Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.     

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.     

非线性弹性杆中纵波传播过程的数值模拟

基于Hamilton空间体系的多辛理论研究了包含材料非线性效应和几何弥散效应的非线性弹性杆中纵波传播问题。导出了其Bridges意义下的多辛形式及其多种守恒律,并构造了等价于Box多辛格式的隐式多辛格式对不考虑材料非线性效应和几何弥散效应、只考虑材料非线性效应、只考虑几何弥散效应、同时考虑材料非线性效应和几何弥散效应四种情况下不同截面参数的圆杆中的纵波传播过程进行数值模拟,模拟结果不仅全面地反映了非线性效应和几何弥散效应对纵波传播的影响,而且也反映了多辛方法的两大优点:精确的保持多种守恒律和良好的长时间数值行为。     

THE MULTI-SYMPLECTIC ALGORITHM FOR "GOOD" BOUSSINESQ EQUATION

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓａｒｅｍａｒｋａｂｌｅｄｅｖｅｌｏｐｍｅｎｔｈａｓｔａｋｅｎｐｌａｃｅｉｎｔｈｅｓｔｕｄｙｏｆｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎａｒｙｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ａｎｅｘａｍｐｌｅｉｓｔｈｅ“Ｇｏｏｄ”Ｂｏｕｓｓｉｎｅｓｑ (Ｇ .Ｂ .)ｅｑｕａｔｉｏｎｕｔｔ =-ｕｘｘｘｘ+ｕｘｘ+ (ｕ2 ) ｘｘ ( 1 )ｗｈｉｃｈｄｅｓｃｒｉｂｅｓｓｈａｌｌｏｗｗａｔｅｒｗａｖｅｓｐｒｏｐａｇａｔｉｎｇｉｎｂｏｔｈｄｉｒｅｃｔｉｏｎｓ.Ｔｈｅａ…     

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION

The multi-symplectic formulations of the “Good” Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that the multi-symplectic scheme have excellent long-time numerical behavior. Foundation items: the Foundation for Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences; the Natural Science Foundation of Huaqiao University. Biography: ZENG Wen-ping (1940-), Professor (E-mail: qmz@1sec.cc.ac.cn)     

Analyzing dynamic response of non-homogeneous string fixed at both ends

To survey the local conservation properties of complex dynamic problems, a structure-preserving numerical method, named as generalized multi-symplectic method, is proposed to analyze the dynamic response of a non-homogeneous string fixed at both ends. Firstly, based on the multi-symplectic idea, a generalized multi-symplectic form derived from the vibration equation of a non-homogeneous string is presented. Secondly, several conservation laws are deduced from the generalized multi-symplectic form to illustrate the local properties of the system. Thirdly, a centered box difference scheme satisfying the discrete local momentum conservation law exactly, named as a generalized multi-symplectic scheme, is constructed to analyze the dynamic response of the non-homogeneous string fixed at both ends. Finally, numerical experiments on the generalized multi-symplectic scheme are reported. The results illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the generalized multi-symplectic scheme well.     

Multi-symplectic methods for membrane free vibration equation

In this paper,the multi-symplectic formulations of the membrane free vi- bration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws—a multi-symplectic conservation law(CLS),a local energy conservation law(ECL)as well as a local momentum conservation law(MCL)—is constructed to discrete the PDEs that are derived from the membrane free vibra- tion equation.The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior.     

Stochastic solution to a time-fractional attenuated wave equation

The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.     

A new framework for studying the stability of genus-1 and genus-2 KP patterns

The Kadomtsev-Petviashvili equation - or KP equation - is a model equation for waves that are weakly two-dimensional in a horizontal plane, and models water waves in shallow water with weak three-dimensionality. It has a vast array of interesting genus—k pattern solutions which can be obtained explicitly in terms of Riemann theta functions. However the linear or nonlinear stability of these patterns has not been studied. In this paper, we present a new formulation of the KP model as a Hamiltonian system on a multi-symplectic structure. While it is well-known that the KP model is Hamiltonian - as an evolution equation in time - multi-symplecticity assigns a distinct symplectic operator for each spatial direction as well, and is independent of the integrability of the equation. The multi-symplectic framework is then used to formulate the linear stability problem for genus-1 and genus-2 patterns of the KP equation; generalizations to genus—k with k > 2 are also discussed.