WANG JUN-JIE AND WANG XIU-YING

(1.School of Mathematics and Statistics,Pu’er University,Pu’er,Yunnan,665000)

(2.School of Mathematics,Northwest University,Xi’an,710127)

(3.Pu’er Meteorological Office of Yunnan Province,Pu’er,Yunnan,665000)

Abstract: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 exempli fied by the multi-symplectic Fourier pseudospectral scheme.The numerical experiments are given,and the results verify the efficiency of the Fourier pseudospectral method.

Key words:the high order wave equation of KdV type,multi-symplectic theory,Hamilton space,Fourier pseudospectral method,local conservation law

1 Introduction

In 2002,Tzirtzilakis et al.[1]considered a class of water wave equation of KdV type(see[2]–[5])

where ρi(i=1,2,3)are free parameters,and α, β are positive real constants.During recent years,more and more experts have paid great attention to(1.1)(see[6]–[8]).Long et al.[6]obtained some exact solutions for some special set of parameters of(1.1)by integral bifurcation method.To the best of the author’s knowledge,no much work has been done to construct the numerical scheme for(1.1)till now.In this paper,we consider numerical method to study(1.1).

Now we use the Weiss-Tabor-Carnevale(WTC)method to test the Painleve property of(1.1).If u(x,t)is a solution of(1.1),then it can be expressed as a Laurent series in ?(x,t)

Inserting(1.2)into(1.1),a leading order analysis uniquely gives

Substituting(1.2)and(1.3)into(1.1),we obtain

When ρ2=2ρ3,(1.1)is integrable.In this paper,we consider multi-symplectic method to study the integrable equation of KdV type as follows:

Multi-symplectic methods have been used to compute multi-symplectic Hamiltonian PDEs from the view point of symplectic geometry(see[9]–[32]).A lot of multi-symplectic methods for designing symplectic integrators have appeared,including the box scheme(see[15]),Rung-kutta collocation scheme(see[17]),Preissmann scheme(see[18]),splitting scheme(see[28]),Fourier pseudospectral scheme(see[31]),etc.Simultaneously,some error analysis for multi-symplectic methods were presented and discussed in[13]and[14],etc.The numerical solutions of many nonlinear wave equations such as KdV equation(see[15]and[18]),Schr¨odinger equation(see[22],[23]and[29]),Klein-Gordon equation(see[21])and KP equation(see[20]),etc.have been studied by the multi-symplectic methods.One the most popular multi-syplectic meyhod is Fourier pseudospectral method which has been successfully applied to modeling wave propagation.After elimination of some auxiliary variables,new multi-symplectic schemes have been obtained,and preserve very well the mass,energy and momentum in long-time evolution.

The outline of this paper is as follows.In Section 2,we give the multi-symplectic Hamiltonian structure of(1.5),and prove that the structure satis fies the multi-symplectic conservation law,local energy and momentum conservation laws.In Section 3,we give the multi-symplectic Fourier pseudospectral method and error estimates of local conservation laws.In Section 4,numerical experiments are given.Finally,a conclusion and some discussions are given in Section 5.

2 Multi-symplectic Structure for the High Order Wave Equation of KdV Type

By the multi-symplectic theory(see[9]–[11]),a wide range of conservative PDEs can be written as a multi-symplectic Hamiltonian system

where M,K ∈ Rd×dare the skew-symmetric matrices,z(x,t)is the vector of state variables,S:Rn→ R is a scalar-valued smooth function,?zS(z)denotes the gradient of the function S=S(z)with respect to variable z.

The system(2.1)has multi-symplectic conservation law(MSCL)

where

the local energy conservation law(LECL)

where

and local momentum conservation law(LMCL)

where

By introducing new variables

(1.5)can be written as the first-order PDEs

If we de fine the state variable z=(u,?,w,Φ,ψ),the system(2.6)can be rewritten as(2.1),where the skew-symmetric matrices are

and the smooth Hamiltonian is

By direct calculations,(2.6)satis fies the local conservation laws(2.2),(2.4)and(2.5),

3 Multi-symplectic Fourier Pseudospectral Method for the High Order Wave Equation of KdV Type

The Fourier pseudospectral methods have been proven very powerful for periodic boundary problem.In this section,we apply Fourier pseudospectral method to the high order wave equation of KdV type(1.5)with periodic boundary condition.

The discretization of the system(2.1)and the conservation law(2.2)can be solved numerically by

De finition 3.1The numerical scheme(3.1)of the system(2.1)is said to be multisymplectic if(3.2)is a discrete conservation law of(2.2).

where Dkrepresents Fourier pseudospectral differential matrix with the elements at the collocation points xiare obtained by

De finition 3.2Let.The Hadamard product of vectors is de fined by

Theorem 3.1LetandWe can obtainwherei=0,1,2···,N ? 1.

Theorem 3.2To the differential matrix Dkand D1,we can obtain

If k is an even number,then differential matrix Dkare skew-symmetric matrices and Dk=(D1)k.

By the above analysis,we can obtain a semi-discrete system for(2.6)

We rewrite(3.3)in a compact form as

Applying the implicit midpoint rule to(3.4),we can obtain a fully discrete system for the system(2.6)

Islas and Schober[13]have proved that if S(z)is the quadratic functional in z,then the scheme(3.5)conserves the LECL exactly.In order to evaluate the local conservation laws of energy and momentum,we use the discretizations of the form

where

4 Numerical Experiment

The high order wave equation of KdV type(1.5)is a kind of important mathematical physics equation.When ρ2=0, α =6, β =1,(1.5)becomes KdV equation as follows:

(4.1)has been solved numerically by multi-symplectic Preissmann method(see[18]).

Here,after elimination of some auxiliary variables,we obtain multi-symplectic Fourier pseudospectral scheme

We propose the following Predictor-Corrector algorithm which is much easier to implement and much more efficient in computation to(4.2).

Predictor is

and corrector predictor is

4.1 Numerical Experiment A

We consider the soliton with the initial and boundary conditions

where ρ2=1,α =1,β =1,L1=0,L2=25, ρ1=1,

We simulate the periodic solitary wave solution with x∈[L1,L2],t∈[0,20],?t=0.001,?x=0.01.Fig.4.1 shows the errors of the local energy and momentum conservation laws.Fig.4.2 shows the pro files of a soliton under the periodic boundary conditions with t=0,2,4,8 by Fourier pseudospectral scheme(3.5).

Fig.4.1

Fig.4.2

4.2 Numerical Experiment B

We consider the collisions two solitons with the initial condition and boundary condition

where ρ2=1,α =1,β =1,L1=0,L2=25,

We simulate the periodic wave solution(4.4)with the initial condition and boundary condition using the implicit multi-symplectic Fourier pseudospectral method(3.5)with x∈[L1,L2],t∈[0,20],?t=0.0001 and?x=0.01.Fig.4.3 shows the errors of the local energy and momentum conservation laws.Fig.4.4 shows the pro files of two solitons collision under the periodic boundary conditions with t=0,2,4,8.

Fig.4.3

Fig.4.4

5 Conclusion

In this paper,we simulate the periodic wave solutions of(4.3)and(4.4)with the initial and boundary conditions.From above the results,we find that the wave-forms keep their amplitudes and velocities invariable throughout the processes of the simulations,which implies that the multi-symplectic Fourier pseudospectral method(3.5)can preserve the local properties of the periodic wave solution perfectly.