非线性Kawahara方程解的存在唯一性

非线性Ｋａｗａｈａｒａ方程是描述不同介质中存在单色非线性扰动时长波的传播问题的一类重要物理模型．本文通过对相应线性问题基本解的估计,导出了一类一般的Ｓｔｒｉｃｈａｒｔｚ－型光滑时空混合范数估计,进而得到了非线性Ｋａｗａｈａｒａ方程解的存在唯一性结果．     

Numerical solutions of the Kawahara type equations using radial basis functions

This study is carried out to investigate the numerical solutions of the Kawahara, KdV‐Kawahara, and the modified Kawahara equations by using the meshless method based on collocation with radial basis functions. Results of the meshless method with different radial basis functions are presented for the travelling wave solution of the Kawahara type equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 542–553, 2012     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

New exact travelling wave solutions of Kawahara type equations

The Auxiliary equation method is used to find analytic solutions for the Kawahara and modified Kawahara equations. It is well known that different types of exact solutions of the given auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, new exact solutions of the auxiliary equation are presented. Using these solutions, many new exact travelling wave solutions for the Kawahara type equations are obtained.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

Solitons and periodic solutions for the Rosenau–KdV and Rosenau–Kawahara equations

In this work we use the sine–cosine and the tanh methods for solving the Rosenau–KdV and Rosenau–Kawahara equations. The two methods reveal solitons and periodic solutions. The study confirms the power of the two schemes.     

New exact travelling wave solutions for the Kawahara and modified Kawahara equations

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

Low regularity solutions of two fifth-order KDV type equations

The Kawahara and modified Kawahara equations are fifth-order KdV type equations that have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for the Kawahara equation in H ^s (R) with s ≥ − 7/4 and the local well-posedness for the modified Kawahara equation in H ^s (R) with s ≥ − 1/4. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the [k; Z]_multiplier norm method of Tao [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.     

Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ‐expansion method

In this paper, variable coefficients Kawahara equation (VCKE) and variable coefficients modified Kawahara equation (VCMKE), which arise in modeling of various physical phenomena, are studied by Lie group analysis. The similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations. Moreover, some exact analytic solutions are considered by the power series method. Further, a generalized ‐expansion method is applied to VCKE and VCMKE for constructing some new exact solutions. As a result, hyperbolic function solutions, trigonometric function solutions and some rational function solutions with parameters are furnished. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation

In this paper, we present the variational iteration method and homotopy perturbation method to solve the modified Kawahara equations. Both methods provide remarkable accuracy for the approximate solutions when compared to the exact solutions. Numerical results demonstrate that the methods provide efficient approaches to solving the modified Kawahara equation.     

ILL-POSEDNESS OF MODIFIED KAWAHARA EQUATION AND KAUP-KUPERSHMIDT EQUATION

In this article,we consider the Cauchy problems for the modified Kawahara equation (6)tu + μ(6)x(u3) + α(6)5xu + β(6)3xu + γ(6)xu =0and the Kaup-Kupershmidt equation (6)tu + μu(6)2xu + α(6)5xu + β(6)3x...     

Comment on: “New exact solutions for the Kawahara equation using Exp-function method” [J. Comput. Appl. Math. 233 (2009) 97-102]

Assas [Laila M.B. Assas, New exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] found some supposedly new exact solutions to the Kawahara equation by means of the Exp-function method. Unfortunately, they are incorrect. We emphasize that the article contains erroneous formulas and resulting relations. In fact, no numerical method was used.     

Rational solutions of fifth-order evolutionary equations for describing waves on water

A method is proposed for obtaining the exact solutions of evolutionary equations in the form of a rational function. Invariant manifolds of the equations are used which have the same form of dependence on the required function and its derivatives as the generalized Riccati equations. Using fifth-order Kawahara and Korteweg–de Vries equations as an example, it is shown that their known particular solutions can be obtained using this method. New solutions of a non-linear fifth-order equation, which is encountered when describing long waves on water, are obtained.     

Error analysis and applications of the Fourier-Galerkin Runge-Kutta schemes for high-order stiff PDEs

An integrating factor mixed with Runge-Kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. In this paper, the novel hybrid Fourier-Galerkin Runge-Kutta scheme, with the aid of an integrating factor, is proposed to solve nonlinear high-order stiff PDEs. Error analysis and properties of the scheme are provided. Application to the approximate solution of the nonlinear stiff Korteweg-de Vries (the 3rd order PDE, dispersive equation), Kuramoto-Sivashinsky (the 4th order PDE, dissipative equation) and Kawahara (the 5th order PDE) equations are presented. Comparisons are made between this proposed scheme and the competing method given by Kassam and Trefethen. It is found that for KdV, KS and Kawahara equations, the proposed method is the best.     

Strichartz estimates for dispersive equations and solvability of the Kawahara equation

We first establish a series of Strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by Kenig, Ponce and Vega. Next we use such estimates to study solvability of the Cauchy problem of the Kawahara equation in the class C(R,Hs(R)). Local existence is proved for s>1/4 and global existence is proved for s?2.     

A Generalized （G＇/G）-Expansion Method to Find the Traveling Wave Solutions of Nonlinear Evolution Equations

In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.     

高阶非线性差分方程的导数收敛及其应用

首先研究高阶线性差分方程的整体收敛性，并证明了高阶非线性差分方程各阶导数的整体收敛；进而得到了关于高阶非线性差分方程整体收敛的一个定理，最后利用这个定理部分解决了Ladas提出的一个猜测。     

带导数项的奇摄动非线性 Schrodinger方程孤波解的存在性及其集中性质

利用Lyapunov-Schmidt方法证明了带有一阶导数项和(V)α势函数的非线性Schrodinger方程半经典孤波解的存在性及其集中性质. 具体地讲,当相当于Planck常数的奇摄动参数趋于零时,证明了该非线性Schrodinger方程的孤波解存在并且这些解在其势函数的非退化临界点处集中. 研究的是椭圆型方程的奇摄动问题,方程带有一阶导数项是本文特征之一.     

Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto–Sivashinsky‐type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three‐dimensional models of a spectral method, which was developed by the authors for the one‐dimensional Kuramoto–Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈C^∞, involving the rate of growth of ?ⁿu, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper–Kawahara, Frenkel–Indireshkumar, and Coward–Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd.