# On the Benjamin and related equations

Christian Klein  Institut de Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France<EMAIL_ADDRESS>,  Felipe Linares  IMPA
Estrada Dona Castorina 110
Rio de Janeiro 22460-320, RJ Brasil<EMAIL_ADDRESS>,  Didier Pilod
Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen,
Norway<EMAIL_ADDRESS>and  Jean-Claude Saut  Laboratoire de
Mathématiques, Bat.307, Université Paris-Saclay et CNRS
91405 Orsay, France<EMAIL_ADDRESS>
(Date: October18th 2023 )

###### Abstract.

We consider in this paper various theoretical and numerical issues on
classical one dimensional models of internal waves with surface tension.They
concern the Cauchy problem, including the long time dynamic, localized
solitons or multisolitons, the soliton resolution property. We survey known
results, present a few new ones together with open questions and conjectures
motivated by numerical simulations.

A major issue is to emphasize the differences of the qualitative behavior of
solutions with those of the same equations without the capillary term.

###### Key words and phrases:

Internal waves, surface tension, Benjamin equation

###### 2010 Mathematics Subject Classification:

Primary 35Q53, 35Q35, 76B15; Secondary 35A01, 76B03

To the memory of Rafael Iório

##  1\. Introduction

###  1.1. The models

This paper is concerned with one-dimensional asymptotic models for weakly
nonlinear internal waves in presence of surface tension.

In absence of surface tension, classical one-dimensional models for long,
weakly nonlinear waves at the interface of a two-layer system when the lower
layer is much larger than the upper one comprise the Benjamin-Ono (BO)
equation derived in [16, 77], see [19, 32, 82, 61] for more details and [80]
for a complete rigorous justification as an internal waves model:

(1.1) $u_{t}+uu_{x}-Hu_{xx}=0,$

where H is the Hilbert transform defined as the convolution with
$PV(\frac{1}{x}),$ that is $\widehat{Hf}(\xi)=-i\text{sgn}(\xi)\hat{f}(\xi),$
(note that here the symbol of $H\partial_{x}$ is $|\xi|$ so that the full
linear symbol is $-i\xi|\xi|)$, and the Intermediate Long Wave (ILW) equation
derived in [65]

(1.2) $u_{t}+uu_{x}+\frac{1}{\delta}u_{x}-\mathcal{T_{\delta}}u_{x}=0,$

where the “Tilbert transform” is defined in Fourier variables by:

$\widehat{\mathcal{T}_{\delta}f}(\xi)=\xi\coth(\delta\xi)\hat{f}(\xi).$

The BO equation corresponds to the limit of an infinite lower layer,
$\delta\to\infty$ in (1.2). In this limit the symbol of
$\widehat{\mathcal{T}_{\delta}}$ loses its regularity.

One also finds in [65] a configuration of the layers where (1.2) should be
replaced by

(1.3)
$u_{t}+uu_{x}+\frac{1}{\delta_{1}}u_{x}-\mathcal{T}_{\delta_{1}}(u_{x})+\frac{1}{\delta_{2}}u_{x}-\mathcal{T}_{\delta_{2}}(u_{x})=0,$

where $\delta_{1}\neq\delta_{2}.$

######  Remark 1.1.

Although the global well-posedness of the ILW equation in $L^{2}$ is now well
understood (see [48, 26]), we recall here known results on the somewhat
unusual equation (1.3). In [7] the global well-posedness of the Cauchy problem
is established in $H^{s}({\mathbb{R}}),s>3/2$ by following the technique used
in [1] for the BO and ILW equations. The results in [76, 75] for the usual ILW
equation (global well-posedness in $H^{s}({\mathbb{R}}),s\geq 1/2$ with
unconditional uniqueness for $s>1/2$ and global well-posedness in
$H^{s}(\mathbb{R})$, $s>\frac{1}{4}$) also apply to (1.3) since the symbol of
this last equation satisfies Hypothesis 1 in [76, 75]. More recently, GWP in
$L^{2}$ both on the line and the torus were proved for (1.3) in [25] (see
Remark 1.3) by relying on the technique introduced in [26] for the ILW
equation.

One also finds in [7] the existence of solitary wave solutions $u(x-ct)$ for
any $c>0$ by minimizing the functional on $H^{1/2}({\mathbb{R}}),$
$J(f)=\int_{-\infty}^{\infty}(Lf)^{2}dx$ where $L$ is the Fourier multiplier
defined by

$\widehat{Lf}(\xi)=(C+m(\xi))^{1/2}\hat{f}(\xi)$

with

$m(\xi)=\xi\coth(\xi\delta_{1})-\frac{1}{\delta_{1}}+\xi\coth(\xi\delta_{2})-\frac{1}{\delta_{2}}$

under the constraint

$\int_{-\infty}^{\infty}f(x)^{3}dx=1.$

Moreover the solitary waves are proven to decay exponentially as
$|x|\to\infty.$ Note that contrary to the original ILW equation the uniqueness
(up to translation) of solitary waves to equation (1.3) seems to be open (see
[5] for the ILW equation).

Concerning the stability issues, the existence of orbitally stable solitary
waves is established in [7] when $\delta_{2}$ is sufficiently close to
$\delta_{1}.$ The stability in the general case is proven in [6].

It is worth noticing that, similarly to the KdV equation, both (1.1) and (1.2)
are completely integrable (see for instance [41, 62, 63, 43], 111The Cauchy
theory on the real line by Inverse Scattering methods is however far from
being complete for those equations, in particular the soliton resolution has
not been proven contrary to the KdV equation, see [85]. On the other hand, the
periodic case was solved for the Benjamin-Ono equation, see [44] and the
references therein. We also refer to [43] for an explicit formula for the
Benjamin-Ono equation on ${\mathbb{R}}$ and ${\mathbb{T}}$ and the surveys
[82, 60] for this aspect). On the other hand, (1.3) is not known to be
integrable.

We also refer to [19] and [32] for a rigorous justification of (1.1) and (1.2)
as models for long internal waves in the sense of consistency.

All the previous equations are derived in absence of surface tension (purely
gravity waves). Actually the effect of surface tension in realistic oceanic
internal waves is tiny and almost negligible. Nevertheless including capillary
effects in classical models of internal waves leads to interesting
mathematical issues and the present paper focusses on this aspect.

The paper is organized as follows. In the next section we present the
derivation (often formal) of the models we will study later. Section 3 deals
with several analytical results on the Cauchy problem and the solitary wave
solutions, while in section 4 we present various numerical simulations that
will illustrate the previous results and help to make relevant conjectures on
the long time behavior of the solutions.

We dedicate this paper to our dear colleague and friend Rafael J. Iorio who
sadly passed away on April 8, 2023.

##  2\. The derivation

We now recall the known derivation of long, weakly nonlinear internal waves
models including surface tension.

In [17] Benjamin derived a version of the BO equation when the interface of
the two layers is subject to surface tension. The modeling depends basically
on the assumption that $T/g(\rho_{2}-\rho_{1})h^{2}\gg 1,$ where $T$ is the
interfacial surface tension, $\rho_{2}-\rho_{1},$ the difference between the
densities and $h$ the undisturbed thickness of the upper layer. This
assumption is satisfied for instance when fluid densities are nearly equal.

