limit ($r_{2}\to r_{1}^{+}$, corresponding to
$m\to 0$), the PDEs for $r_{1}$ and $r_{2}$ coincide, and (4.1) reduces to a
four-component system of PDEs for the vector
${\mathbf{r}}^{\prime}=(r_{1},r_{3},q,p)^{T}$, in which the coefficient
matrices $A_{5}$ and $B_{5}$ are replaced by

$\displaystyle A_{4,o}=\left(\begin{array}[]{ccccc}12r_{1}-6r_{3}-\sigma
q^{2}&0&-qr_{3}\sigma&-q\sigma\\\ 0&6r_{3}-\sigma
q^{2}&-qr_{3}\sigma&-q\sigma\\\ 0&-6q&12r_{1}-6r_{3}-\sigma q^{2}&0\\\
0&q&r_{3}&1\end{array}\right),$ (4.0e) $\displaystyle
B_{4,o}=\left(\begin{array}[]{ccccc}2q\sigma&0&r_{3}\sigma&\sigma\\\
0&2q\sigma&r_{3}\sigma&\sigma\\\ 0&6&2\sigma q&0\\\
0&-1&0&0\end{array}\right).$ (4.0j)

Similarly, in the soliton limit ($r_{2}\to r_{3}^{-}$, corresponding to $m\to
1$), the PDEs for $r_{2}$ and $r_{3}$ coincide, and one obtains a four-
component system for ${\mathbf{r}}^{\prime}=(r_{1},r_{3},q,p)^{T}$, with the
matrices $A_{5}^{\prime}$ and $B_{5}^{\prime}$ replaced by

$\displaystyle A_{4,1}=\left(\begin{array}[]{ccccc}6r_{1}-\sigma
q^{2}&0&-qr_{1}\sigma&-q\sigma\\\ 0&2r_{1}+4r_{3}-\sigma
q^{2}&\frac{1}{3}q\sigma(r_{1}-4r_{3})&-q\sigma\\\
-2q&-4q&2r_{1}+4r_{3}-\sigma q^{2}&0\\\ q&0&r_{1}&1\end{array}\right),$ (4.0e)
$\displaystyle
B_{4,1}=\left(\begin{array}[]{ccccc}2q\sigma&0&r_{1}\sigma&\sigma\\\
0&2q\sigma&-\frac{1}{3}\sigma(r_{1}-4r_{3})&\sigma\\\ 2&4&2\sigma q&0\\\
-1&0&0&0\end{array}\right).$ (4.0j)

These systems coincide with the time-independent reduction of the harmonic and
soliton limits studied in [4, 11], where it was also shown that these systems
are integrable.

###  4.3 Riemann invariant, reduction and integrability

In light of what we learned by studying the YT and XT systems, we expect that,
when considering solutions that are independent of $t$, the frequency $\omega$
will be one of the Riemann invariants. Indeed, in this case the three
compatibility conditions (1.2) yield immediately $\omega_{x}=\omega_{y}=0$. We
now show that this expectation is correct. In this case, however, the
complexity of the system makes it impractical to use the direct approach based
on the use of the characteristic relations and left eigenvectors that was used
in the previous sections. We therefore use an alternative approach, based on
calculating the total differential of $\omega=\omega(\mathbf{r})$ as

${\rm d}\omega={\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}\omega\cdot{\rm
d}{\mathbf{r}}={\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}\omega\cdot\bigg{(}\frac{\partial{\mathbf{r}}}{\partial{x}}{\rm
d}x+\frac{\partial{\mathbf{r}}}{\partial{y}}{\rm d}y\bigg{)}\,,$ (4.0a)

with ${\kern-0.83337pt\lower 0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower
0.20004pt\hbox{\kern 0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}=(\partial_{r_{1}},\partial_{r_{2}},\partial_{r_{3}},\partial_{q},\partial_{p})^{T}$.
The evolution of $\omega$ as dictated by the system (4.1) along the
characteristic coordinates $dy/dx=\lambda$ is then

${\rm d}\omega={\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}\omega\cdot\big{(}\lambda{\mathbb{I}}_{5}-C\big{)}\,\frac{\partial{\mathbf{r}}}{\partial{y}}\,{\rm
d}x\,,$ (4.0b)

with $C=(A_{5})^{-1}B_{5}$. Computing ${\kern-0.83337pt\lower
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\lower 0.20004pt\hbox{\kern
0.50003pt\hbox{$\nabla$}}\kern-8.33336pt\raise
0.20004pt\hbox{$\nabla$}\kern-8.33336pt\raise
0.20004pt\hbox{\kern-0.50003pt\hbox{$\nabla$}}}_{{\mathbf{r}}}\omega$,
substituting in (4.0b) and setting ${\rm d}\omega=0$ then yields a linear
equation that determines the characteristic speed $\lambda$ as

$\lambda=\frac{2\sigma q}{V_{2}-\sigma q^{2}},$ (4.1)

with $V_{2}$ given in (1), which confirms that $\omega$ is indeed a Riemann
invariant for the system (4.1).

As per the above discussion, in order for the KPWS to be compatible, we must
enforce $\omega_{x}=\omega_{y}=0$. Following the procedures of section 2 and
3, we partially diagonalize the $XY$ system (4.1) by performing a dependent
coordinate transformation so that $\omega$ is one of the new dependent
variables. We then solve the resulting PDE by taking $\omega\equiv{}$const and
obtain a one-parameter family of reduced four-component systems, parametrized
by the constant value of $\omega$. Once again, however, the calculations are
more involved than in the previous cases.

