Solitary waves for Korteweg-de Vries equation with small delay

This paper deals with the existence of solitary waves for Korteweg-de Vries equation with time delay. Based upon the inertial manifold theory and differential manifold geometric theory, the existence of solitary wave solution is proved when the delay is small enough. Up to now, studies on solitary wave for such delay differential equation are not available, so the results of this paper are new.     

Scattering of solitons for coupled wave-particle equations

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six-dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.     

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.     

The geodesic hypothesis and non-topological solitons on pseudo-Riemannian manifolds

A class of solitary wave solutions to a semi-linear wave equation on a pseudo-Riemannian manifold is studied. A construction of solutions which concentrate on geodesics is given.     

Schrödinger soliton from Lorentzian manifolds

In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.     

Solitary waves in a cold plasma

We study the existence of soliton-like solutions (solitary waves) to the equations describing the one-dimensional motion of a cold quasi-neutral plasma. It is shown that in some range of the angle between the norperturbed magnetic field and the wave propagation direction there exists a branch of solitary hydromagnetic waves that is a bifurcation of the zero wave number. These solutions lie on a two-dimensional center manifold.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 719–728, May, 1996.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

New explicit solutions for (2 + 1)-dimensional soliton equation

In this Letter, we study (2 + 1)-dimensional soliton equation by using the bifurcation theory of planar dynamical systems. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Bell profile solitary wave solutions, kink profile solitary wave solutions and periodic travelling wave solutions are given. Further, we present the relations between the bounded travelling wave solutions and the energy level h. Through discussing the energy level h, we obtain all explicit formulas of solitary wave solutions and periodic wave solutions.     

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.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

On the solitary wave dynamics,under slowly varying medium,for nonlinear Schrödinger equations

We consider the problem of a solitary wave propagation, in a slowly varying medium, for a variable-coefficients nonlinear Schrödinger equation. We prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media. Moreover, we describe for all time the behavior of these solutions, which include refracted and reflected solitary waves, depending on the initial energy.     

微结构固体中孤立波的演变及非光滑孤立波

给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.     

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.     

On the solitary wave solutions of the CQNLS

In this paper, coexistence and simplified formulations of the solitary waves of the cubic–quintic non-linear Schrödinger equation (CQNLS) are investigated by analyzing the steady bifurcation and the energy integral of the conservative dynamical system satisfied by the wave packet. It is found that the bright solitary waves can coexist with kinks and anti-kinks in a range of the bifurcation control parameter. There exists a critical parameter value at which the dark solitary waves are distinguished from the bright solitary waves, kinks and anti-kinks. All of the simplified solitary wave solutions, kinks and anti-kinks are obtained by using our previously developed approximate method.     

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.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

Geometric integration of the paraxial equation

Evolution of solitary waves in photovoltaic-photorefractive crystal satisfy the paraxial equation. The paraxial equation is transformed into the symplectic structure of the infinite dimensional Hamiltonian system. The symplectic structure of the paraxial equation is discretizated by the symplectic method. The corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation. The symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation. Evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated.     

Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.     

Solitary waves in the nolinear wave equation and in gauge theories

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. This paper is an introduction to the study of solitary waves relative to the nonlinear wave equation and to the Abelian gauge theories. Abelian gauge theories consist of a class of field equations obtained by coupling in a suitable way the nonlinear wave equation with the Maxwell equations. They provide a model for the interaction of matter with the electromagnetic field. One of the motivations of this study lies in the fact that the nonlinear wave equation and the Abelian gauge theories are the simplest equations which satisfy the basic principles of modern physics. Dedicated to the memory of Jean Leray     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.