Solitons and cavitons in a nonlocal Whitham equation

Highlights

• We investigate localized traveling wave solutions to the nonlocal Whitham equation.

• The phase space of the system obtained is two-sheeted leading to necessity of jumps and gluing solutions from different sheets.

• Homoclinic orbits we search for can be to a saddle-center or a saddle-focus, many of them have been found.

• Bifurcation and normal form methods give a way to find the origin of homoclinic orbits existence.

• It is shown methods of symplectic dynamics to work well despite the existence of jumps.

Abstract

Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one sheet represent smooth solutions of the equation but those which perform transitions through the branching plane represent solutions with jumps. They correspond to solutions with singularities of the fourth order differential equation – breaks of the first and third derivatives but continuous even derivatives. The Hamiltonian system can have two types of equilibria on different sheets, they can be saddle-centers or saddle-foci. Using analytic and numerical methods we found many types of homoclinic orbits to these equilibria both with a monotone asymptotics and oscillating ones. They correspond to solitons and cavitons of the initial equation. When we deal with homoclinic orbits to a saddle-center, the values of the second parameter (physical wave speed) are discrete but for the case of a saddle-focus they are continuous. The presence of multiplicity of such solutions displays the very complicated dynamics of the system.

Introduction

Nonlinear nonlocal Whitham equation

$$\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}=\frac{\partial }{\partial x}{\int }_{-\infty }^{\infty }R(x-\xi )V(\xi ,t)d\xi$$

represents a wide class of equations which are of great interest for nonlinear wave theory. It combines the typical hydrodynamic nonlinearity and an integral term descriptive of dispersion of the linear theory. The kernel of the integral term is conventionally defined by the dispersion relation $\omega =k\tilde{R}\left(k\right)$ with

$$R\left(x\right)={\int }_{-\infty }^{\infty }\tilde{R}\left(k\right)e^{-ikx}dk.$$

Eq. (1) with $\tilde{R}={(1+k^2)}^{-1}$ was proposed by G.B. Whitham instead of Korteweg-de Vries equation in order to describe sharp crests of the water waves of a greatest height [1].

The usage of relatively simple Whitham type equations appeared to be very fruitful for various physical applications. A number of special cases of Eq. (1) were examined in detail. Among them are the Benjamin-Ono [2], [3] and Joseph [4] equations describing internal waves in stratified fluids of infinite and finite depth. These equations appeared to be integrable by the inverse scattering technique and the behavior of their solutions has been studied rather well. The Benjamin-Ono and Joseph equations are however the only representatives of the Whitham equations possessing this property [5]. Another widely known equations of that class were studied not so exhaustively, although the literature on the subject is quite extensive. A list of well-known Whitham equations involves the Leibovitz one for the waves in rotating fluid [6], the Klimontovich equation for magnetohydrodynamic waves in non-isothermal collision-less plasma [7], equations for shallow water waves [1], capillary [10] and hydroelastic [11] waves. The review on nonlinear nonlocal equations in theory of waves is presented in detailed monograph [8].

The characteristic feature of above-listed Whitham equations is the existence of solitary wave solutions. For all just listed equations these solutions are smooth except some limiting cases of peaking for the waves of greatest amplitude. Besides, the amplitudes and velocity spectra of solitons can be bounded or not. But in any case the spectra are continuous. These properties are believed to be typical, but, as will be shown below, they are not essential for the solitons of Whitham equations.

Here we examine a particular case of the Whitham equation with a resonance dispersion relation

$$\tilde{R}\left(k\right)=\frac{1}{1-k^2+D^2{k}^4}.$$

That equation has been proposed for nonlinear acoustic waves in simple peristaltic systems [44]. With small D² it is also applicable to the waves in a medium with internal oscillators [12]. A tentative analysis of some peculiarities of solitons to that equation has been performed in a short communication [13].

Dimensionless (1), (2), (3) are written in a coordinate system moving to the right with the speed of sound in an unperturbed fluid. The independent variables x and t represent phase and time; variable V is proportional to the pressure in the fluid. For peristaltic systems the only parameter of (1), (2), (3) is equal

$$D^2=\left(E_x{h}^2\right)/\left(12(1-v^2)p_S^2{a}^2{c}_0^4\right)$$

where E_x, p_s, v are Young modulus, density and Poisson ratio of the shell material, h and a are thickness and radius of the shell, c₀ is the sound speed of the fluid. It should be noted that the form of the integral term in Eq. (1) depends on the poles location of the Fourier transform of the kernel. While D² < 1/4 the poles are located in the real axis so that the causative-type dispersion takes place, $R(x<0)=0$. But for D² > 1/4, the poles are located symmetrically with respect to $Im\left(k\right)=0,R(-x)=R\left(x\right),$ and dispersion therefore takes a space-type form. The space-type dispersion occurs when the sound velocity of a fluid is greater than the minimum phase velocity of bending oscillations of the shell. This is caused by the origination of radiation physically similar to Cherenkov one [44].

We study here specific features of solitary wave solutions to (1), (2), (3). It is shown that this equation possesses both smooth and singularity involving solitons with monotone exponential asymptotics, bound states of solitons and solitary waves with oscillating exponential asymptotics. The velocity spectra of localized solitons with monotone asymptotics turn out to be discrete ones. In fact, we study traveling wave solutions with localization property, the equation for such solutions (see below) is reduced to a Hamiltonian system with two degrees of freedom and we apply various tools to discover and classify the needed solutions. Solutions we seek are homoclinic orbits to equilibria of the Hamiltonian system obtained. The types of these equilibria depend on parameters (D², λ) and we apply different methods to discover homoclinic orbits and present their classification in dependence of their asymptotic properties. Existence of such orbits for different types of equilibria is our main results. To find them we apply methods of the theory of dynamical systems, homoclinic dynamics, bifurcation theory and numerical simulations.