The complication is that the expression (1.1) for $\omega$ does not allow one
to uniquely obtain any one of the dependent variables in terms of the others
(recall (1.0a)). The best one can do is to solve for $q$, which entails a
choice of sign:

$q=\pm\sqrt{\sigma\bigg{(}\frac{\omega}{k}-V\bigg{)}}\,.$ (4.2)

Following the same methods as in the previous sections, one can then obtain a
four-component hydrodynamic system of equations for
${\mathbf{v}}=(r_{1},r_{2},r_{3},p)^{T}$. A single coefficient matrix of the
form

$\frac{\partial{\mathbf{v}}}{\partial{x}}+C_{4}^{\prime}\,\frac{\partial{\mathbf{v}}}{\partial{y}}=0\,,$
(4.3)

is quite complicated. However, the system (4.1) can be transformed, using the
same methods, into the concise form

$A^{\prime}\frac{\partial{\mathbf{u}}}{\partial{x}}+B^{\prime}\frac{\partial{\mathbf{u}}}{\partial{y}}=0\,,$
(4.4)

where $A^{\prime}=AT$ and $B^{\prime}=BT$ and
${\mathbf{u}}=(r_{1},r_{2},r_{3},\omega,p)^{T}$. Once the PDE for $\omega$ is
disregarded, since $\omega\equiv{}$const solves it, we arrive at the system

$A_{4}^{\prime}\frac{\partial{\mathbf{p}}}{\partial{x}}+B_{4}^{\prime}\frac{\partial{\mathbf{p}}}{\partial{y}}=0\,,$
(4.5)

where ${\mathbf{p}}=(r_{1},r_{2},r_{3},p)^{T}$, the coefficient matrices are

$\displaystyle
A_{4}^{\prime}=\left(\begin{array}[]{cccc}q\left(\left(c_{1}+1\right)\nu_{1}+q^{2}(-\sigma)+V_{1}\right)&-\left(c_{1}+c_{2}-1\right)\nu_{1}q&\left(c_{2}+1\right)\nu_{1}q&-q^{2}\sigma\\\
\left(c_{1}+1\right)\nu_{2}q&-q\left(\left(c_{1}+c_{2}-1\right)\nu_{2}+q^{2}\sigma-
V_{2}\right)&\left(c_{2}+1\right)\nu_{2}q&-q^{2}\sigma\\\
\left(c_{1}+1\right)\nu_{3}q&-\left(c_{1}+c_{2}-1\right)\nu_{3}q&q\left(\left(c_{2}+1\right)\nu_{3}+q^{2}(-\sigma)+V_{3}\right)&-q^{2}\sigma\\\
-\sigma\left(c_{1}+1\right)\nu_{5}-\left((\alpha-1)q^{2}\right)&\sigma\left(c_{1}+c_{2}-1\right)\nu_{5}&\alpha
q^{2}-\sigma\left(c_{2}+1\right)\nu_{5}&q\end{array}\right)\,,$ (4.10)

and

$B_{4}^{\prime}=\left(\begin{array}[]{cccc}2\sigma
q^{2}-\left(c_{1}+1\right)\nu_{1}&\left(c_{1}+c_{2}-1\right)\nu_{1}&-\left(\left(c_{2}+1\right)\nu_{1}\right)&q\sigma\\\
-\left(\left(c_{1}+1\right)\nu_{2}\right)&2\sigma
q^{2}+\left(c_{1}+c_{2}-1\right)\nu_{2}&-\left(\left(c_{2}+1\right)\nu_{2}\right)&q\sigma\\\
-\left(\left(c_{1}+1\right)\nu_{3}\right)&\left(c_{1}+c_{2}-1\right)\nu_{3}&2\sigma
q^{2}-\left(c_{2}+1\right)\nu_{3}&q\sigma\\\ (\alpha-1)q&0&-\alpha
q&0\end{array}\right)\,,$ (4.12)

with

$c_{1}=\frac{\omega(E_{m}-K_{m})}{16mk^{3}K_{m}^{3}}\qquad\textrm{and}\qquad
c_{2}=\frac{\omega E_{m}}{16k^{3}K_{m}^{3}(1-m)}\,.$ (4.13)

Note that $q$ is present in the matrices for readability, however its
definition is given in (1.1) in terms of the constant parameter $\omega$ and
the riemann type variables $r_{j}$. Furthermore, one can use computer algebra
software to perform the Haantjies tensor test on the resulting system. Doing
so, we have verified that, as in the case of the XT and YT systems, the
Haantjies tensor of the reduced XY system does indeed vanish identically,
suggesting that the latter system is integrable as well. The reduced system
(4.5) admits hyperbolic or elliptic regimes depending on, for example, the
sign of the argument of the square root in (4.2).

The above reduced XY system possesses a finite harmonic limit, similarly to
those in the previous sections. Specifically, in the limit $r_{2}\to
r_{1}^{+}$, the PDEs for $r_{1}$ and $r_{2}$ coincide, and the four-component
system (4.5) reduces to a $3\times 3$ system in the independent variables $x$
and $y$ for the three-component dependent variable
${\mathbf{r}}_{3}=(r_{1},r_{3},p)^{T}$, with coefficient matrices

$\displaystyle
A_{3}=\left(\begin{array}[]{ccc}(h_{1}+16)r_{1}-\frac{3}{8}(3h_{1}+8)r_{3}&\frac{1}{4}(h_{1}+4)r_{3}&-\sqrt{\sigma}h_{2}\\\
r_{3}-\frac{h_{1}r_{3}}{8}&(h_{1}+4)r_{1}-\frac{3h_{1}r_{3}}{4}+9r_{3}&-\sqrt{\sigma}h_{2}\\\
\frac{(h_{1}-8)r_{3}}{8\sqrt{\sigma}h_{2}}&\frac{3(h_{1}-4)r_{3}-4(h_{1}+4)r_{1}}{4\sqrt{\sigma}h_{2}}&1\\\
\end{array}\right),$ (4.0d) $\displaystyle
B_{3}=\frac{1}{h_{2}}\left(\begin{array}[]{ccc}-\left(\sqrt{\sigma}(16(h_{1}+4)r_{1}+(40-17h_{1})r_{3})\right)&-2(h_{1}+4)r_{3}\sqrt{\sigma}&\sigma
h_{2}\\\
(h_{1}-8)r_{3}\sqrt{\sigma}&-2\sqrt{\sigma}(8(h_{1}+4)r_{1}+(20-7h_{1})r_{3})&\sigma
h_{2}\\\ 0&-h_{2}&0\\\ \end{array}\right),$ (4.0h)

where

$h_{1}=\frac{\pi\omega}{\sqrt{(r_{3}-r_{1})^{3}}},\qquad
h_{2}=8\sqrt{(h_{1}-2)r_{3}-(h_{1}+4)r_{1}}.$ (4.1)

Like with the XT and YT reductions, however, the system (4.5) does not admit a
finite soliton limit in general, since $\omega\to 0$ as $m\to 1$ [cf. (1.0a)],
which is incompatible with having $\omega=\mathrm{const}\neq 0$ in (4.5).

