# On the Kaup-Broer-Kupershmidt systems

Christian Klein  Institut de Mathématiques de Bourgogne, UMR 5584;
Institut Universitaire de France
Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex, France
<EMAIL_ADDRESS>and  Jean-Claude Saut  Laboratoire de
Mathématiques, UMR 8628,
Université Paris-Saclay et CNRS
91405 Orsay, France<EMAIL_ADDRESS>
(Date: February 23rd, 2024)

###### Abstract.

The aim of this paper is to survey and complete, mostly by numerical
simulations, results on a remarkable Boussinesq system describing weakly
nonlinear, long surface water waves. It is the only member of the so-called
(abcd) family of Boussinesq systems known to be completely integrable.

In memoriam Thomas Kappeler (1953-2022)

##  1\. introduction

We are interested in this paper in a particular case of the so-called abcd
Boussinesq systems for surface water waves, see [7, 8, 9] 111Boussinesq [10]
was the first to derive a particular Boussinesq system, not in the class of
those studied here though. We refer to [21, 22, 23] for details and for an
excellent history of hydrodynamics in the nineteenth century.

(1.1) $\left\\{\begin{array}[]{l}\eta_{t}+\nabla\dot{\bf
v}+\epsilon\nabla\cdot(\eta{\bf v})+\mu[a\nabla\cdot\Delta{\bf
v}-b\Delta\eta_{t}]=0,\\\ {\bf
v}_{t}+\nabla\eta+\epsilon\frac{1}{2}\nabla|{\bf
v}|^{2}+\mu[c\nabla\Delta\eta-d\Delta{\bf v}_{t}]=0.\end{array}\right.$

Here $\eta=\eta(x,t),x\in{\mathbb{R}}^{d},d=1,2,t\in{\mathbb{R}}$ is the
elevation of the wave, ${\bf v}={\bf v}(x,t)$ is a measure of the horizontal
velocity, $\mu$ and $\epsilon$ are the small parameters (shallowness and
nonlinearity parameters respectively) defined as

$\mu=\frac{h^{2}}{\lambda^{2}},\quad\epsilon=\frac{\alpha}{h}$

where $\alpha$ is a typical amplitude of the wave, $h$ a typical depth and
$\lambda$ a typical horizontal wavelength.

In the Boussinesq regime, $\epsilon$ and $\mu$ are supposed to be of same
order, $\epsilon\sim\mu\ll 1,$ and we will take for simplicity $\epsilon=\mu,$
writing (1.1) as 222Particular cases are formally derived in [11, 24, 54].

(1.2) $\left\\{\begin{array}[]{l}\eta_{t}+\nabla\cdot{\bf
v}+\epsilon[\nabla\cdot(\eta{\bf v})+a\nabla\cdot\Delta{\bf
v}-b\Delta\eta_{t}]=0,\\\ {\bf
v}_{t}+\nabla\eta+\epsilon[\frac{1}{2}\nabla|{\bf
v}|^{2}+c\nabla\Delta\eta-d\Delta{\bf v}_{t}]=0,\end{array}\right.$

The coefficients (a, b, c, d) are restricted by the condition

$a+b+c+d=\frac{1}{3}-\tau,$

where $\tau\geq 0$ is the surface tension coefficient.

When restricted to one-dimensional, unidirectional motions, (1.2) leads to the
Korteweg-de Vries (KdV) equation, see [45]

$u_{t}+u_{x}+\epsilon\left(\frac{1}{3}-\tau\right)u_{xxx}+\epsilon uu_{x}=0.$

The class of systems (1.1), (1.2) models water waves on a flat bottom
propagating in both directions in the aforementioned regime (see [7, 8, 9]).

It turns out that two particular one-dimensional cases of the abcd systems
have remarkable properties. The first one, we will refer to as the Amick-
Schonbek system, can be viewed as a dispersive perturbation of the Saint-
Venant (shallow water) system333 Actually this system is a particular case of
a system derived by Peregrine in [54] but Schonbek and Amick were the first to
point out its remarkable mathematical properties.. We refer to [37] for a
review of known results together with new numerical simulations.

The second one, referred to as the Kaup-Broer-Kupershmidt system (KBK) was
introduced in [11, 12], as a surface long wave model (see equations (3.2),
(3.3) in [11]) and recently in [14] as an internal wave model. On the other
hand Kaup and Kupershmidt introduced it as an integrable system [35, 43]. It
corresponds in the abcd family to $a=\pm 1,b=c=d=0$ and writes

(1.3) $\left\\{\begin{array}[]{l}\eta_{t}+v_{x}+(\eta v)_{x}+\alpha
v_{xxx}=0,\\\ v_{t}+\eta_{x}+vv_{x}=0.\end{array}\right.$

$\alpha=1$ corresponds to the bad KBK system and $\alpha=-1$ to the good KBK
system. The two systems turn out to be integrable in a sense that will be
detailed below.

######  Remark 1.1.

When viewed as water wave model, the variable $\eta$ in (1.3) represents the
elevation of the wave, so that physically the total depth $\zeta=1+\eta$
should be positive. The non-cavitation condition $\zeta>0$ will be mostly
ignored in what follows. Note that in terms of $(v,\zeta)$ (1.3) writes

(1.4) $\left\\{\begin{array}[]{l}\zeta_{t}+(\zeta v)_{x}+\alpha v_{xxx}=0,\\\
v_{t}+\zeta_{x}+vv_{x}=0.\end{array}\right.$

Setting $U=v_{x}$ in $\eqref{KBKbis}_{1}$ and solving for $\zeta$ leads to the
Boussinesq like equation

(1.5) $U_{tt}+U_{t}U_{xx}+2U_{x}U_{xt}+\frac{3}{2}U_{x}^{2}U_{xx}-\alpha
U_{xxxx}=0.$

######  Remark 1.2.

The KBK system is sometimes written in terms of the velocity potential $\phi$
such that $\phi_{x}(x,t)=v(x,t)$:

(1.6)
$\left\\{\begin{array}[]{l}\eta_{t}+\phi_{xx}+(\eta\phi_{x})_{x}+\alpha\phi_{xxxx}=0,\\\
\phi_{t}+\eta+\frac{1}{2}(\phi_{x})^{2}=0.\end{array}\right.$

######  Remark 1.3.

The KBK system should not be confused with the so-called Kaup-Kupershmidt
equation

(1.7) $v_{t}=v_{xxxxx}+10vv_{xxx}+25v_{x}v_{xx}+20v^{2}v_{x},$

that was introduced in [34, 44], and which is the first equation in a
hierarchy (different from the KdV hierarchy) of integrable equations with Lax
operator $\partial_{x}^{3}+2u\partial_{x}+u_{x}$, see for instance [26]. We
are not aware of a physical application of the Kaup-Kupershmidt equation.

The paper is organized as follows. In section 2 we will describe general facts
on the KBK system. The next section reviews the results obtained by partial
differential equations (PDE) techniques while section 4 focusses on the
integrability side. In section 5 we introduce a numerical approach to the KBK
system and test it for the known soliton. In section 6 the known stability of
the solitons is illustrated. In section 7 the long time behavior of KBK
solutions for localised initial data is studied showing that the soliton
resolution conjecture can be applied. In section 8 we explore the formation of
dispersive shock waves in the vicinity of shocks of the corresponding
dispersionless Saint-Venant system. We add some concluding remarks in section
9.