We briefly describe here Benjamin’s formal argument (see also [52]). One
considers a two-fluid system made of a layer of fluid of thickness $h$ and
density $\rho_{1}$ over an infinitely deep incompressible fluid of density
$\rho_{2}>\rho_{1},$ the less dense fluid being bounded from above by a rigid
horizontal plane, the interface being subject of surface tension with
coefficient $T>0.$ In this situation, the phase velocity
$c(k)=\frac{\omega(k)}{k}$ corresponding to plane waves
$\epsilon\exp\\{i(\omega t-kx)\\})$ reads

$c^{2}(k)=\frac{g(\rho_{2}-\rho_{1})+Tk^{2}}{\rho_{1}k\coth kh+\rho_{2}|k|}.$

For waves propagating in the positive direction ($c>0$) the asymptotic
approximation of $c(k)$ for very long waves ($|k|\ll 1)$ gives

$c(k)=c_{0}\left[1-\frac{1}{2}\frac{\rho_{2}}{\rho_{1}}h|k|+\frac{1}{2}\left\\{\frac{T}{g(\rho_{2}-\rho_{1})}+\left(\frac{3}{4}\frac{\rho_{2}^{2}}{\rho^{2}_{1}}-\frac{1}{6}\right)h^{2}\right\\}k^{2}+O(|k|^{2})\right].$

For long waves, $|k|^{2}$ is much smaller that $|k|$, and thus the term in
brackets can be neglected leading to the Benjamin-Ono equation. On the other
hand, if it is assumed that $\rho_{1}$ and $\rho_{2}$ are very close, that is
$\frac{T}{g(\rho_{2}-\rho_{1})}h^{2}\gg 1,$ Benjamin writes the approximate
dispersion

$c(k)=1-\alpha|k|+\beta k^{2},$

in which

$\alpha=\frac{h}{2}\rho_{2}/\rho_{1},\quad\beta=\frac{h^{2}}{2}T/g(\rho_{2}-\rho_{1}).$

This leads to the modified BO equation, known as the Benjamin equation, again
in the notations of [17]222In a frame moving with velocity
$c_{0}=\sqrt{\frac{\rho_{2}-\rho_{1}}{\rho_{2}}}$ :

(2.1) $u_{t}+uu_{x}-\alpha Hu_{xx}-\beta u_{xxx}=0,$

where $\beta>0$ is proportional to the surface tension coefficient. We have
kept the parameter $\alpha>0$ to compare the effects of the two dispersive
terms. The linear symbol is now $i\xi(\beta\xi^{2}-\alpha|\xi|)$ so that the
phase velocity has no longer a constant sign, contrary to the KdV or BO
equations and apriori solitary waves can travel in both direction.

We refer to [17] and [4] for further discussions of the physical regime of
validity of Benjamin equation.

######  Remark 2.1.

As aforementioned the derivation of Benjamin equation in [17, 18] is formal
and a rigorous derivation in the sense of consistency is still lacking. The
derivation in [52], is also formal but Kalish derived the following system
version of the Benjamin equation for bi-directional waves in the same physical
context

(2.2)
$\left\\{\begin{array}[]{l}\displaystyle\zeta_{t}+u_{x}+\sigma(u\zeta)_{x}=0\\\
\displaystyle u_{t}+\zeta_{x}+\sigma
uu_{x}-\epsilon\frac{\rho_{2}}{\rho_{1}}H\zeta_{xx}-\mu\zeta_{xxx}=0.\\\
\end{array}\right.$

This system appears to be a perturbation of one of the abcd Boussinesq systems
derived in [23] corresponding to $b=d=0,a=0,c<0$ and which belongs to
”Schrödinger type” abcd systems (the dispersion is of order 2).

More precisely this (abcd) Boussinesq system is a particular case of an Euler-
Korteweg system, see the survey article [84] for known results on this system.
We will comment on (2.2) in Chapter 7.

In [14] C.T. Anh, extending the results in [19] for internal waves without
surface tension, derived rigorously in the sense of consistency from the two-
layer system asymptotic systems in various regimes when surface tension is
taken into account. In the Benjamin-Ono regime, he derived in the one-
dimensional case the following system:

(2.3)
$\left\\{\begin{array}[]{l}\displaystyle(1+\alpha\frac{\sqrt{\mu}}{3}|\partial_{x}|)\partial_{t}\zeta+\frac{1}{\gamma}\partial_{x}((1-\epsilon\zeta)v)-\frac{\sqrt{\mu}}{\gamma^{2}}\partial_{x}((1-\epsilon\zeta)|\partial_{x}|v)\\\
\displaystyle+\frac{\epsilon\sqrt{\mu}}{\gamma^{2}}|\partial_{x}|\partial_{x}(\zeta
v)+\left[(1-\alpha)\frac{\mu}{3\gamma}-\frac{\mu}{\gamma^{3}}\right]\partial_{x}^{3}v=0\\\
\displaystyle\partial_{t}v+(1-\gamma)\partial_{x}\zeta-\frac{\epsilon}{\gamma}v\partial_{x}v+\frac{\epsilon\sqrt{\mu}}{\gamma^{2}}\partial_{x}(v(|\partial_{x}|v))-\epsilon\sqrt{\mu}\nu\partial_{x}^{3}\zeta=0,\\\
\end{array}\right.$

where $\epsilon\sim\sqrt{\mu}\ll 1$ are small parameters measuring
respectively nonlinear and dispersive effects, $\gamma<1$ is the ratio of the
densities of the two layers, $\nu>0$ measures the capillarity effects and
$\alpha\geq 0$ is a modeling parameter.

The classical BO system (without surface tension, see [19]) is obtained from
(2.3) when omitting the terms of higher order $\mu,\sqrt{\mu}\epsilon$ and
thus reads:

(2.4)
$\left\\{\begin{array}[]{l}\displaystyle(1+\alpha\frac{\sqrt{\mu}}{3}|\partial_{x}|)\partial_{t}\zeta+\frac{1}{\gamma}\partial_{x}((1-\epsilon\zeta)v)-(1-\alpha)\frac{\sqrt{\mu}}{\gamma^{2}}\partial_{x}|\partial_{x}|v=0\\\
\displaystyle\partial_{t}v+(1-\gamma)\partial_{x}\zeta-\frac{\epsilon}{\gamma}v\partial_{x}v=0.\\\
\end{array}\right.$

Both systems are linearly well-posed when $\alpha\geq 1.$

As noticed in [19], neglecting the $O(\sqrt{\mu})=O(\epsilon)$ terms in (2.4),
one finds that $\zeta$ must solve a wave equation with speed
$c=\sqrt{\frac{1-\gamma}{\gamma}}.$

Under a one-way assumption, (2.4) yields the one-parameter family of
equations:

(2.5)
$\left(1+\sqrt{\mu}\frac{\alpha}{\gamma}|\partial_{x}|\right)\partial_{t}\zeta+c\partial_{x}\zeta-\epsilon\frac{3}{4}c\partial_{x}\zeta^{2}-\frac{\sqrt{\mu}}{2\gamma}c(1-2\alpha)|\partial_{x}|\partial_{x}\zeta=0.$

The usual BO equation is recovered when $\alpha=0.$

Another derivation of a system version of the Benjamin equation is made in
[39] where the following system is derived, here in its one-dimensional
version

(2.6)
$\left\\{\begin{array}[]{l}\displaystyle\left(1+\frac{\alpha\sqrt{\mu}}{\gamma}|\partial_{x}|\right)\partial_{t}\zeta+\frac{1}{\gamma}\partial_{x}((1+\epsilon\zeta)v)-(1-\alpha)\frac{\sqrt{\mu}}{\gamma^{2}}|\partial_{x}|\partial_{x}v=0,\\\
\displaystyle\partial_{t}v+(1-\gamma)\partial_{x}\zeta-\frac{\epsilon}{\gamma}v\partial_{x}v=T\partial_{x}^{3}v,\end{array}\right.$

where the surface tension parameter $T$ is given by

$T=\frac{\sigma\mu}{g\rho_{2}d_{1}^{2}}.$

In absence of surface tension, (2.6) is the Benjamin-Ono system (33) derived
in [19]. When $\mu=0,$ (2.6) is the (abcd) system in [23] corresponding to
$a=b=d=0,c<0.$

As noticed in [39], (2.6) is linearly well-posed if $\alpha\geq 1.$

An interesting point concerning the Benjamin equation (2.1) is that it links
two completely integrable equations (though being probably not integrable
itself), and it thus appears to be a good candidate to test well-known
features of integrable equations such as the soliton resolution.