###  4.4 Stationary solutions of the KP equation and Whitham modulation system
for the Boussinesq equation

Importantly, even though the system (4.1) describes stationary solutions of
the KPWS, the corresponding solutions of the KP equation are not stationary,
unless $\omega=0$. On the other hand, if $\omega=0$, the modulated solutions
of the KP equation described by (4.1) are also stationary. This point is
relevant because stationary solutions of the KP equation (1.2) satisfy
versions of the Boussinesq equation, namely [5, 6]

$u_{\tau\tau}-c^{2}u_{xx}+\sigma(6uu_{x}+\varepsilon^{2}u_{xxx})_{x}=0\,,$
(4.2)

with $\tau=y$ and $c=0$. The case $\sigma=+1$ is the “good” Boussinesq
equation with real linear dispersion. Boussinesq derived the “bad” version
($\sigma=-1$) as a long-wavelength model of water waves, whose linearized
equation is ill-posed [16]. Therefore, the modulation system (4.5) with
$\omega=0$ is also the genus-1 Whitham modulation equations for the above
Boussinesq equations. This is noteworthy because the Boussinesq equations
(4.2) are associated with a $3\times 3$ Lax pair (e.g., see [5, 6]), which
significantly complicates the analysis, and as a result, the development of
Whitham modulation theory via, e.g., finite gap integration, has not been
formulated yet.

We point out that, when $\omega=0$, the modulation system (4.5) greatly
simplifies because $c_{1}=c_{2}=0$ and $q=\pm\sqrt{-\sigma V}$. Moreover, when
$\omega=0$, the system (4.5) remains well-defined both in the harmonic and the
soliton limits.

##  5 On the compatibility and integrability of the full KPWS

Recall that the first conservation of waves equation, namely (1.0a), is one of
the equations that eventually yield (1.0a) (as in the KdV equation), while the
second conservation of waves equation, namely (1.0b), yields the evolution
equation for $q$, namely, (1.0b). However, we have seen that the original,
five-component KPWS (1.1) is not automatically compatible with the third
conservation of waves equation, namely (1.0c). In this section, we investigate
the question of the compatibility and integrability of the full KPWS (1.2).
Specifically, we show that when fields are independent of $x$, $y$ or $t$ the
full six-component KPWS (1.2) becomes compatible, and one recovers the results
of the previous sections. The calculations in this section also provide an
alternative way to obtain those results.

###  5.1 Compatibility and integrability of the full YT system

We begin by studying the compatibility and integrability of the “full YT
system”, namely the reduction of the overdetermined, six-component KPWS (1.2)
when all fields are independent of $x$. As mentioned in section 2, under the
assumption that $k$, $l$ and $\omega$ do not depend on $x$, the closure
conditions (1.2) immediately imply that $k$ is independent of both $y$ and
$t$. For clarity, let us set $k=k_{0}$, with $k_{0}$ a real positive constant.
Then (1.0d) is satisfied trivially. Moreover, the relation $k=k_{0}$ provides
an algebraic constraint among the variables $r_{1}$, $r_{2}$ and $r_{3}$,
which implies that only two of them are independent. Writing the the resulting
system of equations in term of the variables $s_{2}$ and $s_{3}$ defined in
section 2, one then obtains (2.1) together with

$\displaystyle\frac{\partial r_{1}}{\partial{y}}+\alpha\frac{\partial
s_{2}}{\partial{y}}=0,$ (5.0a) $\displaystyle\sigma\frac{\partial
p}{\partial{y}}+\frac{\partial r_{1}}{\partial{t}}+2\sigma q\frac{\partial
r_{1}}{\partial{y}}+\sigma\nu_{1}\frac{\partial q}{\partial{y}}=0\,.$ (5.0b)

Altogether, (2.1) and (5.1) are a system of five equations for the dependent
variables $(s_{3},s_{2},q,r_{1},p)$, which is partially decoupled since the
variables $r_{1}$ and $p$ do not appear in (2.1). Hence, the system can be
solved for the variables $(s_{3},s_{2},q)$, and $r_{1}$ and $p$ obtained from
(5.0a) and (5.0b) by direct integration. Hence, we just need to focus on
equations (2.1) subject to the constraint (2.14).

Note, that, for fixed $k_{0}>0$ the algebraic equation (2.14) gives a one-
parameter family of functions of the form $s_{3}=s_{3}(s_{2};k_{0})$. (Here we
chose to view $s_{3}$ as a function of $s_{2}$, but the results are equivalent
if we interchange $s_{3}\leftrightarrow s_{2}$.) Observe that

$\frac{\partial s_{3}}{\partial{t}}=\frac{{\rm d}s_{3}}{{\rm
d}{s_{2}}}\frac{\partial s_{2}}{\partial{t}},\qquad\frac{\partial
s_{3}}{\partial{y}}=\frac{{\rm d}s_{3}}{{\rm d}{s_{2}}}\frac{\partial
s_{2}}{\partial{y}}.$

Substituting into (2.0b) and using (2.0a), one can verify that (2.0b) is
identically satisfied, which allows us to further reduce the analysis of the
system to the coupled equations (2.0a) and (2.0c). Before we proceed further
with the analysis of these equations, note that the constraint (2.14) can be
equivalently written as

$s_{3}=4k_{0}^{2}mK_{m}^{2}\,,$ (5.0a)

where we used the relation $s_{2}=s_{3}/m$ and the fact that $k_{0}$ is a
positive constant. The advantage of (5.0a) is that it also allows us to
express $s_{2}$ as

$s_{2}=4k_{0}^{2}K_{m}^{2}\,.$ (5.0b)

Equations (5.1) show that $m$ is in fact a “natural” variable for
parametrising both $s_{3}$ and $s_{2}$. Therefore, we now aim to replace
(2.0a) with a corresponding equation containing $m$ and $q$, which is promptly
achieved by noting that

