Explicit Multi-symplectic Method for a High Order Wave Equation of KdV Type

In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constructed for the equation, and the conservation laws of the continuous equation are presented. The multisymplectic discretization of each formulation is exemplified by the multi-symplectic Fourier pseudospectral scheme. The numerical experiments are given, and the results verify the efficiency of the Fourier pseudospectral method.     

Multi-symplectic methods for the Ito-type coupled KdV equation

In this paper, we find that the Ito-type coupled KdV equation can be written as a multi-symplectic Hamiltonian partial differential equation (PDE). Then, multi-symplectic Fourier pseudospectral method and multi-symlpectic wavelet collocation method are constructed for this equation. In the numerical experiments, we show the effectiveness of the proposed methods. Some comparisons between the proposed methods are also made with respect to global conservation properties.     

Zhiber-Shabat方程的多辛几何和多辛隐式格式研究

Zhiber-Shabat方程,描述许多重要的物理现象,是一类重要的非线性方程,有着许多广泛的应用前景.本文给出Zhiber-Shabat方程的多辛几何结构和多辛Fourier拟谱方法.数值算例结果表明多辛离散格式具有较好的长时间的数值稳定性.     

Geometric numerical schemes for the KdV equation

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.     

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov–Kuznetsov (ZK) equation and the Kadomtsev–Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.     

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov-Kuznetsov (ZK) equation and the Kadomtsev-Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.     

A Multi-Symplectic Scheme for RLW Equation

The Hamiltonian and multi-symplectic formulations for RLW equation are considered in this paper. A new twelve-point difference scheme which is equivalent to multi-symplectic Preissmann integrator is derived based on the multi-symplectic formulation of RLW equation. And the numerical experiments on solitary waves are also given. Comparing the numerical results for RLW equation with those for KdV equation, the inelastic behavior of RLW equation is shown.     

带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法

带乘性噪声的空间分数阶随机非线性Schrödinger方程是一类重要的方程,可应用于描述开放非局部量子系统的演化过程.该方程为一个无穷维分数阶随机Hamilton系统,且具有广义多辛结构和质量守恒的性质.针对该方程的广义多辛形式,在空间上采用拟谱方法离散分数阶微分算子,在时间上则采用隐式中点格式,构造出一类保持全局质量的广义多辛格式.对行波解和平面波解等进行数值模拟,结果验证了所构造格式的有效性和保结构性质,时间均方收敛阶约在0.5到1之间.     

An exponential wave integrator Fourier pseudospectral method for the nonlinear Schrödinger equation with wave operator

In this article, an exponential wave integrator Fourier pseudospectral (EWI-FP) method is proposed for solving the nonlinear Schrödinger equation with wave operator. The numerical method is based on a Deuflhard-type exponential wave integrator for temporal integration and the Fourier pseudospectral method for spatial discretizations. The scheme is fully explicit and very efficient thanks to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established by means of the mathematical induction. Numerical results are reported to confirm the theoretical studies.     

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.     

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg‐de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh‐order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite‐order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite‐difference discretization, in which a lattice generalized solitary wave solution is found.     

The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves

Surface wave data from the Adriatic Sea are analysed in the light of new data analysis techniques which may be viewed as a nonlinear generalization of the ordinary Fourier transform. Nonlinear Fourier analysis as applied herein arises from the exact spectral solution to large classes of nonlinear wave equations which are integrable by the inverse scattering transform (IST). Numerical methods are discussed which allow for implementation of the approach as a tool for the time series analysis of oceanic wave data. The case for unidirectional propagation in shallow water, where integrable nonlinear wave motion is governed by the Korteweg-deVries (KdV) equation with periodic/quasi-periodic boundary conditions, is considered. Numerical procedures given herein can be used to compute a nonlinear Fourier representation for a given measured time series. The nonlinear oscillation modes (the IST ‘basis functions’) of KdV obey a linear superposition law, just as do the sine waves of a linear Fourier series. However, the KdV basis functions themselves are highly nonlinear, undergo nonlinear interactions with each other and are distinctly non-sinusoidal. Numerical IST is used to analyse Adriatic Sea data and the concept of nonlinear filtering is applied to improve understanding of the dominant nonlinear interactions in the measured wavetrains.     

Sine-Gordon方程的多辛Leap-frog格式

非线性发展方程由于具有多种形式的解析解而吸引着众多的研究者,借助多辛保结构理论研究了Sine-Gordon方程的多辛算法．利用Hamilton变分原理,构造出了sine-Gordon方程的多辛格式;采用显辛离散方法得到了Leap-frog多辛离散格式,该格式满足多辛守恒律;数值结果表明leap-frog多辛离散格式能够精确地模拟sine-Gordon方程的孤子解和周期解,模拟结果证实了该离散格式具有良好的数值稳定性．     

Local well-posedness for the periodic higher order KdV type equations

Higher order KdV type equations are the equation replaced by a higher order derivative ${\partial_{x}^{2k+1}}$ for the KdV equation. Recently, the local well-posedness result for these equations on torus have been given by Gorsky and Himonas (Math. Comput. Simul. 80:173–183, 2009). We extend this result by improving a bilinear estimate used in the Fourier restriction norm method.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations

In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n) equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K (n, n, n) equations.     

Soliton interaction for the extended Korteweg-de Vries equation

Soliton interactions for the extended Korteweg-de Vries (KdV)equation are examined. It is shown that the extended KdV equationcan be transformed (to its order of approximation) to a higher-ordermember of the KdV hierarchy of integrable equations. This transformationis used to derive the higher-order, two-soliton solution forthe extended KdV equation. Hence it follows that the higher-ordersolitary-wave collisions are elastic, to the order of approximationof the extended KdV equation. In addition, the higher-ordercorrections to the phase shifts are found. To examine the exactnature of higher-order, solitary-wave collisions, numericalresults for various special cases (including surface waves onshallow water) of the extended KdV equation are presented. Thenumerical results show evidence of inelastic behaviour wellbeyond the order of approximation of the extended KdV equation;after collision, a dispersive wavetrain of extremely small amplitudeis found behind the smaller, higher-order solitary wave.     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.     

Exact traveling wave solutions of some nonlinear physical models. I. The fifth order KdV type equations

New exact traveling wave solutions are derived for the fifth order KdV type equations by using a delicate way of rank analysis two-step ansatz method. Solitary shallow-water wates described by the above equation are discussed.     

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.