As Benjamin did for the Benjamin-Ono equation, one also can take into account
surface tension effects in the ILW equations (1.2), (1.3), yielding
respectively

(2.7) $u_{t}+uu_{x}+\frac{1}{\delta}u_{x}-\mathcal{T}_{\delta}(u_{x})-\beta
u_{xxx}=0,$

(2.8)
$u_{t}+uu_{x}+\frac{1}{\delta_{1}}u_{x}-\mathcal{T}_{\delta_{1}}(u_{x})+\frac{1}{\delta_{2}}u_{x}-\mathcal{T}_{\delta_{2}}(u_{x})-\beta
u_{xxx}=0,$

From now on we will refer to (2.8) as the ILW-Benjamin (ILW-B) equation. As
does the Benjamin equation this equation connects two integrable equations,
the KdV and the ILW equations yielding interesting issues on the long time
behavior of solutions.

######  Remark 2.2.

We will not comment more on the derivation of equations (2.7), (2.8) but the
formal approach of Benjamin provides a justification. We refer again to [14]
for a rigorous derivation (in terms of consistency) of system versions (in
both one and two spacial dimensions) of the BO and ILW equations following the
approach of [19] for the purely gravity waves.

As aforementioned their one-way propagation versions do not reduce to the
Benjamin or ILW-B equations, they contain higher order terms that are
neglected in the Benjamin or ILW-B equations. The situation here is somewhat
similar to the formal derivation of the Kawahara equation which modifies the
KdV equation

(2.9)
$\eta_{t}+c_{0}\eta_{x}+\frac{3c_{0}}{2h}\eta\eta_{x}+\frac{1}{2}c_{0}h^{2}(\frac{1}{3}-\tau)\eta_{xxx}=0,$

when the Bond number $\tau$ is very close to $\frac{1}{3}.$ This leads (see
for instance [46]) to the Kawahara equation

(2.10)
$\eta_{t}+c_{0}\eta_{x}+\frac{3c_{0}}{2h}\eta\eta_{x}+\frac{1}{2}c_{0}h^{2}(\frac{1}{3}-\tau)\eta_{xxx}+\frac{1}{4}c_{0}h^{4}(\frac{14}{90}-\tau^{2})\partial_{x}^{5}\eta=0,$

where higher order nonlinear terms are neglected.

The equations (2.7), (2.1) can be seen as perturbations of the Korteweg-de
Vries equation (KdV)

(2.11) $u_{t}+uu_{x}-\beta u_{xxx}=0.$

They can alternatively be written in the form

(2.12) $u_{t}+uu_{x}-\mathcal{L}u_{x}-\beta u_{xxx}=0,$

where $\mathcal{L}$ is defined in Fourier space by

$\widehat{\mathcal{L}f}(\xi)=p(\xi)\widehat{f}(\xi),$

with $p(\xi)=|\xi|$ for equation (1.1),
$p(\xi)=p_{\delta}(\xi)=\xi\coth(\delta\xi)-\frac{1}{\delta}$ for equation
(1.2) and
$p_{\delta_{1},\delta_{2}}(\xi)=\xi\coth(\delta_{1}\xi)+\xi\coth(\delta_{2}\xi)-\frac{1}{\delta_{1}}-\frac{1}{\delta_{2}}$
for equation (2.7).

To make more clear the interaction of the two dispersive terms we introduce a
parameter $\alpha>0$ and rewrite the equations:

(2.13) $u_{t}+uu_{x}-\alpha Hu_{xx}-\beta u_{xxx}=0,$ (2.14)
$u_{t}+uu_{x}+\alpha\left(\frac{1}{\delta}u_{x}-\mathcal{T}_{\delta}(u_{x})\right)-\beta
u_{xxx}=0.$

and

(2.15)
$u_{t}+uu_{x}+\alpha\left(\frac{1}{\delta_{1}}u_{x}-\mathcal{T}_{\delta_{1}}(u_{x})+\frac{1}{\delta_{2}}u_{x}-\mathcal{T}_{\delta_{2}}(u_{x})\right)-\beta
u_{xxx}=0,$

Note, however, that contrary to the KdV equation the phase velocity
$c(\xi)=\frac{\omega(\xi)}{\xi}$ where $\omega$ is the dispersion relation has
a non constant sign. For instance, for equation (2.13) one finds
$c(\xi)=\beta\xi^{2}-\alpha|\xi|$ and
$c(\xi)=\beta\xi^{2}-\alpha(\xi\coth(\delta\xi)-\frac{1}{\delta})$ for (2.14).

This fact has an influence on the admissible velocities of solitary wave
solutions, see [17, 18] and below.

Another important point is that none of those equations appear to be
completely integrable contrary to the KdV, Benjamin-Ono or Intermediate Long
Wave equations. They are thus very good examples to test the expected soliton
resolution conjecture.

######  Remark 2.3.

There are interesting Kadomtsev-Petviashvili (KP) versions of those one-
dimensional models, that is taking into account weak transverse effects, which
should be useful to study the transverse stability of the one-dimensional
solitary waves.

Proceeding as in the formal derivation of the classical Kadomtsev-Petviashvili
equation in [51] (see for instance [57, 56] for a formal derivation of (2.16)
from the two-layer system for the BO case) one obtains respectively

(2.16) $u_{t}+uu_{x}-\alpha Hu_{xx}-\beta u_{xxx}+\partial_{x}^{-1}u_{yy}=0,$
(2.17)
$u_{t}+uu_{x}+\alpha\left(\frac{1}{\delta}u_{x}-\mathcal{T}_{\delta}(u_{x})\right)-\beta
u_{xxx}+\partial_{x}^{-1}u_{yy}=0,$

and

(2.18)
$u_{t}+uu_{x}+\alpha\left(\frac{1}{\delta_{1}}u_{x}-\mathcal{T}_{\delta_{1}}(u_{x})+\frac{1}{\delta_{2}}u_{x}-\mathcal{T}_{\delta_{2}}(u_{x})\right)-\beta
u_{xxx}+\partial_{x}^{-1}u_{yy}=0.$

We refer to a subsequent work for a study of these equations that are
mathematically interesting since they have both a focusing and defocusing
aspect (depending on frequencies). The KP versions of the Benjamin-Ono and ILW
equations are studied in [70] but many issues remain open.