$\frac{\partial m}{\partial{t}}=\frac{{\rm d}m}{{\rm d}{s_{2}}}\frac{\partial
s_{2}}{\partial{t}}\qquad\frac{\partial m}{\partial{y}}=\frac{{\rm d}m}{{\rm
d}{s_{2}}}\frac{\partial s_{2}}{\partial{y}},$

with

$\frac{{\rm d}m}{{\rm
d}{s_{2}}}=\frac{m(1-m)}{4k_{0}^{2}K_{m}(E_{m}-(1-m)K_{m})}.$

Substituting the above expressions into (2.0a) and (2.0c), we obtain the two-
component system

$\displaystyle\frac{\partial m}{\partial{t}}+2\sigma q\frac{\partial
m}{\partial{y}}+\sigma\Phi_{1}(m)\frac{\partial q}{\partial{y}}=0,$ (5.0a)
$\displaystyle\frac{\partial q}{\partial{t}}+2\sigma q\frac{\partial
q}{\partial{y}}+k_{0}^{2}\Phi_{2}(m)\frac{\partial m}{\partial{y}}=0,$ (5.0b)

where

$\displaystyle\Phi_{1}(m)=\frac{2m(1-m)K_{m}\left(3E_{m}^{2}-2(2-m)E_{m}K_{m}+(1-m)K_{m}^{2}\right)}{3E_{m}(E_{m}-K_{m})\left(E_{m}-(1-m)K_{m}\right)}\,,$
(5.0a)
$\displaystyle\Phi_{2}(m)=\frac{8\left(3E_{m}^{2}-2(2-m)E_{m}K_{m}+(1-m)K_{m}^{2}\right)}{m(1-m)}\,.$
(5.0b)

The system (5.1) explicitly contains the constant parameter $k_{0}$, which
cannot be eliminated by a rescaling of the dependent and independent
variables. The system (5.1), which is integrable, can be brought into the
diagonal form

$\frac{\partial R_{\pm}}{\partial{t}}=\lambda_{\pm}\frac{\partial
R_{\pm}}{\partial{y}},\qquad i=1,2$ (5.1)

where the characteristic speeds $\lambda_{\pm}$ (i.e., the eigenvalues of the
$2\times 2$ coefficient matrix associated with the system (5.1)), which are
the same as for (2.0b), are now expressed as

$\lambda_{\pm}=2\sigma q\pm k_{0}\sqrt{\sigma\Phi_{1}(m)\Phi_{2}(m)}\,,$

and the associated Riemann invariants, which also define the change of
variables $(m,q)\mapsto(R_{+},R_{-})$, and which are the same as (2.0a), are
now expressed as

$R_{\pm}=q\pm
k_{0}\int_{0}^{m}\sqrt{\frac{\Phi_{2}(\mu)}{\sigma\Phi_{1}(\mu)}}\,{\rm
d}\mu\,.$

It is well known that systems of the form (5.1) are integrable by the
hodograph method, and the general solution $(R_{+}(y,t),R_{-}(y,t))$ is given
by (locally, i.e., in a neighbourhood of points where
$\partial{R_{\pm}}/\partial{y}\neq 0$)

$x+\lambda_{\pm}(R_{+},R_{-})t+w_{\pm}(R_{+},R_{-})=0,$

where $w_{\pm}((R_{+},R_{-}))$ are solutions of the following system of linear
PDEs:

$\frac{1}{w_{+}-w_{-}}{\frac{\partial
w_{\pm}}{\partial{R_{\mp}}}}=\frac{1}{\lambda_{+}-\lambda_{-}}{\frac{\partial\lambda_{\pm}}{\partial{R_{\mp}}}}\,.$

One can look for further reductions by looking for solutions such that
$m=m_{0}$, with $m_{0}$ a constant value in the interval $[0,1]$. Under this
assumption, equation (5.0a) implies the constraints $\Phi_{1}(m_{0})=0$. The
only solutions to this constraint arise when $m_{0}=0$ and $m_{0}=1$, i.e., in
the harmonic and soliton limit, respectively. However, the case $m_{0}=1$
where $r_{2}=r_{3}$ in the soliton limit needs to be treated separately, since
the system (5.1) has been derived under the assumption that $k_{0}>0$ whereas
the condition $r_{2}=r_{3}$ implies that $k=k_{0}=0$.

###  5.2 Compatibility and integrability of the full XT system

Next we consider the reduction of the six-component full KPWS (1.2) when
fields are independent of $y$. Imposing that $k$, $l$ and $\omega$ are
$y-$independent, the closure conditions (1.2) immediately imply that $l$ is
constant. Hence, setting $l=l_{0}$, with $l_{0}$ a fixed real constant, we can
write

$q=\frac{l_{0}}{k}=\frac{2l_{0}K_{m}}{\sqrt{r_{3}-r_{1}}},$ (5.2)

implying that $q$ is functionally dependent on $r_{1}$, $r_{2}$ and $r_{3}$.
The corresponding reduction of the full KPWS (1.2) then coincides with (3.1),
which reads in component form as

$\displaystyle\frac{\partial r_{i}}{\partial{t}}+(V_{i}+\sigma q^{2}-2\sigma
q^{2})\frac{\partial r_{i}}{\partial{x}}-\sigma\nu_{i}q\frac{\partial
q}{\partial{x}}-\sigma q\frac{\partial p}{\partial{x}}=0\qquad i=1,2,3$ (5.0a)
$\displaystyle\frac{\partial q}{\partial{t}}+(V_{2}+\sigma q^{2}-2\sigma
q^{2})\frac{\partial q}{\partial{x}}-(4-\nu_{4})q\frac{\partial
r_{1}}{\partial{x}}-(2+\nu_{4})q\frac{\partial r_{3}}{\partial{x}}=0$ (5.0b)
$\displaystyle\frac{\partial p}{\partial{x}}+(1-\alpha)q\frac{\partial
r_{1}}{\partial{x}}+\alpha q\frac{\partial
r_{3}}{\partial{x}}+\nu_{5}\frac{\partial q}{\partial{x}}=0.$ (5.0c)

