Traveling solitary wave solutions to evolution equations with nonlinear terms of any order

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature.     

Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation

In this paper, by using the integral bifurcation method, we study a generalized KdV equation which was first derived by Fokas from physical considerations via a methodology of Fuchssteiner. All kinds of soliton-like or kink-like wave solutions and periodic wave solutions with loop or without loop are obtained. Smooth compacton-like periodic wave solution and non-smooth periodic cusp wave solution are also obtained. Their dynamic properties are investigated and their profiles are given by Mathematical software.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Constructing rational and multi‐wave solutions to higher order NEEs via the Exp‐function method

In this paper, we present an application of some known generalizations of the Exp‐function method to the fifth‐order Burgers and to the seventh‐order Korteweg de Vries equations for the first time. The two examples show that the Exp‐function method can be an effective alternative tool for explicitly constructing rational and multi‐wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Time-periodic solution to a nonlinear parabolic type equation of higher order

In this paper, the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a class of parabolic type equation of higher order are proved by Galerkin method.     

The correct traveling wave solutions for the high-order dispersive nonlinear Schrödinger equation

The high-order dispersive nonlinear Schrödinger equation is considered. The exact solutions were obtained by Zhang et al. [J.L. Zhang, M.L. Wang, X.Z. Li, Phys. Lett. A 357 (2006) 188-195] are analyzed. We can demonstrate that some solutions do not satisfy this equation. To obtain the correct solutions, the F-expansion method is applied to solve it.     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

关于带有非线性控制项及强迫项的差分方程的周期解

考虑了带有非线性控制函数及强迫项的差分方程,该方程可应用于神经网络的研究中,建立了该方程存在周期解的一个充分条件,表明如何将其周期解构造出来并给出一个具体例子说明这一结果.     

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|^p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u₀. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.     

Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations

In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equation

     

一类具有非线性阻尼和源项的Petrovsky方程整体解的存在性与渐近性

研究了一类具有非线性阻尼和源项的Petrovsky方程u_(tt) △~2u au_t|u_t|~(m-2)= bu|u|~(p-2)的初边值问题.在对m,p的大小关系不加任何限制的情况下,利用稳定集证明了整体解的存在性,并且得到了整体解的渐近性质.     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

Classification and existence of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form

Partial quasianalyticity of distribution solutions to weakly nonlinear differential equations with weights assigned to derivatives

We obtain conditions sufficient for the set of distribution solutions of a weakly nonlinear differential equation with weights assigned to derivatives to be a quasianalytic class with respect to part of the variables. The class of equations considered in the paper contains a number of equations describing the propagation of nonlinear waves. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 608–619, October, 2000.     

Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay

In this paper we study the existence of periodic and asymptotically periodic solutions of a system of nonlinear Volterra difference equations with infinite delay. By means of fixed point theory, we furnish conditions that guarantee the existence of such periodic solutions.