##  2\. Generalities on the Kaup-Broer-Kupershmidt system

We recall here that the Kaup-Broer-Kupershmidt system is the one-dimensional
version of the 2-D (abcd Boussinesq) system

(2.1) $\left\\{\begin{array}[]{l}\eta_{t}+\nabla\cdot{\bf
v}+\epsilon\nabla\cdot(\eta{\bf v})+\alpha\epsilon\Delta\nabla\cdot{\bf
v}=0,\;\alpha=\pm 1\\\ {\bf v}_{t}+\nabla\eta+\frac{\epsilon}{2}\nabla|{\bf
v}|^{2}=0,\end{array}\right.$

which is linearly well-posed when $\alpha=-1.$ The local well-posedness of the
Cauchy problem is established in [36] (see also [58]). The more difficult
question of long time existence (that is on time scales of order
$O(1/\epsilon))$ is established in [58] under the non-cavitation condition
$1+\epsilon\eta>0.$

On the other hand (2.1) is ill-posed when $\alpha=+1.$ Note that the ill-posed
version can be turned into a well-posed system by using the BBM trick, leading
to

(2.2) $\left\\{\begin{array}[]{l}\eta_{t}+\nabla\cdot{\bf
v}+\epsilon\nabla\cdot(\eta{\bf v})-\epsilon\Delta\eta_{t}=0\\\ {\bf
v}_{t}+\nabla\eta+\epsilon\frac{1}{2}\nabla|{\bf v}|^{2}=0,\end{array}\right.$

for which the long time existence is established in [58].

We will from now on restrict to the one-dimensional case with most of the time
$\epsilon=1$. The linearized ”bad” KBK system ($\alpha=1$ in (1.3)) is
Hadamard ill-posed since the dispersion relation is $\omega^{2}=k^{2}-k^{4}$,
showing the short wave ill-posedness. It was established in [2] that
nonlinearity does not erase the problem, making the Cauchy problem ill-posed
in all Sobolev spaces in the sense that arbitrary small smooth solutions can
blow-up in arbitrary short time in Sobolev spaces norms.

######  Remark 2.1.

Following [43], the change of variable $v=v,\eta=h-v_{x}$ transforms the Kaup-
Broer-Kupershmidt system with $\alpha=1$ into the (ill-posed) system

(2.3) $\left\\{\begin{array}[]{l}h_{t}+v_{x}+h_{xx}+(vh)_{x}=0,\\\
v_{t}+h_{x}-v_{xx}+vv_{x}=0\end{array}\right.$

with dispersion $\omega(k)=\pm ik\sqrt{1-k^{2}}.$

On the other hand, in the well-posed case $\alpha=-1,$ the change of variable
$v=v,\eta=h+iv_{x}$ leads to the (linearly well-posed) ”Schrödinger like”
system

(2.4) $\left\\{\begin{array}[]{l}h_{t}+v_{x}-ih_{xx}+(vh)_{x}=0,\\\
v_{t}+h_{x}+iv_{xx}+vv_{x}=0,\end{array}\right.$

with dispersion $\omega(k)=\pm ik(1+k^{2})^{1/2}$

As recalled in [52], Broer [11] and Kaup [35] derived the system in
dimensional form

(2.5)
$\left\\{\begin{array}[]{l}\eta_{t}+h_{0}\phi_{xx}+(\eta\phi_{x})_{x}+\left(\frac{h_{0}^{3}}{3}-\frac{h_{0}\tau}{\rho
g}\right)\phi_{xxx}=0\\\
\phi_{t}+\frac{1}{2}(\phi_{x})^{2}+g\eta=0,\end{array}\right.$

where $\eta$ is the elevation of the wave, $\phi$ the velocity potential
evaluated at the free surface, the constant $h_{0}$ is the quiescent water
depth, $g>0$ is the gravitational acceleration, $\tau\geq 0$ is the surface
tension coefficient and $\rho$ the density of the fluid.

The ”bad” version corresponds to pure gravity waves ($\tau=0$) or to gravity-
capillarity waves with small surface tension ($\tau/(g\rho)<h_{0}^{2}/3$,
corresponding to $b=d=c=0,a>0$ in (1.2)). The ”good” version occurs with
strong surface tension that is when $\tau/(g\rho)>h_{0}^{2}/3,$ (a not too
physical case…), corresponding to $b=d=c=0,a<0$ in (1.2) . The difference of
nature of the equation depending on the capillarity is reminiscent of that of
the Kadomtsev-Petviashvili equation (KP-II for gravity waves and KP-I for
gravity-capillarity waves with strong surface tension) although both KP-II and
KP-I are well-posed!

######  Remark 2.2.

Despite the fact that the Kaup-Broer-Kupershmidt system with the plus sign is
linearly ill-posed, it possesses solitary wave solutions
$(\eta(x-kt),v(x-kt))$ when $k>1$ 444In the classical version $a=1/3$.. This
has been found by Kaup, [35] using the Inverse Scattering machinery who also
proved the existence of N-solitons. Matveev and Yavor [47], exhibited a vast
class of almost periodic solutions containing N-soliton solutions as a
degenerate case, see also [52]. The existence of undular bores is proven in
[27].

A direct approach to the existence of solitary waves is given in [15] with
$a=\frac{1}{3}$. Actually a solitary wave satisfies the equations

$(v^{\prime})^{2}=R_{1}(u)\equiv\frac{3}{4}v^{2}(v-(2k-2))(v-(2k+2)),\quad\eta=v\left(k-\frac{v}{2}\right),$

where the derivative is taken with respect to $\xi=x-kt.$

By studying the function $R_{1},$ one shows that there exists a unique
solution $v(\xi)$ which is even and monotone for $\xi>0$ for any $k>1$. More
precisely, the solution $v(\xi)$ reads

$v(\xi)=\frac{2(k^{2}-1)}{\cosh\bigl{(}\sqrt{3(k^{2}-1)}\xi\bigr{)}+k}.$

We also have

$\eta(\xi)=\frac{2(k^{2}-1)\Bigl{(}k\cosh\bigl{(}\sqrt{3(k^{2}-1)}\xi\bigr{)}+1\Bigr{)}}{\Bigl{(}\cosh\bigl{(}\sqrt{3(k^{2}-1)}\xi\bigr{)}+k\Bigr{)}^{2}}.$

When $1<k\leq 2,$ the corresponding $\eta(\xi)$ is also monotonically
decreasing for $\xi>0$ while this is no longer true when $k>2.$

One notices that for $1<k\leq 2,$
$\|v\|_{L^{\infty}}=\|\eta\|_{L^{\infty}}=2(k-1).$

##  3\. The Kaup-Broer-Kupershmidt system by PDE techniques

We recall that the Cauchy problem for (2.1) with $\alpha=-1$ is well-posed in
both spatial dimensions one and two under the non-cavitation condition
$1+\epsilon\eta>0$ on time scales of order $O(1/\epsilon),$ see [58].

