Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method

In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.     

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.     

Space-Curved Resonant Line Solitons in a Generalized(2 + 1)-Dimensional Fifth-Order KdV System

On the basis of N-soliton solutions, space-curved resonant line solitons are derived via a new constraint proposed here, for a generalized(2 + 1)-dimensional fifth-order KdV system. The dynamic properties of these new resonant line solitons are studied in detail. We then discuss the interaction between a resonance line soliton and a lump wave in greater detail. Our results highlight the distinctions between the generalized(2+1)-dimensional fifth-order KdV system and the classical type.     

Exact solutions to the time-fractional differential equations via local fractional derivatives

This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa–Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.     

An Explicit Scheme for the KdV Equation

A new explicit scheme for the Korteweg~：le Vries （KdV） equation is proposed. The scheme is more stable than the Zabusky Kruskal scheme and the multi-symplectic six-point scheme. When used to simulate the collisions of multi-soliton, it does not show the nonlinear instabilities and un-physical oscillations.     

The complex multi-symplectic scheme for the generalized sinh-Gordon equation

In this paper,the complex multi-symplectic method and the implementation of the generalized sinhGordon equation are investigated in detail.The multi-symplectic formulations of the generalized sinh-Gordon equation in Hamiltonian space are presented firstly.The complex method is introduced and a complex semi-implicit scheme with several discrete conservation laws(including a multi-symplectic conservation law(CLS),a local energy conservation law(ECL) as well as a local momentum conservation law(MCL)) is constr...     

一类五阶非线性演化方程的新孤波解

利用齐次平衡法给出了一类较广泛的五阶非线性演化方程的孤波解，数学物理中著名的Kaup-Kupershmidt方程、Caudrey-Dodd-Gibbon-Sawada-Kotera方程和五阶Korteweg-de-Vries方程等都可作为该方程的特殊情形而得到相应的孤波解. 关键词：     

Newton-conjugate-gradient methods for solitary wave computations

In this paper, the Newton-conjugate-gradient methods are developed for solitary wave computations. These methods are based on Newton iterations, coupled with conjugate-gradient iterations to solve the resulting linear Newton-correction equation. When the linearization operator is self-adjoint, the preconditioned conjugate-gradient method is proposed to solve this linear equation. If the linearization operator is non-self-adjoint, the preconditioned biconjugate-gradient method is proposed to solve the linear equation. The resulting methods are applied to compute both the ground states and excited states in a large number of physical systems such as the two-dimensional NLS equations with and without periodic potentials, the fifth-order KdV equation, and the fifth-order KP equation. Numerical results show that these proposed methods are faster than the other leading numerical methods, often by orders of magnitude. In addition, these methods are very robust and always converge in all the examples being tested. Furthermore, they are very easy to implement. It is also shown that the nonlinear conjugate gradient methods are not robust and inferior to the proposed methods.     

Approximate solutions of nonlinear PDEs by the invariant expansion

<正>It is difficult to obtain exact solutions of the nonlinear partial differential equations(PDEs) due to their complexity and nonlinearity,especially for non-integrable systems.In this paper,some reasonable approximations of real physics are considered,and the invariant expansion is proposed to solve real nonlinear systems.A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries(KdV) equation with a fifth-order dispersion term,the perturbed fourth-order KdV equation,the KdV-Burgers equation,and a Boussinesq-type equation.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov 方程作为一类重要的非线性方程有着许广泛的应 用前景,基于Hamilton 空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值 解法,讨论了利用Preissmann 方法构造离散多辛格式的途径, 并构造了一种典型的半隐 式的多辛格式, 该格式满足多辛守恒律、局部能量守恒律. 数值算例结果表明该多辛离 散格式具有较好的长时间数值稳定性.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov方程作为一类重要的非线性方程有着许多广泛的应用前景,基于Hamilton空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值解法,讨论了利用Preissmann方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

The novel multi-solitary wave solution to the fifth-order KdV equation

By using Hirota‘s method, the novel multi-solitary wave solutions to the fifth-order KdV equation are obtained. Furthermore, various new solitary wave solutions are also derived by a reconstructed bilinear Bǎcklund transformation.     

Numerical integration of the Ostrovsky equation based on its geometric structures

We consider structure preserving numerical schemes for the Ostrovsky equation, which describes gravity waves under the influence of Coriolis force. This equation has two associated invariants: an energy function and the L² norm. It is widely accepted that structure preserving methods such as invariants-preserving and multi-symplectic integrators generally yield qualitatively better numerical results. In this paper we propose five geometric integrators for this equation: energy-preserving and norm-preserving finite difference and Galerkin schemes, and a multi-symplectic integrator based on a newly found multi-symplectic formulation. A numerical comparison of these schemes is provided, which indicates that the energy-preserving finite difference schemes are more advantageous than the other schemes.     

Local structure-preserving methods for the generalized Rosenau-RLW-KdV equation with power law nonlinearity

Local structure-preserving algorithms including multi-symplectic, local energy-and momentum-preserving schemes are proposed for the generalized Rosenau–RLW–Kd V equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time–space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.     

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.     

变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

Conserved densities of the fifth-order Korteweg-de Vries equation

It is proved that the rank of the non-trivial polynomial conserved density of the fifth-order KdV equation is 3p–2 or 3p (p=1, 2, ...).     

Multiple complex soliton solutions for the integrable KdV,fifth-order Lax,modified KdV,Burgers, and Sharma–Tasso–Olver equations

We show that completely integrable equations give multiple complex soliton solutions in addition to multiple real real soliton solutions. We exhibit complex categories of the simplified Hirota’s method to confirm these new findings. To demonstrate the power of the new complex forms, we test it on integrable KdV, fifth-order Lax, modified KdV, fifth-order modified KdV, Burgers, and Sharma–Tasso–Olver equations.     

A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation

A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein－Gordon equation.     

Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations

In this paper, we propose explicit multi-symplectic schemes for Klein–Gordon–Schrödinger equation by concatenating suitable symplectic Runge–Kutta-type methods and symplectic Runge–Kutta–Nyström-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments.