Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ .     

一类反应扩散方程的孤立周期波和局部临界周期分支

研究了一类含有五次非线性反应项和常数扩散项的反应扩散方程的小振幅孤立周期波解,以及它的行波方程局部临界周期分支问题.运用行波变换将反应扩散方程转换为对应的行波系统,应用奇点量方法和计算机代数软件MATHEMATICA计算出该系统的前8个奇点量,得到该系统奇点的两个中心条件,并证明行波系统原点处可分支出8个极限环,对应的...     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.     

Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein–Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one‐dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

The linear stability of the solitary waves for the one‐dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Some exact solutions for Toda type lattice differential equations using the improved (G′/G)‐expansion method

Nonlinear lattice differential equations (also known as differential‐difference equations) appear in many applications. They can be thought of as hybrid systems for the inclusion of both discrete and continuous variables. On the basis of an improved version of the basic (G′/G)‐expansion method, we focus our attention towards some Toda type lattice differential systems for constructing further exact traveling wave solutions. Our method provides not only solitary and periodic wave profiles but also rational solutions with more arbitrary parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of one non‐trivial anti‐periodic solution for second‐order impulsive differential inclusions

The existence of one non‐trivial solution for a second‐order impulsive differential inclusion is established. More precisely, a recent critical point result is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non‐trivial anti‐periodic solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.     

Perturbation Theory for the Solitary Wave Solutions to a Sasa‐Satsuma Model Describing Nonlinear Internal Waves in a Continuously Stratified Fluid

The adiabatic evolution of perturbed solitary wave solutions to an extended Sasa‐Satsuma (or vector‐valued modified Korteweg–de Vries) model governing nonlinear internal gravity propagation in a continuously stratified fluid is considered. The transport equations describing the evolution of the solitary wave parameters are determined by a direct multiple‐scale asymptotic expansion and independently by phase‐averaged conservation relations for an arbitrary perturbation. As an example, the adiabatic evolution associated with a dissipative perturbation is explicitly determined. Unlike the case with the dissipatively perturbed modified Korteweg–de Vries equation, the adiabatic asymptotic expansion for the Sasa‐Satsuma model considered here is not exponentially nonuniform and no shelf region emerges in the lee‐side of the propagating solitary wave.     

二维浅水波方程的间断周期解和间断孤立波解讨论

本文讨论地球流体运动浅水波模式中间断周期解与间断孤立波解.在系统的非平衡点即奇点附近考虑轨线性质时,我们发现只要引入广义解的概念(分片光滑连续解),就会产生间断周期解并得到了间断周期解的条件.当系统在退化的过程中,发现系统此时会产生间断的孤立波解,与此同时其它物理量也产生了间断.这里我们发现,一般认为在超高速情况下解会产生间断,然而在非超高速时也会产生间断现象.本文讨论了上述一系列问题得到了间断解的解析解表达式,并把这一事实与飑线的实例进行比较,得到了不少类似之处.