Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as _∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t_∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.     

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.     

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.     

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.

     

Interval oscillation criteria for a forced second-order nonlinear differential equations with damping

In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results.     

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.     

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

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.     

The lifespan for nonlinear wave equations with multiple propagation speeds in 3D

In this paper, we consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. On the basis of some Klainerman–Sideris type weighted estimates, we get the lifespan of small amplitude solutions for systems with nonlinearities depending on both the unknown functions and their derivatives.