Instability of bound states for abstract nonlinear Schrödinger equations

We study the instability of bound states for abstract nonlinear Schrödinger equations. We prove a new instability result for a borderline case between stability and instability. We also reprove some known results in a unified way.     

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.     

Free and forced vibrations of nonlinear wave equations in ball

First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation (∂_t²−△)u=f(r,u) in n-dimensional ball B_a with center at the origin and radius a, where f is smooth, monotone decreasing in u, and satisfies f(r,0)=0. f(r,u) has asymptotic properties . For n=1,3 we shall show the existence of infinitely many radially symmetric time-periodic solutions with different periods of irrational multiple of a. Second, we shall deal with BVP for a forced nonlinear wave equation (∂_t²−△)u=εg(r,t,u), where g is T-periodic in t and ε is a small parameter. Under some Diophantine condition on a/T we shall show the existence of time-periodic solutions of the BVP. For 1?n?5 we shall construct infinitely many T satisfying the above Diophantine inequality, using asymptotic expansions of the zero points of the Bessel functions.     

The existence of ground states for fourth-order wave equations

Focusing on the fourth-order wave equation u_tt+Δ²u+f(u)=0, we prove the existence of ground state solutions u=u(x+ct) for an optimal range of speeds c∈Rⁿ and a variety of nonlinearities f.     

New explicit traveling wave solutions for three nonlinear evolution equations

In this paper, using the extended tanh-function method, new explicit traveling wave solutions including rational solutions for three nonlinear evolution equations are obtained with the aid of Mathematica.     

Quasi-periodic solutions of completely resonant nonlinear wave equations

It is shown that there are many elliptic invariant tori, and thus quasi-periodic solutions, for the completely resonant nonlinear wave equation subject to periodic boundary conditions via KAM theory.     

非线性发展方程行波解的符号计算

Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation.     

Positive steady states for a prey-predator model with some nonlinear diffusion terms

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof.     

A note on convection with nonlinear heat flux

The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory.      

Nonlinear wave equations and singular solutions

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface, which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.

     

Exp-function method for traveling wave solutions of nonlinear evolution equations

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations. The proposed technique is tested on the modified Zakharov-Kuznetsov (ZK) and Zakharov-Kuznetsov-Modified-Equal-Width (ZK-MEW) equations. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed exp-function method.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

The extended tanh method for solving systems of nonlinear wave equations

The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.     

SPHERICAL NONLINEAR PULSES FOR THE SOLUTIONS OF NONLINEAR WAVE EQUATIONS Ⅱ, NONLINEAR CAUSTIC

This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows that the nonlinearities have a strong effect at the focal point. Scattering operator is introduced to describe the caustic crossing. With the aid of the L∞norms, it analyzes the relative errors in approximate solutions.     

A remark on triviality for the two-dimensional stochastic nonlinear wave equation

We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer et al. (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the weak noise regime, we show that solutions to SdNLW with regularized noises converge to a solution to a deterministic damped nonlinear wave equation with an additional mass term.     

Ground states for nonlinear fractional Choquard equations with general nonlinearities

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian: where the nonlinearity satisfies the general Berestycki–Lions‐type assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations

We investigate scattering theory in the energy space for fourth-order nonlinear defocusing wave equations and prove the Levandosky-Strauss conjecture stating that scattering holds true for such equations and arbitrary initial data.     

Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α?5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2<p?10. And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.