We now focus on the one-dimensional ”good” KBK system with $\epsilon=1$ which
has the Hamiltonian structure

$\eta_{t}+\partial_{x}\frac{\delta\mathcal{H}}{\delta v}=0,\quad
v_{t}+\partial_{x}\frac{\delta\mathcal{H}}{\delta\eta}=0,$

where the Hamiltonian $\mathcal{H}$ is given by

$\mathcal{H}=\frac{1}{2}\int[\eta^{2}+(1+\eta)v^{2}+(v_{x}^{2})]dx.$

######  Remark 3.1.

The Hamiltonian of the ”bad” KBK system is

$\mathcal{H}=\frac{1}{2}\int[\eta^{2}+(1+\eta)v^{2}-(v_{x}^{2})]dx.$

######  Remark 3.2.

Both the ”bad” and ”good” KBK system have the conserved quantity:

$H_{0}=\int_{\mathbb{R}}\eta vdx.$

Angulo, [3], proved that the Cauchy problem for the ”good” KBK system is
locally well-posed in $H^{s-1}({\mathbb{R}})\times
H^{s}({\mathbb{R}}),s>3/2.$555A local well-posedness result in a weighted
space with less regularity was obtained in [36].

Surprisingly he also found an additional conservation law namely

(3.1)
$\begin{split}I_{3}(\eta,v)=&\frac{1}{8}\int[4(v_{xx}^{2})+8(v_{x}^{2})+4v^{2}+4(\eta_{x}^{2})+4\eta^{2}+6v^{2}(v_{x})^{2}\\\
&-16\eta vv_{xx}-4\eta(v_{x})^{2}+10\eta
v^{2}+2\eta^{3}+v^{4}+6\eta^{2}v^{2}+\eta v^{4}]dx.\end{split}$

The existence of such a conservation law suggests that the ”good” Kaup-Broer-
Kupershmidt system is integrable in the sense of Inverse Scattering. We refer
to the next section for further information on this issue.

Actually $I_{3}$ can be viewed as the Hamiltonian of the (linearly well-posed)
higher order system

(3.2)
$\left\\{\begin{array}[]{l}v_{t}-\eta_{xxx}+\eta_{x}+\\{\frac{5}{4}v^{2}+\frac{3}{4}\eta^{2}+\frac{3}{2}\eta
v^{2}+\frac{1}{8}v^{4}-2vv_{xx}-\frac{1}{2}v_{x}^{2}\\}_{x}=0,\\\
\eta_{t}+v_{xxxxx}-2v_{xxx}+v_{x}+\\{\frac{3}{2}vv^{2}_{x}-\frac{3}{2}(v^{2}v_{x})_{x}-2\eta
v_{xx}-2(\eta v)_{xx}+\frac{1}{2}(\eta v_{x})_{x}+\frac{5}{2}v\eta\\\
+\frac{1}{2}v^{3}+\frac{3}{2}(v\eta^{2})+\frac{1}{2}v^{3}\eta\\}_{x}=0\end{array}\right.$

that can be written

$\eta_{t}+\partial_{x}\frac{\delta I_{3}}{\delta v}=0,\quad
v_{t}+\partial_{x}\frac{\delta I_{3}}{\delta\eta}=0.$

We are not aware of results on the Cauchy problem for (3.2).

In order to get a further insight into the ”good” KBK system, it is useful to
diagonalize the linear part to obtain the equivalent system, see [58]:

(3.3)
$\left\\{\begin{array}[]{l}\eta_{t}+J_{\epsilon}\zeta_{x}+\frac{\epsilon}{2}N_{1}(\eta,w)=0,\\\
w_{t}-J_{\epsilon}w_{x}+\frac{\epsilon}{2}N_{2}(\eta,w)=0.\end{array}\right.$

where $J_{\epsilon}=(I-\epsilon\partial_{x}^{2})^{1/2}$ and

$N_{1}(\eta,w)=\partial_{x}[(\zeta+w)J_{\epsilon}^{-1}(\eta-w)]+J_{\epsilon}[J_{\epsilon}^{-1}(\eta-w)J_{\epsilon}^{-1}(\eta_{x}-w_{x})],$

and

$N_{2}(\eta,w)=\partial_{x}[(\eta+w)J_{\epsilon}^{-1}(\zeta-w)]-J_{\epsilon}[J_{\epsilon}^{-1}(\eta-w)J_{\epsilon}^{-1}(\eta_{x}-w_{x})].$

Since

$(1+\epsilon\xi^{2})^{1/2}-\epsilon^{1/2}|\xi|=\frac{1}{(1+\epsilon\xi^{2})^{1/2}+\epsilon^{1/2}|\xi|},$

(3.3) writes

(3.4)
$\left\\{\begin{array}[]{l}\eta_{t}+\epsilon^{1/2}\mathcal{H}\eta_{xx}+R_{\epsilon}\eta+\frac{\epsilon}{2}N_{1}(\eta,w)=0,\\\
w_{t}-\epsilon^{1/2}\mathcal{H}w_{xx}-R_{\epsilon}w+\frac{\epsilon}{2}N_{2}(\eta,w)=0.\end{array}\right.$

where $R_{\epsilon}$ is the (order zero) skew-adjoint operator with symbol
$\frac{i\xi}{(1+\epsilon\xi^{2})^{1/2}+\epsilon^{1/2}|\xi|}.$

Thus the ”good” KBK system is equivalent to a system having a decoupled
”Benjamin-Ono type” linear part. This property allows to apply the techniques
used to obtain the local well-posedness of the Benjamin-Ono equation in low
regularity spaces. Note that this approach would yield existence on the
”short” time range $O(1/\sqrt{\epsilon}).$ See the discussion in [58].

######  Remark 3.3.

In a recent paper, Melinand [48] derived various dispersive estimates
(Strichartz, local Kato smoothing, Morawetz) for a large class of (abcd)
systems including the Kaup-Broer-Kupershmidt system, both in one and two
spatial dimensions. These estimates should play an important role for the
local resolution of the Cauchy problem in ”large” functional spaces and in the
proof of the possible scattering of small solutions.

We conclude this section by alluding to results in [18], concerning the
evolution of discontinuous initial data under the flow of the good KBK system.

Actually a Riemann-type problem is considered with initial data

$\eta(x,t=0)=\eta_{L},\quad\text{and}\quad v(x,t=0)=v_{L}\quad\text{for}\quad
x<0,$ $\eta(x,t=0)=\eta_{R},\quad\text{and}\quad
v(x,t=0)=v_{R}\quad\text{for}\quad x>0,$

and one finds in [18] a classification of the wave patterns evolving from
these initial discontinuities.

###  3.1. Traveling wave solutions

Following Angulo [3] who considered the case $\epsilon=1,$ we now consider the
traveling wave solutions of (1.3) that is solutions of the form ${\bf
U}_{c}=(\eta,v)$ with

$\eta(x,t)=n(x-ct),\;v(x,t)=\phi(x-ct)$

with $c\in\mathbb{R}$ constant so that $(n,\phi)$ satisfies the system (we
have assumed that $(n,\phi)$ vanish at infinity which is natural in the water
waves context):

(3.5) $\left\\{\begin{array}[]{l}-cn+\phi+\epsilon
n\phi-\epsilon\phi^{\prime\prime}=0,\\\
-c\phi+n+\epsilon\frac{\phi^{2}}{2}=0,\end{array}\right.$

where ′ denotes the derivative with respective to $\xi=x-ct$.