Note also that due to the second order nonlocal dispersive operator, the KP-
Benjamin type equations do not possess the scaling invariance property

$u_{\lambda}(t,x,y)=\lambda^{2}u(\lambda^{3}t,\lambda
x,\lambda^{2}y),\;\lambda>0$

of the usual KP equation.

##  3\. Analysis

The aim of this section is to review the known results on the Cauchy problem
and on solitary wave solutions for the Benjamin equation and to extend some of
them to the ILW-Benjamin equation.

###  3.1. The Cauchy problem

Concerning the Cauchy problem for the Benjamin equation, Linares has proven in
[69] that (2.1) is globally well-posed for initial data in
$L^{2}({\mathbb{R}})$ and also, in a periodic setting, in
$L^{2}({\mathbb{T}}).$ Well-posedness below $L^{2}({\mathbb{R}})$ was obtained
in [30, 64, 68] leading to global well-posedness in
$H^{s}({\mathbb{R}}),s\geq-3/4.$

We now briefly explain how these results extend to (2.7), (2.8). By writing
these equations on the form (1.3) in [76], we observe that the symbol of the
respective linear symbols of these equations
$p_{ILW-B}=\beta\xi^{3}-(\coth{\delta xi}\xi-\frac{1}{\delta})\xi$ and
$p_{mILW-B}=\beta\xi^{3}-(\coth{\delta_{1}xi}\xi-\frac{1}{\delta_{1}})\xi-(\coth{\delta_{2}xi}\xi-\frac{1}{\delta_{2}})\xi$
satisfy Hypothesis 1 in [76] (this is a direct consequence of Lemma 2.1 in
[76]). Therefore, we deduce from Theorem 1.5 and Corollary 1.10 in [76] that
the Cauchy problems associated with (2.7) and (2.8) are globally well-posed in
$L^{2}(M)$ with $M=\mathbb{R}$ or $\mathbb{T}$.

While the global well-posedness is well established for those equations, the
global behavior of solutions is unknown and challenging. Being in some sense
an “interpolate” equation between two integrable ones, one might expect that
the soliton resolution holds. A first preliminary step towards this conjecture
would be the existence and stability of multi-soliton solutions but this seems
to be an open question as we will discuss later on.

######  Remark 3.1.

Many papers have been devoted to controllability properties of the KdV or
Benjamin-Ono equations, see for instance [67, 73] for the BO case. Concerning
the Benjamin equation, the controllability and stabilization were proven in
[78] for the linearized equation and in [79] for the equation itself, both on
a periodic domain.

It would be interesting to extend those results to (2.7) and (2.8).

###  3.2. Solitary waves

We refer to [7, 4, 12] for existence and stability properties of solitary
waves to (2.1), and to the end of the present chapter for extensions to (2.7),
(2.8).

Note that due to the lack of regularity of the symbol of (2.1) the associated
(non explicit) solitary waves have only an algebraic decay while the decay is
exponential for the two other equations.

We first review and complete known results on the existence and properties of
solitary waves, starting by the BO case that is for the equation (2.13). A
solitary wave solution $u(x,t)=v(x-ct)$ of (2.13) satisfies the equation

(3.1) $-cv-\beta v_{xx}-\alpha|D|v+\frac{v^{2}}{2}=0$

yielding the energy equality

(3.2) $\int_{{\mathbb{R}}}\left(-cv^{2}+\beta
v_{x}^{2}-\alpha|D^{1/2}v|^{2}+\frac{v^{3}}{2}\right)dx=0.$

A Pohojaev identity is obtained by multiplying (3.1) by $xv_{x}$ and
integration (justification by the usual smoothing and truncation process). We
observe by Plancherel and integration by parts that

$\int_{\mathbb{R}}xv_{x}|D|vdx=0$

and obtain

(3.3)
$\int_{{\mathbb{R}}}\left(\frac{c}{2}v^{2}+\frac{\beta}{2}v_{x}^{2}-\frac{v^{3}}{6}\right)dx=0$

and by combination of those identities

(3.4)
$\int_{{\mathbb{R}}}\left(\frac{c}{2}v^{2}+\frac{5\beta}{2}v_{x}^{2}-\alpha|D^{1/2}v|^{2}\right)dx=0,$

that is in Fourier variables

(3.5)
$\int_{{\mathbb{R}}}\left(\frac{c}{2}+\frac{5\beta}{2}|\xi|^{2}-\alpha|\xi|\right)|\hat{v}|^{2}d\xi=0.$

We recover here the well-known fact that for the pure KdV equation
($\alpha=0$) there exists a unique depression solitary wave with velocity c
strictly negative (mind the minus sign in front of $u_{xxx}!$) while for the
pure Benjamin-Ono equation ($\beta=0$) there exists a unique elevation
solitary wave with a strictly positive velocity.

In the present case, (3.5) implies that no solitary wave with large velocities
exists, that is when $c>\frac{\alpha^{2}}{5\beta}$ and a natural conjecture is
that solitary waves solutions of (3.1) exist with velocities
$c\in(-\infty,\frac{\alpha^{2}}{5\beta})$ .

The existence of solitary wave solutions of the Benjamin equation was sketched
in [17] by minimizing the Hamiltonian with fixed $L^{2}$ norm. Using a Leray-
Schauder degree theory, Benjamin [18] proved the existence of solitary waves
(and also of space periodic travelling waves) for small values of the
parameter $\alpha/\sqrt{\beta c}$, by perturbation of a KdV soliton.

Chen and Bona, [27], using a concentration-compactness approach, proved the
existence of solitary waves of velocity c provided
$\min_{x\in{\mathbb{R}}}\\{x^{2}-\alpha|x|+c\\}>0,$ which implies in our
notations $c<-\frac{\alpha^{2}}{4\beta}.$

The later paper contains also a rigorous asymptotics of the solitary waves
following the general theory of Bona and Li, [21], namely as the (explicit)
soliton of the Benjamin-Ono equation, any solitary wave solution of the
Benjamin equation has the asymptotics

$\lim_{x\to\pm\infty}x^{2}\phi(x)=C,$

for some constant $C\in{\mathbb{R}},C\neq 0.$

In [12] Angulo minimizes the energy functional

$E(\psi)=\frac{1}{2}\int_{\mathbb{R}}\left(\beta\psi_{x}^{2}-\psi
H\psi_{x}-\frac{1}{3}\psi^{3}\right)dx$

under the constraint

$\int_{-\infty}^{\infty}\psi^{2}dx=\lambda>0.$

This yields, for any $\lambda>0$ small enough, the existence of a nonempty set
$G_{\lambda}$ of solitary waves with positive stability which is orbitally
stable.

######  Remark 3.2.

In [13], Angulo proved the instability of solitary waves of the generalized
Benjamin equation, extending similar known results for the generalized KdV
equation.

To summarize the above results, existence and stability of solitary waves for
the Benjamin equation is known in the velocity range

$-\infty<c<-\frac{\alpha^{2}}{4\beta}$

and non existence in the range

$c>\frac{\alpha^{2}}{5\beta}.$

Existence for velocities satisfying

$-\frac{\alpha^{2}}{4\beta}<c<\frac{\alpha^{2}}{5\beta}$

is therefore an open problem that will be addressed numerically in the present
paper. Note again that the possibility of existence of solitary waves with
positive velocities is due to the BO term in the Benjamin equation.