Expanding the derivatives in (5.0b) as

$\frac{\partial q}{\partial{t}}=\sum_{i=1}^{3}\frac{\partial
q}{\partial{r_{i}}}\frac{\partial r_{i}}{\partial{t}},\qquad\frac{\partial
q}{\partial{x}}=\sum_{i=1}^{3}\frac{\partial q}{\partial{r_{i}}}\frac{\partial
r_{i}}{\partial{x}},$

where $q=q(r_{1},r_{2},r_{3})$ is given by (5.2), and substituting the
expressions for $\partial{p}/\partial{x}$ and $\partial{r_{i}}/\partial{t}$
for $i=1,2,3$ obtained from the remaining equations (5.0a) and (5.0c), one can
directly check that equation (5.0b) is identically satisfied. Therefore, the
reduction (5.2) of the full KPWS reduces to equations (5.0a) and (5.0c), which
are equivalent to the $3\times 3$ diagonalisable sytem (3.1) plus the equation
(5.0c). This equation, given the solutions $r_{1}$, $r_{2}$ and $r_{3}$ of the
system (3.1), allows one to recover $p$ by direct integration.

###  5.3 Compatibility and integrability of the full XY system

Similarly to the previous reductions, if $k$, $l$ and $\omega$ do not depend
on $t$, the closure conditions (1.2) immediately imply that $\omega$ is
constant. Then we set $\omega=\omega_{0}$, where $\omega_{0}$ is a real
constant. Hence, the definition of $\omega$ (1.0c) implies

$q^{2}=\sigma\left(\frac{\omega_{0}}{k}-V\right),$

i.e., $q=q(r_{1},r_{2},r_{3})$ is a function of the variables $r_{1}$,
$r_{2}$, $r_{3}$ only. Then, when all fields are independent of $t$, the full
six-component KPWS (1.2) reduces to (4.1), which, in component form, is

$\displaystyle(V_{i}-\sigma q^{2})\frac{\partial
r_{i}}{\partial{x}}-q\nu_{i}\frac{\partial q}{\partial{x}}-\sigma
q\frac{\partial p}{\partial{x}}+2\sigma q\frac{\partial
r_{i}}{\partial{y}}+\sigma\nu_{i}\frac{\partial
q}{\partial{y}}+\sigma\frac{\partial p}{\partial{y}}=0\qquad i=1,2,3,$ (5.0a)
$\displaystyle(V_{2}-\sigma q^{2})\frac{\partial
q}{\partial{x}}-(4-\nu_{4})q\frac{\partial
r_{1}}{\partial{x}}-(2+\nu_{4})q\frac{\partial r_{3}}{\partial{x}}+2\sigma
q\frac{\partial q}{\partial{y}}+(4-\nu_{4})\frac{\partial
r_{1}}{\partial{y}}+(2+\nu_{4})\frac{\partial r_{3}}{\partial{y}}=0,$ (5.0b)
$\displaystyle\frac{\partial p}{\partial{x}}+(1-\alpha)q\frac{\partial
r_{1}}{\partial{x}}+\alpha q\frac{\partial
r_{3}}{\partial{x}}+\nu_{5}\frac{\partial
q}{\partial{x}}-(1-\alpha)\frac{\partial
r_{1}}{\partial{y}}-\alpha\frac{\partial r_{3}}{\partial{y}},$ (5.0c)
$\displaystyle\frac{\partial k}{\partial{y}}-q\frac{\partial
k}{\partial{x}}-k\frac{\partial q}{\partial{x}}=0\,.$ (5.0d)

Rearranging (5.0a) and (5.0c) with respect to the $y$-derivatives of $r_{1}$,
$r_{2}$, $r_{3}$ and $p$, and substituting into the remaining equations, we
verify that, under the assumptions above, both equations (5.0b) and (5.0d) are
identically satisfied. Therefore, the system (5.3) reduces to a diagonalizable
system of four equations for the variables $r_{1}$, $r_{2}$, $r_{3}$ and $p$.

##  6 Concluding remarks

In conclusion, in this work we investigated the two-dimensional reductions of
the KPWS (1.2) obtained when all fields are independent of one of the spatial
or temporal coordinates. We have also seen that, even though the reductions of
the original five-component KPWS (1.1) are not integrable, adding the sixth
equation, namely (1.0d), which enforces the compatibility with the
conservation of waves, results in an additional constant of motion, which not
only makes the reductions of the full KPWS compatible, but it also makes each
reduction integrable.

The fact that the original KPWS (1.1) is not integrable might seem surprising,
since it is an asymptotic reduction of the KP equation, which is integrable.
It is important to realize, however, that not all solutions of the KPWS (1.1)
describe modulated solutions of the KP equation. This is because not all
solutions of the KPWS (1.1) automatically satisfy the third conservation of
waves $k_{y}=l_{x}$. In other words, the original KPWS (1.1) describes
modulated one-phase solutions of the KP equation only if its initial
conditions are such that this condition is satisfied at $t=0$ [4].

Turning to the full, six-component KPWS (1.2), in general one does not expect
an overdetermined quasi-linear system to be either compatible or integrable,
so some mechanism of enforcing compatibility is required. In our previous
work, we enforced the compatibility, and thereby obtained integrable systems,
by considering the harmonic or soliton limit, either of which results in a
reduction in the number of modulation equations. In this work, we added to the
catalog of integrable reductions of the KPWS by characterizing two-dimensional
reductions of the KPWS.

The results of this work and the above discussion lead to the natural issue of
whether there are other integrable reductions of the KPWS, and whether it is
possible to identify all such integrable reductions. In other words, the
question is whether it is possible to identify suitable conditions that ensure
that the full KPWS is compatible. We plan to investigate this question in
future work.

We reiterate that, even though both the reduced YT, XT and XY systems admit a
finite harmonic limit, none of these systems admits a well-defined soliton
limit in general. However, setting the constant values of $k$, $l$, or
$\omega$ for the YT, XT, or XY system, respectively, to zero does result in
well-defined soliton limits.

