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.

     

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.     

Existence and Orbital Stability of Solitary-Wave Solutions for Higher-Order BBM Equations

This paper discusses the existence and stability of solitary-wave solutions of a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involves pseudo-differential operators for the linear part. One of such equations can be derived from water-wave problems as second-order approximate equations from fully nonlinear governing equations. Under some conditions on the symbols of pseudo-differential operators and the nonlinear terms, it is shown that the general higher-order BBM equation has solitary-wave solutions. Moreover, under slightly more restrictive conditions, the set of solitary-wave solutions is orbitally stable. Here, the equation has a nonlinear part involving the polynomials of solution and its derivatives with different degrees (not homogeneous), which has not been studied before. Numerical stability and instability of solitary-wave solutions for some special fifth-order BBM equations are also given.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Solitary-wave families of the Ostrovsky equation: An approach via reversible systems theory and normal forms

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Limited functional analytic results exist for the occurrence of one family of solitary-wave solutions of this equation, as well as their approach to the well-known solitons of the famous Korteweg–de Vries equation in the limit as the rotation becomes vanishingly small. Since solitary-wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary-wave solutions and are thus entirely new. Directions for future work are also mentioned.     

广义Pochhammer-Chree方程的显式精确孤波解

首先对广义Pochhammer-Chre方程(PC方程)u_tt-u_ttxx+ru_xxt-(a₁u+a₂u2+a₃u3)_xx=0(r≠0)(Ⅰ)的孤波解u(ξ)建立了公式∫_-∞+∞[u'(ξ)]2dξ=1/12rv(C₊-C_-)3[3a₃(C₊+C_-)+2a₂]。由此推知:广义PC方程(Ⅰ)不可能有钟状孤波解,只可能有扭状孤波解;而广义PC方程u_tt-u_ttxx-(a₁u+a₂u2+a₃u3)_xx=0(Ⅱ)可能既有钟状孤波解又有渐近值满足3a₃(C₊+C_-)+2a₂=0的扭状孤波解。进一步求出了广义PC方程(Ⅰ)的扭状孤波解,求出了广义PC方程(Ⅱ)的钟状孤波解和渐近值满足2a₃(C₊+C_-)+2a₂=0的扭状孤波解。最后给出了广义PC方程u_tt-u_ttxx-(a₁u+a₃u3+a₅u5)_xx=0(Ⅲ)的显式孤波解。     

Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type

We consider solitary-wave solutions of equations of Benjamin-Bona-Mahony type. We show that for a large class of equations of BBM type, there do exist stable sets consisting of solitary-wave profile functions. In the case of generalized BBM equations, we found that there are profile functions of stable solitary waves that are not the minimizers of the associated variational problem. Such a phenomenon is not known to exist for equations of Korteweg-de Vries type.     

Exact solutions for two nonlinear wave equations with nonlinear terms of any order

In this paper, based on a variable-coefficient balancing-act method, by means of an appropriate transformation and with the help of Mathematica, we obtain some new types of solitary-wave solutions to the generalized Benjamin–Bona–Mahony (BBM) equation and the generalized Burgers–Fisher (BF) equation with nonlinear terms of any order. These solutions fully cover the various solitary waves of BBM equation and BF equation previously reported.     

Asymptotic stability of solitary wave solutions to the regularized long-wave equation

We study the asymptotic stability of solitary wave solutions to the regularized long-wave equation (RLW) in . RLW is an equation which describes the long waves in water. To prove the result, we make use of the monotonicity of the local H¹-norm and apply the Liouville property of (RLW) as in Merle and Martel (J. Math. Pures Appl. 79 (2000) 339; Arch. Rational Mech. Anal. 157 (2001) 219).     

Cosine expansion‐based differential quadrature algorithm for numerical solution of the RLW equation

The differential quadrature method based on cosine expansion is applied to obtain numerical solutions of the RLW equation. The propagation of single solitary wave is studied to validate the efficiency of the algorithm. Then, test problems including interaction of two and three solitary waves, undulation, and evolution of solitary waves are implemented. Solutions are compared with earlier results. Discrete conservation quantities are computed for test experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.     

RLW方程的一个新的显式多辛格式

以多辛Euler-box格式为基础对正则长波(RLW)方程的初边值问题进行了讨论,推导了一个新的显式10点格式.模拟孤立波的数值实验表明,这个新的多辛格式是行之有效的,能很好的反映出RLW方程的非弹性性质.     

二维RLW方程和二维SRLW方程的显式精确解

本文讨论了二维RLW方程和二维SRLW方程孤立波解的性态，通过直接积分的方法求出了这两个方程的显式精确孤立波解，并通过选取初始条件的方法求出了二维RLW方程和二维SRLW方程的另一类精确行波解．     

Two dimensional solitary waves in shear flows

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.     

广义修正Boussinesq方程椭圆函数分式形式的精确周期解

本文利用假设待定法求出了广义修正Boussinesq方程的具有Jacobi椭圆函数分式形式的精确周期解,据此还求出了它的若干新精确孤波解．     

Exact chirped solitary-wave solutions for Ginzburg–Landau equation

In this short letter, by applying specially envelope transform and direct ansatz approach to (1 + 1)D Ginzburg–Landau equation the authors obtain a new type of exact solitary wave solution including chirped bright solitary-wave and chirped dark solitary-wave solutions.     

Solving unsteady Korteweg–de Vries equation and its two alternatives

This paper aims to present the generalized Kudryashov method to find the exact traveling wave solutions transmutable to the solitary wave solutions of the ubiquitous unsteady Korteweg–de Vries equation and its two famed alternatives, namely, the regularized long‐wave equation and the time regularized long‐wave equation. The exact analytic solutions of the studied equations are constructed explicitly in three forms, namely, hyperbolic, trigonometric, and rational function. The validity of our solutions is verified with MAPLE by putting them back into the original equation and found correct. Moreover, it has shown that the generalized Kudryashov method is an easy and reliable technique over the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

The collocation method based on quartic B‐spline interpolation is studied for numerical solution of the regularized long wave (RLW) equation. The time‐split RLW equation is also solved with the quartic B‐spline collocation method. Numerical accuracy is tested by obtaining the single solitary wave solution. Then, interaction, undulation and evolution of solitary waves are studied. Solutions are compared with available results. Conservation quantities are computed for all test experiments. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

应用指数函数展开法求解非线性发展方程

利用指数函数展开法,研究BBM方程与KG方程,在一个特定的变换下,借助Maple软件的符号运算功能,获得BBM方程与KG方程指数函数型新的孤立波解与周期解.这种方法用于求解非线性发展方程是简单而有效的.     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.