######  Remark 3.3.

A stationary solution of the Benjamin equation should satisfy

(3.6) $-\beta v_{xx}-\alpha|D|v+\frac{v^{2}}{2}=0.$

While no such non trivial solutions exist by (3.5) in the pure KdV or
Benjamin-Ono case, the Pohajaev type argument does not exclude the possibility
of existence of steady solutions for the Benjamin equation. This issue appears
to be open.

######  Remark 3.4.

Concerning the qualitative properties of the Benjamin solitary waves we
already mentioned that Bona and Chen [27] gave the algebraic decay rate of the
solitary waves. They also conjecture that the solitary waves are even, have a
finite number of oscillations and decay in a monotone way at $\pm\infty.$ This
issue appears to be open.

######  Remark 3.5.

An interesting open issue is that of the uniqueness (up to the trivial
symmetries) of the solitary waves. Actually the techniques used in [10, 11, 5]
does not work because of the non-positivity of the symbol
$-\alpha|\xi|+\beta\xi^{2}.$

Also, since contrary to the KdV or Benjamin-Ono equations the Benjamin
equation is not integrable, the possible soliton decomposition property is an
important open problem.

We refer to [36, 37] and to section 4 for numerical simulations of solitary
waves of the Benjamin and Benjamin type equations.

######  Remark 3.6.

By an implicit functions argument from the cnoidal wave solutions of the KdV
equation the existence of an analytic curve of periodic solitary waves of the
Benjamin equation was proven in [9] together with their orbital stability. We
briefly sketch here how to extend those results to the ILW-B equation written
as

(3.7) $u_{t}+uu_{x}+u_{xxx}-l\mathcal{T}_{\delta}(u_{xx})=0$

where $l\in{\mathbb{R}}.$

When acting on $2L-$ periodic functions of $x$, the nonlocal operator
$\mathcal{T}_{\delta}$ is defined by (see [2, 1])

$\mathcal{T}_{\delta}f(x)=\frac{1}{\delta}f_{x}+i\sum_{k\in{\mathbb{Z}}^{*}}\coth\left(\frac{k\pi\delta}{L}\right)f_{k}e^{ik\pi
x/L}.$

We are looking for real even periodic traveling wave solutions of the form
$u(x,t)=\phi(x-ct)$ of (3.7), where $\phi=\phi_{l,c}$ satisfies the equation:

(3.8)
$\phi^{\prime\prime}-l\mathcal{T}_{\delta}\phi+\frac{1}{2}\phi^{2}-c\phi=A_{\phi_{l,c}},$

where $A_{\phi_{l,c}}$ is a constant of integration which will be different
from zero. Actually if $\phi_{l,c}$ has mean zero over $[-T,T],$ the constant
of integration satisfies

$A_{\phi_{l,c}}=\frac{1}{4L}\int_{-L}^{L}\phi^{2}_{l,c}(x)dx.$

When $l=0,$ $\phi_{0,c}$ represents the well known periodic solution of the
equation

$\psi^{\prime\prime}_{c}+\frac{1}{2}\psi_{c}^{2}-c\psi_{c}=A_{\psi_{c}},$

the so-called cnoidal wave of the KdV equation.

It is very likely that the implicit function approach used in [9] to prove
existence of a continuous curve of periodic solitary waves of the Benjamin
equation can be used to get a similar result for the ILW-Benjamin equation and
thus solutions to equation (3.8).

We now comment on the existence of solitary waves to the Benjamin-ILW
equation. A natural approach is to look for solutions to the following
variational problem:

(3.9) $\inf\\{\mathcal{H}(u);\int_{\mathbb{R}}u^{2}dx=constant\\},$

where

$\mathcal{H}(u)=\frac{1}{2}\int_{{\mathbb{R}}^{2}}[\beta
u_{x}^{2}-|T_{\delta}^{1/2}u|^{2}+\frac{1}{3}u^{3}],$

as sketched by Benjamin [17] and implemented in [12] for the Benjamin
equation.

Another approach used in [27] for the Benjamin equation would consist in
considering the minimization problem for $\lambda>0:$

$\inf\\{J(u),u\in H^{1}({\mathbb{R}});\
\int_{\mathbb{R}}u^{3}(x)dx=\lambda\\},$

where

$J(u)=\int_{\mathbb{R}}u(c+\mathcal{L})udx,$

$\mathcal{L}$ being the operator with symbol
$|\xi|^{2}-l(\xi\coth(\xi\delta)-1/\delta).$ We refer to a subsequent work for
an implementation of those methods.

The symbol $\xi\coth(\xi\delta)-1/\delta$ of the ILW-Benjamin equation being
smooth, contrary to that of the Benjamin equation, one can expect exponential
decay of the solitary waves of the ILW-Benjamin equation.

A natural idea is to follow the strategy of Bona-Li, [21], writing the
equation of a solitary wave $\phi$ as

$-\phi^{\prime\prime}+\mathcal{T}_{\delta}\phi+c\phi=\frac{1}{2}\phi^{2}$

or

$\phi=\frac{1}{2}K\star\phi^{2},$

where $K$ is the inverse Fourier transform of

Concerning the decay property, the first step would be to obtain the
asymptotic behavior of the Fourier transform of the function

$p(\xi)=\left(c+\beta\xi^{2}-\left(\xi\coth(\delta\xi)-\frac{1}{\delta}\right)\right)^{-1}.$

####  3.2.1. Symmetry and uniqueness of solitary waves

The profile of Benjamin solitary waves appears to be symmetrical, see the
simulations below.

While the solitary waves of the KdV and Benjamin-Ono equation are known to be
unique (see [10] for the delicate case of the BO equation) no such result
seems to be known for the Benjamin equation.

######  Remark 3.7.

1\. The existence and possible stability of multi-solitons is an interesting
open question for the Benjamin type equations.

2\. The non existence of small solitary waves suggests the scattering of small
solution which to our knowledge is an open question.

.

###  3.3. Unique continuation

In this subsection we are concerned with unique continuation principles (UCP)
satisfied for solutions to the equations considered in this paper.

In [54] Kenig, Ponce and Vega proved the following linear result for
generalizations of the BO equation. Suppose that
$k,j\in{{\mathbb{Z}}}^{+}\cup\\{0\\}$ and that

$a_{m}:{\mathbb{R}}\times[0,T]\mapsto{\mathbb{R}},\hskip
5.0ptm=0,1,\dots,k,\hskip 5.0pt\text{and}\hskip
5.0ptb:{\mathbb{R}}\times[0,T]\mapsto{\mathbb{R}}$

are continuous functions with $b(\cdot)$ never vanishing on
$(x,t)\in{\mathbb{R}}\times[0,T]$, and consider the IVP

(3.10)
$\begin{cases}\partial_{t}w-b(x,t)H\partial_{x}^{j}w+\underset{m=0}{\overset{k}{\sum}}a_{m}(x,t)\,\partial_{x}^{m}w=0,\\\
w(x,0)=w_{0}(x).\end{cases}$

######  Theorem 3.8 (Theorem 1.4 [54]).

Let $w\in C([0,T]:H^{s}({\mathbb{R}}))\cap
C^{1}((0,T):H^{s-2}({\mathbb{R}}))$, $s>\max\\{k\,;j\\}+1/2$ be a real
solution to the IVP (3.10). If there exists an open set
$\Omega\subset{\mathbb{R}}\times[0,T]$ such that

(3.11) $w(x,t)=0,\hskip 10.0pt(x,t)\in\Omega,$

then

(3.12) $w(x,t)=0,\hskip 10.0pt(x,t)\in{\mathbb{R}}\times[0,T].$

As pointed out in [54] the following result can be seen as a corollary of
Theorem 3.8.