Eliminating $n$ from the second equation yields

(3.6)
$-\epsilon\phi^{\prime\prime}+(1-c^{2})\phi+\frac{3}{2}C\epsilon\phi^{2}-\frac{\epsilon^{2}}{2}\phi^{3}=0.$

From Theorem 5 in [6] non trivial solitary waves exist if and only if $|c|<1.$
Actually (3.6) can be integrated by standard methods, leading, for $|c|<1,$ a
unique even solution (up to translations) which reads when $\epsilon=1$, see
[3]:

(3.7)
$v_{c,1}(\xi)=\frac{2(1-c^{2})}{\cosh(\sqrt{1-c^{2}}\xi)-c},\quad\eta_{c,1}(\xi)=cv_{c,1}(\xi)-\frac{1}{2}v_{c,1}(\xi)$

from which we deduce the expression of $v_{c,\epsilon}:$

(3.8)
$v_{c,\epsilon}(\xi)=\frac{2(1-c^{2})}{\epsilon[{\cosh(\sqrt{1-c^{2}}\epsilon^{-1/2}\xi)}-c]}$

We now turn to stability issues for the above solitary wave, following Angulo
[3] who solved the problem with $\epsilon=1$ and actually proved using the
method in [28] that the solitary wave is orbitally stable in
$L^{2}({\mathbb{R}})\times H^{1}({\mathbb{R}})$ for $|c|<1.$

This uses the fact that ${\bf U}_{c}$ can be viewed as a critical point of the
functional $F=\Phi_{1}-c\Phi_{0},$ that is it satisfies

$(\Phi_{1}-c\Phi_{0})^{\prime}({\bf U}_{c})=0$

where

$\Phi_{1}(v,\eta)=\frac{1}{2}\int_{\mathbb{R}}[v^{2}+\eta^{2}+v^{2}\eta+(v_{x})^{2}]dx,$

and

$\Phi_{0}(v,\eta)=\int_{\mathbb{R}}v\eta dx.$

Angulo proved moreover, using the higher order conservation law that the
Cauchy problem is globally well-posed in $H^{s-1}({\mathbb{R}})\times
H^{s}({\mathbb{R}}),s\geq 2$ provided the initial data $(\eta_{0},v_{0})$ are
$L^{2}({\mathbb{R}})\times H^{1}({\mathbb{R}})$-close to some translation of
the solitary wave. This is one of the few known results on the global well-
posedness of the Cauchy problem for abcd Boussinesq systems.

######  Remark 3.4.

We are not aware of explicit formulas for N-solitons of the good Kaup-Broer-
Kupershmidt system.

###  3.2. Remarks on the stationary solutions

We comment here on the stationary solutions of the good KBK system (here we
keep the dependance in $\epsilon$). Assuming that such a solution vanishes at
infinity. In one spatial dimension $(\eta,v)$ should satisfy

(3.9) $-v^{\prime\prime}+v-\frac{\epsilon^{2}}{2}v^{3}=0,$
$\eta=-\frac{\epsilon}{2}v^{2}.$

and $v$ can be expressed in terms of the profile of the soliton of the
focusing cubic nonlinear Schrödinger equation, leading to the explicit form

$v(x)=\frac{2}{\epsilon\cosh(\epsilon^{-1/2}x)}.$

In the two-dimensional case, one has

(3.10) $\nabla\cdot[-\Delta{\bf v}+{\bf v}-\frac{\epsilon^{2}}{2}|{\bf
v}|^{2}{\bf v}]=0,$ $\eta=-\frac{\epsilon}{2}|{\bf v}|^{2}.$

Any solution ${\bf v}$ of

(3.11) $-\Delta{\bf v}+{\bf v}-\frac{\epsilon^{2}}{2}|{\bf v}|^{2}{\bf v}=0,$

solves (3.10).

Equation (3.11) is a particular case of the equation of a bound state solution
of a vector nonlinear Schrödinger equation. It is proven in [17] that (3.11)
has a solution whose components are constant multiples of the ground state of
the corresponding scalar nonlinear Schrödinger equation.

In both cases it would be interesting to investigate the stability of those
solutions with respect to the good KBK system.

##  4\.  The Kaup-Broer-Kupershmidt system by IST techniques

The connection of (1.3) with the theory of integrable systems was first
noticed in [43, 35] for both the ”bad” and ”good” cases. Actually, Kaup [35],
proved that the KBK system is the compatibility condition for a pair of linear
equations.

Writing the KBK system as

(4.1) $\zeta_{t}+(\zeta v)_{x}\pm v_{xxx}=0,\quad
v_{t}+vv_{x}+\zeta_{x}=0,\quad\zeta=1+\eta,$

the two aforementioned linear equations are:

(4.2)
$\left\\{\begin{array}[]{l}4\psi_{xx}=\pm[(\lambda-\frac{1}{2}v)^{2}-\zeta]\psi,\\\
\psi_{t}=\frac{1}{4}v_{x}\psi-(\lambda+\frac{1}{2}v)\psi_{x}.\end{array}\right.$

Kaup also wrote the formulation of the direct and inverse scattering problems
and the formula for the soliton solution.

Kupershmidt [43] showed how the KBK system can be derived in bi-Hamiltonian
form. He considered in fact a more general class of systems that he described
as the ”richest integrable known system known to date”:

(4.3) $\left\\{\begin{array}[]{l}u_{t}+h_{x}+\beta u_{xx}+uu_{x}=0,\\\
h_{t}+\alpha u_{xxx}-\beta h_{xx}=0,\end{array}\right.$

where $\alpha$ and $\beta$ are arbitrary real constants. Note that the system
is linearly well-posed if and only if $\alpha<-\beta^{2}.$ The KBK systems
correspond to $\beta=0,$ $\alpha>0$ for the ”bad” one, $\alpha<0$ for the
”good” one. Kupershmidt derived a corresponding hierarchy by making use of the
theory of non standard integrable systems.

As already alluded to, the integrable system structure of the KBK systems
yields existence of various special solutions, both in the ”good” and the
”bad” case.

For instance Sachs [57] wrote an infinite family of rational solutions of the
”bad” KBK system. Clarkson [16] obtains a larger class of rational solutions
in terms of the generalized Hermite and generalized Okamoto polynomials for
both the ”good” and ”bad” version of the KBK system.

Furthermore, Ito in [32] (see also [57]) exhibited an infinite family of
commuting Hamiltonian flows associated to an infinite set of commuting
integrals $F_{n}$ such that

$\\{F_{n},\mathcal{H}\\}=0$

where $\\{,\\}$ is the Poisson bracket and $\mathcal{H}$ the Hamiltonian of
the KBK system.

This leads to a hierarchy of KBK Boussinesq systems.

The second one in the ”bad” case is

(4.4)
$\left\\{\begin{array}[]{l}v_{t}+\frac{1}{4}(v^{3}+6v\zeta+4v_{xx})_{x}=0,\\\
\zeta_{t}+\frac{1}{4}(3v^{2}\zeta+3\zeta^{2}+3v^{2}_{x}+6vv_{xx}+4\zeta_{xx})_{x}=0,\end{array}\right.$

where again $\zeta=1+\eta.$

Note that, contrary to the ”bad” KBK system, the linear part in (4.4) is
decoupled and well-posed.

