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.     

带有阻尼项的广义对称正则长波方程的指数吸引子

考虑了带有阻尼项的广义对称正则长波方程的整体快变动力学．证明了与该方程有关的非线性半群的挤压性质和指数吸引子的存在性．对指数吸引子的分形维数的上界也进行了估计．     

Elliptic function solutions for some nonlinear pdes in mathematical physics

In this work, we have constructed various types of soliton solutions of the generalized regularized long wave and generalized nonlinear Klein-Gordon equations by the using of the extended trial equation method. Some of the obtained exact traveling wave solutions to these nonlinear problems are the rational function, 1-soliton, singular, the elliptic integral functions $F, E, \Pi$ and the Jacobi elliptic function sn solutions. Also, all of the solutions are compared with the exact solutions in literature, and it is seen that some of the solutions computed in this paper are new wave solutions.     

应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程

研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.     

Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

阻尼波动方程在模空间上的一些估计

本文研究分数次阻尼波动方程解在模空间上的估计及相应的时空估计, 作为应用, 我们将得到小初值条件下一类Cauchy问题的全局解.     

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.     

具有局部时间分数阶导数的非线性Zoomeron方程的精确解

In this paper, we are concerned with the nonlinear Zoomeron equation with local conformable time-fractional derivative. The concept of local conformable fractional derivative was newly proposed by R. Khalil et al. The bifurcation and phase portrait analysis of traveling wave solutions of the nonlinear Zoomeron equation are investigated. Moreover, by utilizing the exp(-?(ε))-expansion method and the first integral method, we obtained various exact analytical traveling wave solutions to the Zoomeron equation such as solitary wave, breaking wave and periodic wave.     

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

In this paper, we adopt the homotopy analysis method (HAM) to obtain solutions of linear and nonlinear fractional diffusion and wave equation. The fractional derivative is described in the Caputo sense. Some illustrative examples are presented.     

Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.     

IMPROVED FRACTIONAL SUB-EQUATION METHOD AND ITS APPLICATIONS TO FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Based on an improved fractional sub-equation method involving Jumarie's modified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound Kd V-Burgers equation and coupled Burgers' equations. These results not only reveal that the method is very effective and simple in studying solutions to the fractional partial differential equation, but also include some new exact solutions.     

Nonlinear waves in friedman-robertson-walker space-times

In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman-Robertson-Walker space-time, □ being the D'Alambertian in local coordinates of space-time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.     

Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters北大核心CSCD

利用多项式完全判别系统法求得非线性光学中带参数时空分数阶Fokas-Lenells方程在一般情况下的精确解,包括有理函数解、周期解、孤波解、Jacobi椭圆函数解和双曲函数解等,绘制了精确解的相关图像,并由此分析了参数对解的结构的影响。     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method

Recently the unified method for finding traveling wave solutions of nonlinear evolution equations was proposed by one of the authors. It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries equation with space-time dependent coefficients.     

Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

Long Time Behavior of the Dissipative Generalized Symmetric Regularized Long Wave Equations

This paper deals with the long time behavior of solutions for the dissipative generalized symmetric regularized long wave equations. We show the existence of global weak attractors for the periodic initial value problem of the equations in H¹ × L². The finite dimensionality of the global attractors is also established.     

The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time-dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time-dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow-up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time-dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.     

耗散的广义对称正则长波方程周期初值问题的整体吸引子

该文考虑了带有耗散项的广义对称正则长波方程解的长时间性态.证明了周期初值问题整体吸引子的存在性,用关于时间的一致先验估计获得整体吸引子的Hausdorff和分形维数的上界估计.     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.