######  Theorem 3.9 (Theorem 1.1 [54]).

Let $u_{1},u_{2}$ be real solutions to the IVP associated to the BO equation
for $(x,t)\in{\mathbb{R}}\times[0,T]$ such that $u_{1},u_{2}\in
C([0,T]:H^{s}({\mathbb{R}}))\cap C^{1}((0,T):H^{s-2}({\mathbb{R}}))$, $s>5/2$.
If there exists an open set $\Omega\subset{\mathbb{R}}\times[0,T]$ such that

(3.13) $u_{1}(x,t)=u_{2}(x,t),\hskip 10.0pt(x,t)\in\Omega,$

then

(3.14) $u_{1}(x,t)=u_{2}(x,t),\hskip 10.0pt(x,t)\in{\mathbb{R}}\times[0,T].$

In particular, if $u_{1}$ vanishes in $\Omega$, then $u_{1}\equiv 0$.

An application of Theorem 3.8 to the difference of two real solutions
$u_{1},u_{2}$ of the Benjamin equation (2.13) implies that the result in
Theorem 3.9 with $s>7/2$, holds for the IVP associated to the Benjamin
equation (2.13).

In a similar direction, solutions associated to IVP of the ILW equation were
proved to have the same properties as the solutions of the BO equation
mentioned above. For details see Theorem 1.8 and the remark 1.9 in [54]. We
can deduce then that solutions of the IVP associated to the equations (2.14)
and (2.15) for $s>7/2$ satisfy a Unique Continuation Principle (UCP) as the
one described in Theorem 3.9.

A stronger UCP has been obtained for the Benjamin equation (2.13). Consider
the functional space $Z_{s,r}({\mathbb{R}}):=H^{s}({\mathbb{R}})\cap
L^{2}(|x|^{2r}\,dx)$. We recall that for solutions of the BO equation Iorio
([49]) proved that if $u\in C([0,T],Z_{2,2}({\mathbb{R}}))$, and there exist
three distinct times $t_{1},t_{2},t_{3}$ such that the solutions satisfy

(3.15) $u(\cdot,t_{j})\in Z_{4,4}({\mathbb{R}}),\hskip 10.0ptj=1,2,3,$

then $u\equiv 0$. This result is sharp as it was shown in [42]. A similar
result was established in [50] for solutions of the Benjamin equation in
weighted spaces.

######  Theorem 3.10 (Theorem 3 in [50]).

Suppose that $u\in C([0,T]:\dot{Z}_{7,7/2^{-}})$ is a solution of the IVP
associated to the Benjamin equation. If there exist three different times
$t_{1},t_{2},t_{3}\in[0,T]$ such that $u(\cdot,t_{j})\in\dot{Z}_{7,7/2^{-}}$
for $j=1,2,3$, then $u(x,t)\equiv 0$. Here $\dot{Z}_{s,r}:=\\{f\in
Z_{s,r}:\hat{f}(0)=0\\}$.

As far as we know no such result is available for solutions of the ILW-
Benjamin equations (2.14) and (2.15).

##  4\. Numerical simulations

We illustrate here qualitatively the behavior of solutions to the Benjamin
equation aiming to make relevant conjectures. We first construct numerically
solitary waves, then discuss their stability and their appearence in the long-
time behavior of solutions for general initial data.

###  4.1. Solitary waves

To study solitary waves of the Benjamin equation, i.e., localized solutions of
(3.1),

(4.1) $-cQ+\frac{1}{2}Q^{2}-\alpha HQ_{x}-\beta Q_{xx}=0,$

we first construct them numerically. Note that $Q$ depends on three
parameters, $c$, $\alpha$, $\beta$, and only one of them can be scaled out by
rescaling $Q$ and $x$ appropriately. Therefore we will consider the dependence
on some of these parameters whilst the others are kept fixed.

The numerical approach is as in [61] a Fourier method with a Newton-Krylov
iteration. The idea is to study (4.1) in Fourier space

(4.2) $\mathcal{F}(\hat{Q}):=\mathcal{L}\hat{Q}+\frac{1}{2}\widehat{Q^{2}}=0,$

where $\mathcal{L}=\beta k^{2}-\alpha|k|-c$. The Fourier transform will be
approximated in standard way with a discrete Fourier transform (DFT); for
simplicity both are denoted with the same symbol. This means we approximate a
situation on the real line by one on the torus with period $2\pi L$, i.e., we
choose $x\in L[-\pi,\pi]$ with $L>0$ large enough such that the solutions
decay to machine precision (here $10^{-16}$) at the boundaries of the
computational domain. Below we always use $L=20$ for the solitary waves. On
the interval we use the standard discretisation of the DFT, which is computed
via a Fast Fourier transform (FFT). This leads for (4.2) to a system of $N$
(the number of discrete Fourier modes) nonlinear equations $\mathcal{F}=0$
which are solved with a Newton iteration,

$\hat{Q}^{(n+1)}=\hat{Q}^{(n)}-\left(\mbox{Jac}(\mathcal{F})|_{\hat{Q}^{(n)}}\right)^{-1}\mathcal{F}(\hat{Q}^{(n)}),\quad
n=0,1,\ldots,$

where $\mbox{Jac}(\mathcal{F})$ is the Jacobian of $\mathcal{F}$. To compute
the action of the Jacobian on a vector, we use the Krylov subspace technique
GMRES.

To obtain the solitary waves, we apply a tracing technique. We start with the
well known KdV soliton for $\beta=1$, $c=-1$ and $\alpha=0$,
$Q=3c\mbox{sech}(x/2/\sqrt{(}-c/\beta))^{2}$. We do not change the value of
$\beta$ and $c$ in the following, but slowly increase the value of $\alpha$ up
to close to the limiting value 2. The equation for the soliton with $c=-1$ is
then solved for a slightly larger value of $\alpha$ with the solitary wave for
the previous value of $\alpha$ as an initial iterate $\hat{Q}^{(0)}$. Note
that we always use $N=2^{10}$ Fourier modes here, and that the coefficients of
the DFT decrease for all shown examples to $10^{-10}$ and below which gives an
indication of the numerical error. We show the resulting solitons for several
values of $\alpha$ in Fig. 1. It can be seen that for larger values of
$\alpha$, the soliton develops some sort of elevation which turns into
additional oscillations.

Figure 1. Benjamin soliton for $c=-1$, $\beta=1$ and various values of
$\alpha$.

For even larger values of $\alpha$, more and more oscillations appear in the
limit $\alpha\to 2$, see Fig. 2. Note also that the amplitude of the solitons
decreases with $\alpha$.

Figure 2. Benjamin soliton for $c=-1$, $\beta=1$ and $\alpha=1.95$ on the left
and $\alpha=1.99$ on the right.

Note that the tracing approach is important to find the solutions shown in
Fig. 1. If we do for instance the same iteration as above for $\alpha=\beta=1$
with the KdV soliton as the initial iterate, we get the solution shown in Fig.
3 on the left. The residual of the Newton iteration, i.e.,
$||\mathcal{F}(\hat{Q})||_{\infty}$ is as for the solutions in Fig. 1 smaller
than $10^{-10}$. This could indicate that the critical points are not unique.
But if we compute the energy, it is roughly 3.5 times the energy of the
solitary wave in Fig. 1. Thus the ground states appear to be of the form shown
in Fig. 1.