The Hamiltonian corresponding to (4.4) is

(4.5)
$H_{2}(v,\zeta)=\frac{1}{4}\int_{\mathbb{R}}\\{v^{3}\zeta+3v\zeta^{2}+2v_{xx}\zeta+2v\zeta_{xx}+\frac{3}{2}v^{2}v_{xx}\\}dx.$

######  Remark 4.1.

The system (4.4) is different from another integrable system introduced by
Kupershmidt [44] as the s-mKdV system and which writes, in the notations of
[63] where one also finds a bi-Hamiltonian formulation of the equation:

(4.6)
$\left\\{\begin{array}[]{l}v_{t}-v_{xxx}+6v^{2}v_{x}-3(\eta_{xx}\eta)_{x}-6(\eta_{x}\eta
v)_{x}=0,\\\
\eta_{t}-4\eta_{xxx}+6(v^{2}-v_{x})+3(2v_{x}v-v_{xx})\eta=0.\end{array}\right.$

Actually (4.6) is obtained after a Miura type transform

$u=-v_{x}-v^{2}+\eta_{x}\eta,\quad\xi=\eta_{x}+\eta v$

from the so-called super-KdV equation (s-KdV), see [44, 63]

(4.7) $\left\\{\begin{array}[]{l}u_{t}-u_{xxx}-6uu_{x}+12\xi\xi_{xx}=0,\\\
\xi_{t}-4\xi_{xxx}-6u\xi_{x}-3\xi u_{x}=0.\end{array}\right.$

We refer to [55, 5, 49] for ill-posedness and well-posedness results on the
Cauchy problem for a variant of (4.7). On the other hand we are not aware of
mathematical results on the Cauchy problem for (4.6).

The next system in the ”bad” KBK hierarchy is in the $(v,\zeta)$ variables

(4.8)
$\left\\{\begin{array}[]{l}v_{t}+\frac{1}{8}(v^{4}+12v^{2}\zeta+6\zeta^{2}+6v_{x}^{2}+16vv_{xx}+8\zeta_{xx})_{x}=0,\\\
\zeta_{t}+\frac{1}{8}(4v^{3}\zeta+12v\zeta^{2}+16v\zeta_{xx}+20v_{x}\zeta_{x}+20v_{xx}\zeta+8v_{xxxx}+12vv_{x}^{2}+12v^{2}v_{xx})_{x}=0,\end{array}\right.$

which is linearly ill-posed.

More generally the higher order flows are defined by the equations

(4.9) $\begin{pmatrix}v\\\ \zeta\end{pmatrix}_{t}+J\nabla H_{m}=0,$

where

$J=\begin{pmatrix}0&\partial_{x}\\\ \partial_{x}&0\end{pmatrix},$

and where the Hamiltonian $H_{m}$ are defined inductively by

$H_{2}=\frac{1}{2}\int_{\mathbb{R}}(v^{2}\zeta+\zeta^{2}-v_{x}^{2})dx,$

$J\nabla H_{m}=K\nabla H_{m-1},$

where K is the skew-symmetric operator

$K=\begin{pmatrix}\partial_{x}&\frac{1}{2}\partial_{x}v\\\
\frac{1}{2}v\partial_{x}&\partial_{x}^{3}+\zeta\partial_{x}+\frac{1}{2}\zeta_{x}\end{pmatrix}.$

The same strategy can be applied to the ”good” KBK system, starting now from
the Hamiltonian

$H_{2}=\frac{1}{2}\int_{\mathbb{R}}(v^{2}\zeta+\zeta^{2}+v_{x}^{2})dx,$

and the new K defined as

$K=\begin{pmatrix}\partial_{x}&\frac{1}{2}\partial_{x}v\\\
\frac{1}{2}v\partial_{x}&-\partial_{x}^{3}+\zeta\partial_{x}+\frac{1}{2}\zeta_{x}\end{pmatrix}.$

The second equation in the hierarchy is thus:

(4.10)
$\left\\{\begin{array}[]{l}v_{t}+\frac{1}{4}(v^{3}+6v\zeta-4v_{xx})_{x}=0,\\\
\zeta_{t}+\frac{1}{4}(3v^{2}\zeta+3\zeta^{2}-3v_{x}^{2}-6vv_{xx}-4\zeta_{xx})_{x}=0,\end{array}\right.$

which again is linearly well-posed and correspond to the Hamiltonian

(4.11)
$H_{2}(v,\zeta)=\frac{1}{4}\int_{\mathbb{R}}\\{v^{3}\zeta+3v\zeta^{2}-2v_{xx}\zeta-2v\zeta_{xx}-\frac{3}{2}v^{2}v_{xx}\\}dx.$

######  Remark 4.2.

In terms of the variables $(v,\eta),$(4.10) writes

(4.12)
$\left\\{\begin{array}[]{l}v_{t}+\frac{1}{4}(v^{3}+6v+6v\eta-4v_{xx})_{x}=0,\\\
\zeta_{t}+\frac{1}{4}(3v^{2}+3v^{2}\eta+6\eta+3\eta^{2}-3v_{x}^{2}-6vv_{xx}-4\eta_{xx})_{x}=0,\end{array}\right.$

######  Remark 4.3.

Equations (4.4), (4.10) and (4.12) are systems of KdV type involving second
and third order nonlinear terms making the study of the local Cauchy problem
delicate. We refer to [20, 46] for a study of the Cauchy problem for scalar
dispersive equations involving a nonlinear dispersive third order term.

The next system in the ”good” KBK hierarchy is

(4.13)
$\left\\{\begin{array}[]{l}v_{t}+\frac{1}{8}(v^{4}+12v^{2}\zeta+6\zeta^{2}-6v_{x}^{2}-16vv_{xx}-8\zeta_{xx})_{x}=0,\\\
\zeta_{t}+\frac{1}{8}(4v^{3}\zeta+12v\zeta^{2}-16v\zeta_{xx}-20v_{x}\zeta_{x}-20v_{xx}\zeta+8v_{xxxx}-12vv_{x}^{2}-12v^{2}v_{xx})_{x}=0,\end{array}\right.$

which is linearly well-posed and the $(v,\zeta)$ version of (3.2).

It corresponds to the Hamiltonian

(4.14)
$H(v,\zeta)=\frac{1}{8}\int_{\mathbb{R}}(v^{4}\zeta+6v^{2}\zeta^{2}-8v^{2}\zeta_{xx}+4v^{2}_{xx}+10\zeta
v_{x}^{2}+4\zeta_{x}^{2}-2v^{3}v_{xx})dx.$

We now briefly describe some recent progress on the Kaup-Broer-Kupershmidt
system.