We should also mention that one could equivalently carry out all calculations
by replacing the PDE for $q$ with the following simplified PDE, as derived in
[4, 3]:

$\frac{\partial q}{\partial{t}}+(V+\sigma q^{2})\frac{\partial
q}{\partial{x}}+\frac{D}{Dy}(V+\sigma q^{2})=0\,.$

For brevity, however, we omit the details.

Finally, we reiterate that the XY reduction of the KPWS allowed us to
explicitly obtain the Whitham modulation system for the Boussinesq equation,
which had not been derived before. It is hoped that this novel system will
prove to be as useful as the other reductions of the KPWS.

Another potential application of the results of this work are to situations in
which initial or boundary data for the KPWS are chosen to be independent of
one independent variable. In order to use the reduced YT, XT, or XY systems,
the soliton limit will not be available except in specialized situations,
namely when $k\equiv 0$, $l\equiv 0$, or $\omega\equiv 0$. Nevertheless, one
interesting class of problems are generalized Riemann problems consisting of
abrupt transitions between two periodic traveling waves. The reduced KPWS
obtained here could be used to study certain generalized Riemann problems.

## Acknowledgments

The authors thank the Isaac newton Institute for Mathematical Sciences for its
support and hospitality during the program Dispersive Hydrodynamics when parts
of the present work were undertaken. This work was supported by: EPSRC Grant
Number EP/R014604/1. GB and AB were partially supported by the National
Science Foundation under grant number DMS-2009487. MH was partially supported
by NSF under the grant DMS-1816934.

## Appendix

###  A.1 Coefficients matrices, harmonic and soliton limits, relations between
elliptic integrals

The coefficient matrices $A_{5}$ and $B_{5}$ of the KPWS (1.1) are:

$\displaystyle A_{5}=\left(\begin{array}[]{ccccc}V_{1}-\sigma
q^{2}&0&0&-\sigma\nu_{1}q&-\sigma q\\\ 0&V_{2}-\sigma
q^{2}&0&-\sigma\nu_{2}q&-\sigma q\\\ 0&0&V_{3}-\sigma
q^{2}&-\sigma\nu_{3}q&-\sigma q\\\ -(4-\nu_{4})q&0&-(2+\nu_{4})q&V_{2}-\sigma
q^{2}&0\\\ -(1-\alpha)q&0&\alpha q&0&0\end{array}\right)\,,$ (A.0f)
$\displaystyle B_{5}=\left(\begin{array}[]{ccccc}2\sigma
q&0&0&\sigma\nu_{1}&\sigma\\\ 0&2\sigma q&0&\sigma\nu_{2}&\sigma\\\
0&0&2\sigma q&\sigma\nu_{3}&\sigma\\\ 4-\nu_{4}&0&2+\nu_{4}&2\sigma q&0\\\
1-\alpha&0&\alpha&0&0\end{array}\right)\,.$ (A.0l)

The definitions of all the coefficients appearing in (A.1) are given in (1)
through (1).

Next, for convenience, we list the limiting values of the coefficients
appearing in the harmonic and soliton limits of the KP-Whitham system, since
these coefficients appear in all reductions. Recal that, in the harmonic
limit, the elliptic parameter $m$ tends to $0$ and $r_{2}\mapsto r_{1}^{+}$.
In this limit, the various coefficients then become

$\displaystyle m=0\,,\quad V=4r_{1}+2r_{3}\,,$ (A.0a) $\displaystyle
V_{1}=V_{2}=12r_{1}-6r_{3}\,,\quad V_{3}=6r_{3}\,,$ (A.0b)
$\displaystyle\nu_{1}=\nu_{2}=\nu_{3}=\nu_{5}=r_{3},\quad\nu_{4}=4,\quad\alpha=1\,.$
(A.0c)

Conversely, in the soliton limit the elliptic parameter tends to $1$ and
$r_{2}\mapsto r_{3}^{-}$. The limiting value of the various coefficients in
this case is

$\displaystyle m=1\,,\quad V=r_{1}+2r_{3}\,,$ (A.0a) $\displaystyle
V_{1}=6r_{1}\,,\quad V_{2}=V_{3}=2r_{1}+4r_{3}\,,$ (A.0b)
$\displaystyle\nu_{1}=\nu_{5}=r_{1}\,,\quad\nu_{2}=\nu_{3}={\textstyle\frac{1}{3}}(4r_{3}-r_{1})\,,\quad\nu_{4}=2\,,\quad\alpha=0\,.$
(A.0c)

In this work we use the elliptic parameter $m$ as opposed to the elliptic
modulus $k$. Recall that the two are related as $m=k^{2}$. The complementary
modulus is then simply ${k^{\prime}}^{2}=1-k^{2}=1-m$. While this choice is in
line with modern works, it differs from the convention in [28] and its
associated references. Thus the various ODEs from[28] must be transformed
accordingly. Specifically, the derivatives of $K$ and $E$ with respect to the
elliptic parameter $m$ are

$\displaystyle\frac{{\rm d}K_{m}}{{\rm
d}{m}}=\frac{E_{m}-(1-m)K_{m}}{2m(1-m)}\,,$ (A.0a) $\displaystyle\frac{{\rm
d}E_{m}}{{\rm d}{m}}=\frac{E_{m}-K_{m}}{2m}\,.$ (A.0b)

In addition, we have

$\frac{{\rm d}^{2}E_{m}}{{\rm d}{m}^{2}}=-\frac{1}{2m}\frac{{\rm d}K_{m}}{{\rm
d}{m}}\,.$ (A.1)

###  A.2 Invariances, traveling wave and stationary solutions of the KP
equation and KPWS

Here we show how, using the invariances of the KP equation and the KPWS, one
can map all traveling wave solutions of the KP equation and the KPWS (i.e.,
solutions that are stationary in a comoving reference frame) into solutions
that are stationary in a slanted but fixed reference frame.

