Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.     

Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water

We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly nonlinear long-wave model. We investigate higher order nonlinear effects on the evolution of solitary waves by comparing our numerical solutions of the model with weakly nonlinear solutions. We carry out the local stability analysis of solitary wave solution of the model and identify an instability mechanism of the Kelvin–Helmholtz type. With parameters in the stable range, we simulate the interaction of two solitary waves: both head-on and overtaking collisions. We also study the deformation of a solitary wave propagating over non-uniform topography and describe the process of disintegration in detail. Our numerical solutions unveil new dynamical behaviors of large amplitude internal solitary waves, to which any weakly nonlinear model is inapplicable.     

Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.     

Stability of Solitary Waves and Global Existence of a Generalized Two-Component Camassa–Holm System

We study here the existence of solitary wave solutions of a generalized two-component Camassa–Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global strong solutions to the equation in some special case.     

PEAKED WAVE SOLUTIONS TO A GENERALIZED HYPERELASTIC-ROD WAVE EQUATION

We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation method. The existence domain of the peaked solitary waves are found. The analytic expressions of peaked solitary wave solutions are obtained. Our numerical simulation and qualitative results are identical.     

Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively.     

Solitary Wave Solutions and Kink Wave Solutions for a Generalized PC Equation

We present in this paper a generalised PC (GPC) equation which includes several known models. The corresponding traveling wave system is derived and we show that the homoclinic orbits of the traveling wave system correspond to the solitary waves of GPC equation, and the heteroclnic orbits correspond to the kink waves. Under some parameter conditions, the existence of above two types of orbits is demonstrated and the explicit expressions of the two solutions are worked out.     

Solitary wave solutions of the generalized two-component Hunter–Saxton system

The existence of solitary wave solutions of the generalized two-component Hunter–Saxton system is determined. It is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions.     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

Coexistence of the solitary and periodic waves in convecting shallow water fluid

The existence of a solitary wave for the shallow water model in convecting circumstance was established in previous works. It is still unknown that whether there exist periodic waves. In this paper, we prove that the models possess periodic waves with a fixed range of wave speed. The amplitude and wave speed are explicitly given. Moreover, the coexistence of the solitary wave and one periodic wave is established.     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

Exact solitary water waves with capillary ripples at infinity

We prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity

We investigate the KdV like equation with higher order nonlinearity ut + a(1 +bun)unux + uxxx = 0with n ≥ 1, a, b ∈ R and α≠ 0. The bifurcations and explicit expressions of solitary wave solutions for theequation are discussed by using the bifurcation method and qualitative theory of dynamical systems. Thebifurcation diagrams, existence and number of the solitary waves are given.     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equation:ut + c0ux + 3uux - α2(uxxt + uuxxx + 2uxuxx) + γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves: periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

Travelling waves of the KP equations with transverse modulations

Kadomtsev-Petviashvili (KP) equations arise genetically in modelling nonlinear wave propagation for primarily unidirectional long waves of small amplitude with weak transverse dependence. In the case when transverse dispersion is positive (such as for water waves with large surface tension) we investigate the existence of transversely modulated travelling waves near one-dimensional solitary waves. Using bifurcation theory we show the existence of a unique branch of periodically modulated solitary waves. Then, we briefly discuss the case when the transverse dispersion is negative (such as for water waves with zero surface tension).     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.