The paper [51] focusses on the periodic ”good” KBK system and provides an
explicit form of the periodic traveling waves in terms of the Weierstrass
elliptic function $\wp:$

$v(\xi)=\frac{a_{11}\wp(\xi+\phi)+a_{12}}{a_{21}\wp(\xi+\phi)+a_{22}},\quad\eta(\xi)=cv(\xi)-\frac{1}{2}v(\xi)^{2};\;\;\
\xi=(x-ct).$

The authors also exhibit a matrix Lax pair that is used it to prove the
existence of an infinite number of complex conservation laws:

######  Theorem 4.1.

-[51]

$(\rho_{n})_{t}+(j_{n})_{x}=0,$

where the conserved densities $\rho_{n}$ are determined by the recursion:

$\rho_{1}=\frac{1}{2}\eta+\frac{i}{2}v_{x},\quad\rho_{2}=iv\rho
1-2\rho_{1x}+\frac{i}{2}v,$

$\rho_{n+1}=iv\rho_{n}-\rho_{n-1}-2\rho_{nx}-2\sum_{k=1}^{n-1}\rho_{k}\rho_{n-k},\quad
n>1,$

and the conserved currents are determined by the conserved densities:

$j_{1}=\frac{1}{4}v+\frac{1}{2}v\rho_{1}-\frac{i}{2}\rho-2,$

$j_{n}=\frac{1}{2}v\rho_{n}-\frac{i}{2}\rho_{n+1}+\frac{i}{2}\rho_{n-1},\quad
n>1.$

Each complex conservation law gives rise to two real conservation laws that
are polynomials in $\eta,v$ and their higher order derivatives with respect to
x.

The Lax pair is also used to construct the forward scattering transform for
periodic solutions and to find exact formulas for finite gap solutions.

Two-dimensional integrable generalizations of the Kaup-Broer-Kupershmidt
system are given in [56, 52, 50]. None of them seem physically relevant,
however.

On the other hand for capillary waves such that $\tau/(g\rho)>h_{0}^{2}/3,$
(2.5) is analogous to the good BSK system.

######  Remark 4.4.

The ”bad” Kaup-Broer-Kuperschmidt system is studied from the Inverse
Scattering point of view in [64] under the form

(4.15) $\left\\{\begin{array}[]{l}v_{t}=\frac{1}{2}(v^{2}+2w-v_{x})_{x},\\\
w_{t}=(vw+\frac{1}{2}w_{x})_{x},\end{array}\right.$

which is an alternative form of (2.3). Using this version of the system,
(4.15) is the compatibility condition of the following Lax pair

(4.16) $\psi_{x}=U\psi,\quad\psi_{t}=V\psi,$

with

(4.17) $\left\\{\begin{array}[]{l}U=-ik\sigma_{3}+Q,\quad
Q=\frac{v}{2}\sigma_{3}+\sigma_{+}-w\sigma_{-},\\\
V=k^{2}\sigma_{3}+\widehat{Q},\quad\widehat{Q}=-\frac{1}{4}(v_{x}-v^{2})\sigma_{3}+(ik+\frac{v}{2})\sigma_{+}-\frac{1}{2}(w_{x}+wv)\sigma_{-}\end{array}\right.$

where $k$ is a spectral parameter, $\sigma_{j},j=1,2,3$ are the classical
Pauli matrices and
$\sigma_{+}=\frac{1}{2}(\sigma_{1}+i\sigma_{2}),\;\sigma_{-}=\frac{1}{2}(\sigma_{1}-i\sigma_{2}).$

The Inverse Scattering mechanism described in [64] allows the construction of
special solutions such as kinks, see [64] for details.

######  Remark 4.5.

There is so far no rigorous result on the complete resolution of the Cauchy
problem for the well-posed Kaup-Broer-Kupershmidt system ($\alpha=-1$) under
an appropriate functional setting for the initial data and the solution by
using inverse scattering techniques, and giving possibly insight into the
qualitative behavior of the solutions.This would include a rigorous theory for
the direct and inverse scattering problem. We refer to [51] for progress in
this direction.

##  5\. Numerical approach to the KBK system

In this section, we will detail the numerical approach used to solve the
‘good’ KBK system in the numerical experiments.

We recall the conserved energy for this equation is

(5.1) $E=\frac{1}{2}\int_{\mathbb{R}}(\eta^{2}+(1+\eta)v^{2}+v_{x}^{2})dx.$

Solitons of the KBK system (with location $x_{0}$ of the maximum for $t=0$)
can be written for the velocity $C\in\mathbb{R}$, $|C|<1$ with (3.7) in the
form

$\displaystyle v$ $\displaystyle=\frac{2(1-C^{2})}{\cosh(\sqrt{1-C^{2}}(x-Ct-
x_{0}))-C},$ (5.2) $\displaystyle\eta$ $\displaystyle=Cv-\frac{1}{2}v^{2}.$

To numerically solve the system (1.3) we essentially use the diagonalisation
approach for the linear part of (3.3), (3.4). We consider the KBK system in
Fourier space,

$\displaystyle\hat{\eta}_{t}$
$\displaystyle=-ik(1+k^{2})\hat{v}-ik\widehat{\eta v},$ (5.3)
$\displaystyle\hat{v}_{t}$
$\displaystyle=-ik\hat{\eta}-\frac{ik}{2}\widehat{v^{2}}.$

Here we use the standard definition for a Fourier transform for integrable
functions $u(x)$ denoted by $\hat{u}(k)$ with dual variable $k$ and its
inverse (in the sense of tempered distributions),

$\displaystyle\hat{u}(k)$ $\displaystyle=\int_{\mathbb{R}}u(x)e^{-ikx}dx,\quad
k\in\mathbb{R},$ (5.4) $\displaystyle u(x)$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\hat{u}(k)e^{ikx}dk,\quad
x\in\mathbb{R}.$

Introducing

(5.5) $\hat{u}_{\pm}=\hat{v}\pm\frac{\hat{\eta}}{\sqrt{1+k^{2}}},$

we can write the system (5.3) in the form

(5.6) $(u_{\pm})_{t}=\mp
u_{\pm}-ik\left(\frac{1}{2}\widehat{v^{2}}\pm\frac{\widehat{\eta
v}}{\sqrt{1+k^{2}}}\right).$

Obviously the dispersion relation is as in (2.4), i.e., for large $|k|$ as in
the Schrödinger equation.

For the numerical treatment, the Fourier transform in (5.6) will be
approximated in standard way by the discrete Fourier transform (DFT) which can
be conveniently computed by the _Fast Fourier Transform_ (FFT). This is a
_spectral method_ which means that the numerical error in approximating smooth
periodic functions with $N$ modes in the DFT leads to a numerical error
exponentially decreasing with $N$, see the discussion in [61] and references
therein. Thus we will in the following always work on a torus of period $2\pi
L$ with $L>0$, i.e., we will consider values of $x\in L[-\pi,\pi]$, where we
will apply $N$ DFT modes. In an abuse of notation, we will denote the discrete
Fourier transform with the same symbol as the standard Fourier transform.
Since the numerical error in approximating a function with a DFT is of the
order of the highest DFT coefficients, we always use values of $L$ and $N$
such that the DFT coefficients decrease to machine precision which is here of
the order of $10^{-16}$.