To begin, it is useful to consider how the KPWS (1.1) with coefficient
matrices $I_{4}$, $A_{5}$ and $B_{5}$ is affected by affine transformations of
the independent variables. Recall first that the KP equation (1.2) is
invariant under Galilean boosts,

$\displaystyle u(x,y,t)\mapsto u^{\prime}(x,y,t)=c+u(x^{\prime},y,t)\,,$
(A.0a) $\displaystyle v(x,y,t)\mapsto v^{\prime}(x,y,t)=v(x^{\prime},y,t)\,,$
(A.0b)

with $x^{\prime}=x-6ct$, and “pseudo-rotations”,

$\displaystyle u(x,y,t)\mapsto
u^{\prime}(x,y,t)=u(x^{\prime},y^{\prime},t)\,.$ (A.0a) $\displaystyle
v(x,y,t)\mapsto
v^{\prime}(x,y,t)=v(x^{\prime},y^{\prime},t)+au(x^{\prime},y^{\prime},t)\,,$
(A.0b)

with $x^{\prime}=x+ay-\sigma a^{2}t$ and $y^{\prime}=y-2\sigma at$, and with
$a$ and $c$ arbitrary real parameters. Namely, if the $u(x,y,t)$ and
$v(x,y,t)$ comprises any solution of the KP equation, so does the pair
$u^{\prime}(x,y,t)$ and $v^{\prime}(x,y,t)$. Also recall that the above
transformations are mapped respectively into

$\displaystyle{\mathbf{r}}(x,y,t)\mapsto{\mathbf{r}}^{\prime}(x,y,t)={\mathbf{1}}_{3}c+{\mathbf{r}}(x^{\prime},y,t),\qquad$
(A.0a) $\displaystyle q(x,y,t)\mapsto q^{\prime}(x,y,t)=q(x^{\prime},y,t),$
(A.0b) $\displaystyle p(x,y,t)\mapsto
p^{\prime}(x,y,t)=p(x^{\prime},y,t)-cq(x^{\prime},y,t)\,,$ (A.0c)
$\displaystyle{\mathbf{r}}(x,y,t)\mapsto{\mathbf{r}}^{\prime}(x,y,t)={\mathbf{r}}(x^{\prime},y^{\prime},t),\quad$
(A.0d) $\displaystyle q(x,y,t)\mapsto
q^{\prime}(x,y,t)=a+q(x^{\prime},y^{\prime},t),$ (A.0e) $\displaystyle
p(x,y,t)\mapsto p^{\prime}(x,y,t)=p(x^{\prime},y^{\prime},t).$ (A.0f)

with $x^{\prime}=x+ay-\sigma a^{2}t$ and $y^{\prime}=y-2\sigma at$. Finally,
recall that both of these transformations leave the original KPWS (1.1)
invariant [4]. Namely, if ${\mathbf{r}}(x,y,t)$, $q(x,y,t)$ and $p(x,y,t)$ are
any solutions of the KPWS, so are ${\mathbf{r}}^{\prime}(x,y,t)$,
$q^{\prime}(x,y,t)$ and $p^{\prime}(x,y,t)$.

We now show that, using the above invariances, all one- and two-phase
traveling wave solutions of the KP equation can be transformed to a stationary
reference frame. These are the solutions of the KP equation that can be
written in the form

$\displaystyle u(x,y,t)=U(z_{1},z_{2})\,,$ (A.0a) $\displaystyle
z_{n}=k_{n}x+l_{n}y-\omega_{n}t,\quad n=1,2\,.$ (A.0b)

We show below that this formulation includes both classes of non-resonant
elastic two-soliton solutions, the genus-2 solutions, as well as the Miles
resonance solution, the one-soliton solutions and the genus-1 solutions as
special cases. Starting with the two-phase solution (A.2), we apply a pseudo-
rotation and Galilean boost, to obtain the new solution

$\displaystyle u^{\prime}(x,y,t)=c+U(z_{1}^{\prime},z_{2}^{\prime})\,,$ (A.0a)
$\displaystyle
z_{n}^{\prime}=k_{n}x+l_{n}^{\prime}y-\omega_{n}^{\prime}t\,,\quad
l_{n}^{\prime}=l_{n}-ak_{n}\,,\quad\omega_{n}^{\prime}=\omega_{n}+(6c+\sigma
a^{2})k_{n}+2\sigma al_{n}\,,\quad$ (A.0b)

for $n=1,2$. The new solution $u^{\prime}(x,y,t)$ is obviously stationary if
$\omega_{1}^{\prime}=\omega_{2}^{\prime}=0$. In turn, it is trivial to see
that it is always possible to achieve
$\omega_{1}^{\prime}=\omega_{2}^{\prime}=0$ by choosing

$\displaystyle
a=-\frac{k_{2}\omega_{1}-k_{1}\omega_{2}}{2\sigma(k_{2}l_{1}-k_{1}l_{2})}\,,$
(A.0a) $\displaystyle
c=\frac{4\sigma\,(k_{1}l_{2}-k_{2}l_{1})(l_{2}\omega_{1}-\,l_{1}\omega_{2})+(k_{2}\omega_{1}-k_{1}\omega_{2})^{2}}{24\sigma(k_{2}l_{1}-k_{1}l_{2})^{2}}\,.$
(A.0b)

[Note that the denominators in (A.2) are always non-zero for genuine two-phase
solutions. Conversely, if $k_{2}l_{1}=k_{1}l_{1}$ the expression
$u(x,y,t)=U(z_{1},z_{2})$ describes a one-phase solution, in which case it is
sufficient to simply apply a Galilean boost.]

By definition, the two-phase representation (A.2) obviously includes all the
genus-2 solutions of the KP equation (e.g., see [8]), of which the genus-1
solutions are a special case. It should then be clear that both of the non-
resonant elastic two-soliton solutions as well as the Miles resonance solution
and the one-soliton solutions are also included (since the former are obtained
as a degeneration of the genus-2 solutions [1, 2], and the latter are in turn
a degeneration of the former [9]). Nonetheless, we can give a simple proof of
this fact. Recall that general soliton solutions of the KP equation can be
obtained through the Wronskian formalism as [10]

$\displaystyle u(x,y,t)=6\,\frac{\partial^{2}}{\partial
x^{2}}[\log\tau(x,y,t)]\,,\qquad\tau(x,y,t)=\mathop{\rm
Wr}\nolimits(f_{1},\dots,f_{N})\,,$ (A.0a) $\displaystyle
f_{n}(x,y,t)=\sum_{m=1}^{M}C_{n,m}{\rm
e}^{\theta_{m}}\,,\quad\theta_{m}(x,y,t)=K_{m}x+\sqrt{3}\,K_{m}^{2}y-4K_{m}^{3}t\,,\quad
m=1,\dots,M\,.$ (A.0b)