Figure 3. On the left the solution to (4.1) for $c=-1$, $\beta=1$ and
$\alpha=1$ for the initial iterate $Q=3c\mbox{sech}^{2}(x/2/\sqrt{-c/\beta})$
(KdV soliton), on the right solitary waves for $\alpha=1$, $\beta=1$ and
various values of $c$.

If we fix $\alpha=\beta=1$ and vary $c$ starting from $c=-1$ to larger values
of $c$ — again taking the solution at the slightly smaller value of $c$ as the
initial iterate for a larger value of $c$ — we get the sequence of solutions
shown in Fig. 3 on the right. Again the solutions become smaller and more
oscillatory when $c$ approaches the critical velocity $-1/4$.

The situation is slightly different, if we look for solitary waves in the
vicinity of the BO soliton, $Q=4c/(1+c^{2}x^{2})$ for $\alpha=1$, $\beta=0$.
Using the BO soliton for $c=\alpha=1$ and $\beta=10^{-2}$ as the initial
iterate with $N=2^{12}$ and $L=50$, we find the solution on the left of Fig.
4. It appears that there are solitary waves of a similar form as for BO for
small $\beta$. The maximum is only slightly larger than the one of the BO
soliton.

Figure 4. Solution to (4.1) for $c=1$, $\alpha=1$ and $\beta=10^{-2}$ on the
left and for $c=1$, $\alpha=1$ and $\beta=6*10^{-2}$ on the right.

However, this changes for values of $\beta\gtrsim 5*10^{-2}$. As shown on the
right of Fig. 4, there appears to be only a solution to (4.1) with an
oscillatory singularity at infinity. This is not due to the chosen numerical
approach since the behavior does not depend strongly on the chosen period.
Even if we apply the method of [58] where equation (4.1) is considered on the
compactified real line without any periodicity assumption, these oscillations
are observed (note that in this case a first order zero of $Q$ at infinity is
enforced). Thus it appears that solitary waves only exist for very small
$\beta$ in the vicinity of the BO soliton.

######  Remark 4.1.

The iterative approach applied in this section implies that an initial iterate
in the vicinity of the wanted solution has to be chosen since the convergence
of Newton iterations is local, see for instance Fig. 3 on the left.
Consequently we cannot decide with this method whether there are stationary
solutions to the Benjamin equation since we do not have an appropriate initial
iterate and since the branches of solutions we could study above do not have a
(non-trivial) limit for $c\to 0$.

The same numerical approach will be used for solitary waves of the KdV-ILW
equation (2.7). In Fig. 5 we show solitary waves obtained by deforming the KdV
soliton as in Fig. 1. We put $c=-1$, $\beta=1$, $N=2^{10}$ for $x\in
3[-\pi,\pi]$, and $\delta=0.1$. The solitary waves are much more localised
than the ones with a BO term in the Benjamin equation in Fig. 1.

Figure 5. Solitary waves for the KdV-ILW equation (2.7) for $c=-1$, $\beta=1$,
$\delta=0.1$ and various values of $\alpha$.

Again the waves become smaller in amplitude and more oscillatory for larger
values of $\alpha$, which is even more visible in Fig. 6. The limiting value
of $\alpha$ beyond which there are no more localised solitary waves in this
sequence is unknown, but must be close to the shown values.

Figure 6. Solitary waves for the KdV-ILW equation (2.7) for $c=-1$, $\beta=1$,
$\delta=0.1$ and $\alpha=31.3$ on the left and $\alpha=31.5$ on the right.

For larger values of $\delta$, here $\delta=0.9$, the behavior is similar, but
closer to the KdV soliton as can be seen in Fig. 7. The solitary wave becomes
once more oscillatory for $\alpha=4.7$ as shown on the right of the same
figure. The precise limiting value of $\alpha$ for solitary waves in this
sequence is again unknown but close to 5.

Figure 7. Solitary waves for the KdV-ILW equation (2.7) for $c=-1$, $\beta=1$,
$\delta=0.9$ and several values of $\alpha$ on the left and for $\alpha=4.7$
on the right.

The ILW equation has explicit solitons where only parameters have to be
determined for given velocity $c$ and $\delta$ as the solution of a
transcendental equation, see for instance [60]. As in Fig. 4 one can thus look
for solitary waves in the vicinity of the ILW soliton. We put $c=1$,
$\alpha=1$ and consider small values of $\beta$. We get solitary waves for
small values of $\beta$ in this case that are close to the ILW solitons, see
Fig. 8. For larger values of $\beta$ than those shown in the figure, rapid
oscillations appear quickly as in Fig. 4 on the right.

Figure 8. Solitary waves to KdV-ILW equation to (2.7) for $c=1$, $\alpha=1$,
$\delta=0.1$ and $\beta=10^{-2}$ on the left and for $c=1$, $\alpha=1$,
$\delta=0.9$ and $\beta=4*10^{-2}$ on the right.

###  4.2. Time evolution

For the time evolution of initial data for the Benjamin equation, we use the
same discretisation in $x$ and FFT techniques as in the construction of the
soliton. This leads to a system of the form

$\hat{Q}_{t}=\mathbb{L}\hat{Q}+\mathcal{N}(\hat{Q}),$

where $\mathbb{L}=ik(\alpha|k|-\beta k^{2})$ and where
$\mathcal{N}(\hat{Q})=ik\widehat{Q^{2}}/2$. Since $\mathbb{L}$ is cubic in
$k$, the resulting system is _stiff_ which means that explicit time stepping
schemes are inefficient. Therefore we apply an exponential time differencing
(ETD) scheme, see [45] for a review. Since in [59] the studied fourth order
ETD schemes performed similarly, we apply here the scheme by Cox and Matthews
[33]. The numerical accuracy of the solution is controlled as discussed in
[59] via the decrease of the Fourier coefficients333It is known that the DFT
of analytic periodic functions decreases as the Fourier coefficients
exponentially, and for simplicity we speak in the following of the Fourier
coefficients when we refer to the DFT. and the conservation of the numerically
conserved energy. The latter will depend on time due to unavoidable numerical
errors, and its conservation will thus allow to estimate the numerical
accuracy of the time integration.

The code is tested at the example of a solitary wave, for instance the one for
$\alpha=1.95$, $\beta=1$ and $c=-1$ shown in Fig. 2 on the left. We use
$N=2^{10}$ Fourier modes for $x\in 20[-\pi,\pi]$ with $N_{t}=10^{4}$ time
steps for $t\in[0,4]$. The energy is conserved relatively to the order
$10^{-12}$ during the whole computation. The difference between the numerical
solution and the initial data propagated with $c=-1$ is of the order of
$10^{-14}$. This tests both the time evolution code as well as the code for
the numerical construction of the solitary waves. It also shows that the
energy conservation can be used as an indicator of the acuracy of the time
integration.