The resulting $2N$ dimensional system of equations (5.6) are of the form

(5.7) $u_{t}=\mathcal{L}u+\mathcal{N}(u),$

where $\mathcal{L}$ is a linear diagonal operator, here proportional to $\pm
ik\sqrt{1+k^{2}}$, whereas $\mathcal{N}(u)$ is a nonlinear term in the
$u_{\pm}$. Due to $|\mathcal{L}|$ being rapidly increasing with $|k|$, the
system is _stiff_ which means that explicit time integration schemes are not
efficient for such systems due to stability conditions, see for instance the
discussion in [31] and references therein. However, there are efficient time
integration schemes to address systems of the form (5.7) with a stiff diagonal
linear term, see [31] for so-called exponential time differencing schemes
(ETD). The idea of ETD schemes is to use equidistant time steps $h$ and to
integrate equation (5.7) between the time steps $t_{n}$ and $t_{n+1}$,
$n=1,2,\ldots$ with an exponential integrator with respect to $t$. We get

$u(t_{n+1})=e^{\mathcal{L}h}u(t_{n})+\int_{0}^{h}e^{\mathcal{L}(h-\tau)}\mathcal{N}(u(t_{n}+\tau),t_{n}+\tau)d\tau.$

The integral will be computed in an approximate way for which different
schemes exist. In [40] we have compared for dispersive PDEs various Runge-
Kutta schemes of classical order 4 which all showed similar performance.
Therefore we apply in the following the one by Cox-Matthews [19]. Since the
method is explicit (only information of the functions at the previous time
step $t_{n}$ is needed), it is not important that the nonlinear part is not
diagonal in (5.6) as the linear part. As discussed in [39, 40], the exactly
conserved energy can be used to control the numerical error in the time
integration. Whereas the energy (5.1) is exactly conserved for the KBK system,
the numerically computed energy will depend on time due to unavoidable
numerical errors. The relative energy

(5.8) $\Delta:=|E(t)/E(0)-1|$

typically overestimates the numerical error by 1-2 orders of magnitude. We
will always aim at a $\Delta$ considerably smaller than $10^{-3}$.

To test the code, we consider the soliton (5.2) for $C=0.8$ as initial data.
We use $N=2^{11}$ DFT modes for $x\in 15[-\pi,\pi]$ with $N_{t}=4000$ time
steps for $t\leq 1$. The DFT coefficients decrease to machine precision during
the whole computation in this case, and the relative energy is conserved to
better than $10^{-12}$. In Fig. 1 we show the difference between numerical and
exact solution at the final time. It can be seen that it is of the order of
$10^{-12}$ which shows that the code can propagate the soliton with
essentially machine precision.

Figure 1. Difference between numerical and exact solution for soliton initial
data (5.2) with $C=0.8$ for $t=1$, on the left $\eta$, on the right $v$.

##  6\. Perturbed solitons

Angulo [3] showed that the solitons of the KBK system are stable. In this
section we illustrate this by considering perturbations of the solitons.

We study perturbations of the form

(6.1) $v(x,0)=\lambda v_{C}(x),\quad\eta(x,0)=\mu\eta_{C}(x),$

where $v_{C}$, $\eta_{C}$ are the solitons (5.2) for a given real velocity $C$
with $|C|<1$, and where $\lambda$, $\mu$ are real numbers in the vicinity of
1. We use $N=2^{12}$ DFT modes for $x\in 30[-\pi,\pi]$ and $N_{t}=4000$ time
steps for $t\in[0,5]$. Note that the maximum of $v$ in (5.2) is given by $1+C$
and thus strictly smaller than 2. This means that the value of $\lambda$ in
(6.1) has to be chosen such that this condition is satisfied. To study
numerically perturbations and to see an effect in finite time, one has to
consider values of $\lambda$ and or $\mu$ that have a finite difference to 1.
This implies that the resulting solution will be close to the original
soliton, but not of identical velocity. We fit the soliton in the following
way: in the numerical solution for $v$ at the final time $t=5$, we identify
the location $x_{0}$ and the value $v_{0}$ of the maximum. Since the maximum
value for $v$ in (5.2) is given by $2(C+1)$, we can get the value of $C$ from
this maximum.

We show the solution to the KBK system for $C=0.8$, $\lambda=1.01$ and $\mu=1$
at the final time together with the fitted soliton in green (the fitted
velocity is 0.819) in Fig. 2. It can be seen that the solution is very close
to the fitted soliton, but that there is also some small radiation. The
soliton is thus as expected stable.

Figure 2. The solution to the KBK system for perturbed soliton initial data of
the form (6.1) with $\lambda=1.01$ and $\mu=1$ at the final time $t=5$ in blue
and a fitted soliton in green, on the left $v$, on the right $\eta$.

The situation is very similar for initial data of the form (6.1) with
$\lambda=0.99$ and $\mu=1$ as can be seen in Fig. 3. The fitted soliton has
velocity $0.7811$. Again the final state is a soliton plus radiation.

Figure 3. The solution to the KBK system for perturbed soliton initial data of
the form (6.1) with $\lambda=0.99$ and $\mu=1$ at the final time $t=5$ in blue
and a fitted soliton in green, on the left $v$, on the right $\eta$.

The same stability aspects are observed if perturbations of the form (6.1)
with $\lambda=1$ and $\mu\sim 1$ are studied as shown in Fig. 4. The fitted
values of the velocity are $0.8155$ for $\mu=1.01$ and $0.7846$ for
$\mu=0.99$. In all cases the final state of the solution is a soliton plus
radiation.

Figure 4. The solution to the KBK system for perturbed soliton initial data of
the form (6.1) with $\lambda=1$ and $\mu=1.01$ in the upper row and $\mu=0.99$
in the lower row fitted soliton in green, on the left $v$, on the right
$\eta$.

For $C=0$, the solutions (5.2) become stationary. If we apply the same
perturbations (6.1) to this solution, we find that the stationary solution is
also stable. For $\mu=1$ in the initial data (6.1), we get the solutions shown
in Fig. 5.

Figure 5. The solution to the KBK system for the perturbed stationary solution
of the form (6.1) with $\lambda=1$ and $\mu=1.01$ in the upper row and
$\mu=0.99$ in the lower row at the final time $t=5$ in blue and a fitted
stationary solution in green, on the left $v$, on the right $\eta$.

The situation is similar for $\lambda=1$ as shown in Fig. 6.

Figure 6. The solution to the KBK system for the perturbed stationary solution
of the form (6.1) with $\lambda=1$ and $\mu=1.01$ in the upper row and
$\mu=0.99$ in the lower row fitted stationary solution in green, on the left
$v$, on the right $\eta$.

##  7\. Localised initial data

In this section we will study KBK solutions for localised initial data. The
results confirm the applicability of the soliton resolution conjecture to the
KBK system, that the long time behavior of solutions is given by solitons plus
radiation.

In this section we always use $N=2^{12}$ DFT modes for $x\in 30[-\pi,\pi]$ and
$N_{t}=4000$ time steps. The DFT coefficients decrease to machine precision in
all examples, and the relative conservation of the energy is of the order of
$10^{-10}$ and better during the computations. First we consider initial data
of the form

(7.1) $\eta(x,0)=0,\quad v(x,0)=3\exp(-x^{2}).$

The resulting solution for $v$ can be seen in Fig. 7. There are strong
oscillations propagating to the right and smaller ones propagating to the
left. And there is a solitary wave traveling towards $-\infty$ to be discussed
in more detail below.

Figure 7. Solution $v$ to the KBK system for the initial data (7.1).

The corresponding solution $\eta$ can be seen in Fig. 8. There is similar
radiation as in the solution $v$, and again a solitary structure traveling to
the left.

Figure 8. Solution $\eta$ to the KBK system for the initial data (7.1).