In particular, the Miles resonance solution is obtained by taking $N=1$ and
$M=3$, so that $\tau(x,y,t)={\rm e}^{\theta_{1}}+{\rm e}^{\theta_{2}}+{\rm
e}^{\theta_{3}}$, and the two classes of non-resonant elastic two-soliton
solutions are obtained by taking $N=2$ and $M=4$ and the following: (i) for
the “ordinary” two soliton solutions, $f_{1}={\rm e}^{\theta_{1}}+{\rm
e}^{\theta_{2}}$ and $f_{2}={\rm e}^{\theta_{3}}+{\rm e}^{\theta_{4}}$; (ii)
for the “asymmetric” two soliton solutions, $f_{1}={\rm e}^{\theta_{1}}-{\rm
e}^{\theta_{4}}$ and $f_{2}={\rm e}^{\theta_{2}}+{\rm e}^{\theta_{3}}$. The
Miles resonance solution is then cast in the framework of (A.2) by simply
writing $\tau(x,y,t)={\rm e}^{\theta_{1}}\,(1+{\rm e}^{z_{1}}+{\rm
e}^{z_{2}})$, with $z_{1}=\theta_{2}-\theta_{1}$ and
$z_{2}=\theta_{3}-\theta_{1}$, since the common factor ${\rm e}^{\theta_{1}}$
disappears from the solution (because the $\theta_{j}$ are linear in $x$)
[10]. Similarly, for the ordinary two-soliton solution we have
$\tau(x,y,t)={\rm e}^{\theta_{1}+\theta_{3}}+{\rm
e}^{\theta_{1}+\theta_{4}}+{\rm e}^{\theta_{2}+\theta_{3}}+{\rm
e}^{\theta_{2}+\theta_{4}}=2{\rm
e}^{\frac{1}{2}(\theta_{1}+\theta_{2}+\theta_{2}+\theta_{4})}(\cosh
z_{1}+\cosh z_{2})$, where
$z_{1}=\frac{1}{2}(\theta_{1}+\theta_{3}-\theta_{2}-\theta_{4})$ and
$z_{2}=\frac{1}{2}(\theta_{2}+\theta_{3}-\theta_{1}-\theta_{4})$, and a
similar representation works for the asymmetric two-soliton solution.

Finally, to complete our proof, we now show that no solutions containing more
than two independent phases can be traveling wave solutions of the KP
equation. (In fact, this statement applies to general nonlinear evolution
equations in two spatial dimensions.) To see this, consider a generic
$N$-phase solution $u(x,y,t)=U(z_{1},\dots,z_{N})$, with $z_{n}$ still given
by (A.0b) for $n=1,\dots,N$. If $u(x,y,t)$ is a traveling wave solution, there
exists a coordinate transformations $(x,y,t)\mapsto(X,Y,T)$ with $X=x-ct$,
$Y=y-dt$ and $T=t$, such that $u(x,y,t)=u^{\prime}(X,Y)$. But the
transformation yields $z_{n}=k_{n}(X+ct)+l_{n}(Y+dt)-\omega_{n}t$, so in order
for $u^{\prime}(X,Y)$ to be independent of $T$, we need $c$ and $d$ such that

$k_{n}c+l_{n}d=\omega_{n}\,,\qquad n=1,\dots,N\,.$ (A.1)

If $N=1$, there are an infinite number of solutions to (A.1). (In particular,
one can set $d=0$ and take $c=\omega_{1}/k_{1}$.) If $N=2$, (A.1) admits a
unique solution, given by
$c=(\omega_{1}l_{2}-\omega_{2}l_{1})/(k_{1}l_{2}-k_{2}l_{1})$ and
$d=-(\omega_{1}k_{2}-\omega_{2}k_{1})/(k_{1}l_{2}-k_{2}l_{1})$. If $N>2$,
however, the system (A.1) is overdetermined, and no solution exists. (Here we
assume that all phases are truly independent, which implies
$k_{n}l_{n^{\prime}}-k_{n^{\prime}}l_{n}\neq 0$ for all
$n,n^{\prime}=1,\dots,N$ with $n\neq n^{\prime}$. If this condition is
violated, one can express the same solution with a smaller number of
independent phases.)

###  A.3 Haantjes tensor test for integrability

An efficient criterion to test the diagonalizabiliy for a hydrodynamic system
that does not require the computation of the eigenvalues and eigenvectors of
the coefficient matrix was outlined in [19], involving the vanishing of the
Haantjes tensor associated with the coefficient matrix. Specifically, for
strictly hyperbolic systems, [19] gives the the following theorem as a
necessary condition for diagonalizability: “A hydrodynamic type system with
mutually distinct characteristic speeds is diagonalizable if and only if the
corresponding Haantjes tensor is identically zero.”

The calculation of the Haantjes tensor requires calculation of the Nijenhuis
tensor first. The Nijenhuis tensor of a matrix $M^{i}_{j}$ is defined as

$N^{i}_{jk}=M^{p}_{j}\partial_{u^{p}}M^{i}_{k}-M^{p}_{k}\partial_{u^{p}}M^{i}_{j}-M^{i}_{p}(\partial_{u^{j}}M^{p}_{k}-\partial_{u^{k}}M^{p}_{j})$
(A.2)

where $\partial_{u^{k}}=\partial/\partial{u^{k}}$. In our case, the matrix
$M^{i}_{j}$ is the corresponding coefficient matrix of the system for which
diagonlizability is being tested. Once the Nijenhuis tensor is known, the
Haantjes tensor can be obtained as

$H^{i}_{jk}=N^{i}_{pr}M^{p}_{j}M^{r}_{k}-N^{p}_{jr}M^{i}_{p}M^{r}_{k}-N^{p}_{rk}M^{i}_{p}M^{r}_{j}+N^{p}_{jk}M^{i}_{r}M^{r}_{p}\,.$
(A.3)

The calculation of the various tensors below as applied to the various systems
discussed in this work was performed using the Mathematica software package.

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