####  4.2.1. Stability of solitary waves

The Benjamin solitary wave is known to be stable. To illustrate this, we
consider perturbations of the soliton with $c=-1$, $\alpha=1.95$ and $\beta=1$
as above and use the same numerical parameters as there. In Fig. 2, we show
the solution to the Benjamin equation for $t=4$ for the initial data
$u_{0}(x)=Q(x)\pm 0.05\exp(-x^{2})$, where $Q$ is the solitary wave for these
parameters. This means we consider a solitary wave with a small Gaussian
perturbation. The numerically computed energy is conserved to the order of
$10^{-12}$. The solutions at the final time are shown in Fig. 9. In red we
give the unperturbed solitary wave. Since the perturbation is small, but
finite (in order to allow for numerically visible effects of the perturbation
for finite times), the final state is slighly different from the unperturbed
solitary wave, but obviously as expected close. The perturbation leads to
radiation emitted towards infinity.

Figure 9. Solution to the Benjamin equation for $t=4$ for initial data
$u_{0}(x)=Q(x)\pm 0.05\exp(-x^{2})$ where $Q$ is the solitary wave for
$\alpha=1.95$, $\beta=1$ and $c=-1$, in blue and the unperturbed solitary wave
in red; on the left for the $+$ sign of the perturbation, on the right for the
$-$ sign.

To explore solitary waves in the vicinity of BO, we consider the case
$\alpha=c=1$ and $\beta=0.02$. The solitary wave is constructed with
$N=2^{14}$ Fourier modes for $x=50[-\pi,\pi]$. Initial data corresponding to
this solitary wave are propagated with $N_{t}=10^{4}$ time steps for
$t\in[0,4]$ with a relative energy of the order of $10^{-12}$ to a solution
with a difference to the travelling wave of the same order at the final time
of the computation.

We apply the same numerical parameters for the studies presented below. A
perturbation of the order of $10\%$ seems to lead to instabilities, but these
are presumably outside the domain of applicability of stability theory. A
perturbation of the order of $1\%$ leads to similar results as in Fig. 9.
Concretely we consider initial data of the form $u_{0}(x)=Q(x)\pm
0.04\exp(-x^{2})$ where $Q$ is this time the solitary wave for $\alpha=c=1$
and $\beta=0.02$. The solutions at time $t=4$ are shown in Fig. 10. They are
again close to the unperturbed travelling wave which indicates the stability
of these solitary waves.

Figure 10. Solution to the Benjamin equation for $t=4$ for initial data
$u_{0}(x)=Q(x)\pm 0.05\exp(-x^{2})$ where $Q$ is the solitary wave for
$\alpha=1$, $\beta=0.02$ and $c=1$, in blue and the unperturbed solitary wave
in red; on the left for the $+$ sign of the perturbation, on the right for the
$-$ sign.

####  4.2.2. The soliton resolution conjecture

Since the solitary waves are stable, it is expected that they appear in the
long time behavior of solutions of the Benjamin equation for initial data of
sufficient mass. As an example we consider $u_{0}(x)=-10\exp(-x^{2})$ with
$N=2^{12}$ Fourier modes and $L=50$ (the other numerical parameters are
unchanged). The solution for $t=4$ can be seen in Fig. 11. In the shown close-
up at least one soliton appears, possibly two. Since the solitary wave does
not have a simple scaling in $c$, it is difficult to work out the related $c$
and to compare with the corresponding solution of (4.1).

Figure 11. A close-up of the solution to the Benjamin equation with
$\alpha=\beta=1$ for $t=4$ for initial data $u_{0}(x)=-10\exp(-x^{2})$.

The interpretation of the larger hump as a solitary wave is in accordance with
the $L^{\infty}$-norm of the solution on the left of Fig. 12. It appears to
settle on a plateau which is generally an indication that a stable solitary
wave is the final state of the largest hump. The oscillations in the
$L^{\infty}$ norm are due to radiation emitted to the left which reappears on
the other side of the computational domain since we approximate a situation on
$\mathbb{R}$ by a situation on $\mathbb{T}$. The radiation can be seen on the
right of Fig. 12 where the solution of Fig. 11 is shown on the full
computational domain.

Figure 12. $L^{\infty}$ norm of the solution to the Benjamin equation with
$\alpha=\beta=1$ for initial data $u_{0}(x)=-10\exp(-x^{2})$ in dependence of
$t$ on the left, and the solution of Fig. 11 on the full computational domain
on the right.

The situation is slightly different in the vicinity of BO, for instance for
$\alpha=1$ and $\beta=0.02$, where we found a solitary wave for $c=1$, see
Fig. 3 on the left. We use $N=2^{14}$ Fourier modes for $x\in 50[-\pi,\pi]$
and $N_{t}=10^{4}$ time steps for $t\in[0,20]$. The solution for the initial
data $u_{0}=5\exp(-x^{2})$ can be seen on the right of Fig. 13. There appears
to be a solitary wave plus radiation traveling to the left. This
interpretation is backed by the $L^{\infty}$ norm shown on the left of the
same figure seemingly reaching a plateau for large times.

Figure 13. $L^{\infty}$ norm of the solution to the Benjamin equation with
$\alpha=1$, $\beta=0.02$ for initial data $u_{0}(x)=5\exp(-x^{2})$ in
dependence of $t$ on the left, and the solution for $t=20$ on the right.

If we apply the same numerical parameters for times smaller than 5 and the
same initial data to the case $\alpha=1$, $\beta=0.06$, the solution behaves
differently. The $L^{\infty}$ norm on the left appears to decrease without
reaching a plateau. The main difference to Fig. 14 is, however, in the
formation of a strong dispersive shock wave to the right of the initial hump.
There is still some radiation emanating to the left, but it appears that the
main part of the initial hump will end up in the modulated oscillations
propagating towards the right. The behavior is similar to what is known from
the Kawahara equations, a fifth order KdV-type equation, see for instance [38]
and references therein. The dispersive term has the Fourier symbol
$ik(\alpha|k|-\beta k^{2})$. For $k>0$, plane waves with wave number
$k<\alpha/\beta$ travel to the left, whereas those with $k>\alpha/\beta$ have
positive phase velocity.

Figure 14. $L^{\infty}$ norm of the solution to the Benjamin equation with
$\alpha=1$, $\beta=0.06$ for initial data $u_{0}(x)=5\exp(-x^{2})$ in
dependence of $t$ on the left, and the solution for $t=5$ on the right.

The situation is similar for the KdV-ILW equation (2.7). The solitary waves
appear to be stable. Since the figures are very similar, we concentrate on the
case shown in Fig. 14. When the dispersion and thus the phase velocity has
different signs for different wave numbers, one gets a similar behavior as for
the Benjamin equation. We use the same numerical parameters and the same
initial data and get similar solutions in Fig. 15. Once more there are
oscillations traveling to the left and rapid oscillations to the right. This
explains to an extent why we could not find solitary waves in this regime.

Figure 15. Solution to the KdV-ILW equation (2.7) with $\alpha=1$ for initial
data $u_{0}(x)=5\exp(-x^{2})$, on the left for $\delta=0.1$ and $t=1$, on the
right for $\delta=0.9$ and $t=5$.

##  5\. Outlook

######  Acknowledgements.

F.L. was partially supported by CNPq and FAPERJ/Brazil. C.K. and J.-C.S. were
partially supported by the ANR project ISAAC - ANR-23-CE40-0015-01. C.K.
thanks for support by the ANR-17-EURE-0002 EIPHI and by the European Union
Horizon 2020 research and innovation program under the Marie Sklodowska-Curie
RISE 2017 grant agreement no. 778010 IPaDEGAN. D.P. was supported by a Trond
Mohn Foundation grant.

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