To check whether the solitary structure near the origin is in fact evolving
into a soliton, we fit it to the soliton (5.2) as described in the previous
section. We show the result of this fitting for $v$ on left of Fig. 9. The
numerical solution is given in blue, the fitted soliton in green. The
corresponding plot for $\eta$ is shown on the right of the same figure. It can
be seen that the agreement is already very good though there is still a
considerable amount of radiation in the vicinity of the soliton.

Figure 9. The solutions shown in Fig. 7 and Fig. 8 at the final time $t=5$ in
blue and a fitted soliton in green.

Note that there is no criterion known which initial data lead to which number
of KBK solitons for large times. If we consider for instance the initial data

(7.2) $\eta(x,0)=A\exp(-x^{2}),\quad v(x,0)=0,\quad A\in\mathbb{R},$

we get with $A=3$ for $v$ the solution shown in Fig. 10. In this case there is
no indication of solitons.

Figure 10. Solution $v$ to the KBK system for the initial data (7.2) with
$A=3$.

The corresponding solution $\eta$ is shown in Fig. 11. It appears that the
initial data are simply dispersed in this case. Note that the solution looks
qualitatively similar for larger values for $A$, for instance $A=20$, there is
no indication of solitons in this class of initial data.

Figure 11. Solution $\eta$ to the KBK system for the initial data (7.2) with
$A=3$.

However, the situation is different for negative $A$ in (7.2) as can be seen
for $A=-3$ in Fig. 12. There appears to be a solitary structure near the
origin.

Figure 12. Solution $v$ to the KBK system for the initial data (7.2) with
$A=-3$.

The corresponding solution $\eta$ can be seen in Fig. 11.

Figure 13. Solution $\eta$ to the KBK system for the initial data (7.2) with
$A=-3$.

As in Fig. 9, we fit the left maximum to the soliton (5.2). We show the
solution for $v$ at $t=8$ with the fitted soliton in green on the left of Fig.
14. The corresponding plot for $\eta$ can be seen on the right of the same
figure. There is clearly a second soliton moving to the right in this case for
symmetry reasons.

Figure 14. The solutions shown in Fig. 12 and Fig. 13 at the final time $t=8$
in blue and a fitted soliton in green.

Note that these initial data do not satisfy the non-cavitation condition, but
that there is no indication of a blow-up in this case. The solution appears to
be smooth for all times.

##  8\. Dispersive shock waves

Dispersive shock waves (DSWs) are zones of rapid modulated oscillations in
solutions to dispersive nonlinear PDEs in the vicinity of shocks of the
corresponding dispersionless system, here the Saint-Venant system. A possible
way to study such zones for the KBK system is to consider solutions on $x$
scales of order $1/\varepsilon$ for $\varepsilon\ll 1$ on time scales of order
$1/\varepsilon$. This can be conveniently done by rescaling $x$ and $t$ by a
factor $1/\varepsilon$. This leads for (1.3) (we use the same notation as
before)

(8.1) $\left\\{\begin{array}[]{l}\eta_{t}+v_{x}+(\eta
v)_{x}+\varepsilon^{2}v_{xxx}=0\\\ v_{t}+\eta_{x}+vv_{x}=0.\end{array}\right.$

In the formal limit $\varepsilon\to 0$, this leads to the Saint-Venant system

(8.2) $\left\\{\begin{array}[]{l}\eta_{t}+v_{x}+(\eta v)_{x}=0\\\
v_{t}+\eta_{x}+vv_{x}=0.\end{array}\right.$

the solutions of which will have shocks in finite time for hump-like initial
data. The system (8.1) can be seen as a dispersive regularization of the
system (8.2). For the same initial data leading to shocks for the Saint-Venant
system, DSWs are expected in the vicinity of the former for the system (8.1).

We use $N=2^{14}$ DFT modes for $x\in 3[-\pi,\pi]$ and $10^{4}$ time steps for
$t\leq 3$ for the initial data $\eta(x,0)=\exp(-x^{2})$, $v(x,0)=0$. The
solution $\eta$ for $\varepsilon=0.1$ can be seen in Fig. 15. The initial hump
splits into two humps developing strong gradients at the outer edges where
oscillations can be observed.

Figure 15. Solution $\eta$ to the KBK system (8.1) for $\varepsilon=0.1$ and
for the initial data $\eta(x,0)=\exp(-x^{2})$, $v(x,0)=0$.

The corresponding solution $v$ is shown in Fig. 16. The solution is odd in
$x$, but shows oscillations at the same $x$-values as $\eta$.

Figure 16. Solution $\eta$ to the KBK system (8.1) for $\varepsilon=0.1$ and
for the initial data $\eta(x,0)=\exp(-x^{2})$, $v(x,0)=0$.

We show a close up of the oscillatory zone in Fig. 17. The smaller
$\varepsilon$, the more rapid the oscillations and the more they are localised
to a zone sometimes called the Whitham zone.

Figure 17. On the left a close-up of the solutions in Fig. 15 and Fig. 16 at
the final time $t=3$, in the upper row $\eta$, in the lower row $v$; on the
right the oscillatory zone for the same initial data for $\varepsilon=0.01$.

The situation is similar for other hump-like initial data as $\eta(x,0)=0$ and
$v(x,0)=\exp(-x^{2})$. The solution $\eta$ for $\varepsilon=0.1$ can be seen
in Fig. 18. The solution is not symmetric, but there are oscillatory zones as
before on both sides of the humps.

Figure 18. Solution $\eta$ to the KBK system (8.1) for $\varepsilon=0.1$ and
for the initial data $v(x,0)=\exp(-x^{2})$, $\eta(x,0)=0$.

The corresponding solution $v$ is shown in Fig. 19. Note that due to the
rescaling of the original KBK system (1.3), the form of the solitons of (8.1)
is slightly different. It appears that the oscillations traveling to the left
in Fig. 19 are solitons.

Figure 19. Solution $v$ to the KBK system (8.1) for $\varepsilon=0.1$ and for
the initial data $v(x,0)=\exp(-x^{2})$, $\eta(x,0)=0$.

##  9\. Conclusion

In this paper we have presented a detailed numerical study of solutions to the
good KBK system (1.3). The stability of the solitons as proven by Angulo [3]
was illustrated for several examples, also for the stationary solution. It was
shown that the long time behavior of solutions for localized initial data is
given by solitons plus radiation. In the vicinity of shocks to the
corresponding dispersionless Saint-Venant system, dispersive shock waves were
observed. No indication of a blow-up was found even in cases where the non-
cavitation condition is not satisfied.

It is an interesting question whether these features can also be observed in
the 2D variant of the KBK system which is most probably not integrable. An
important point is whether there are 2D localised solitary waves called lumps
as in the KP I equation, or whether there are no localized 2D structures as
for KP II. The 1D solitons discussed in the present paper are infinitely
extended exact solutions to the 2D KBK system, so-called line solitons. There
stability in the 2D equation has to be studied as well as the question whether
there can be blow-up in 2D. This will be the subject of future work.

######  Acknowledgements.

We thank Patrik Nabelek for useful discussions on the KBK system.

The work of the first author was partially supported by the ANR project
ANR-17-EURE-0002 EIPHI.

Both authors were partially supported by the ANR project ISAAC-
ANR-23-CE40-0015-01.

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