Derivation of Lagrangian density for the “good” Boussinesq equation and multisymplectic disretizations

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Abstract

The Lagrangian density for the “good” Boussinesq equation is derived by introducing a new variable in this paper. Based on this density, we consider the corresponding multisymplectic formulation and discretizations. Numerical experiments are also reported.

Introduction

We consider the “good” Boussinesq equation in the formutt=-uxxxx+uxx+u2xx.This nonlinear partial differential equation possesses an interesting soliton-interaction mechanism: solitary waves only exist for a finite range of velocities. They can retain their shape and velocity after collision for small amplitude solitons. However, for large amplitude solitons, they may develop into so-called antisolitons [8], [9], [12], [6]. Using the approach in Ref. [10], the multisymplectic geometry, local conservation laws, and the corresponding disretizations for (1.1) were presented in Ref. [4].

The starting point of the framework in Ref. [10] is the Lagrangian of a given differential equation. Once we know the corresponding Lagrangian for a differential equation, we can formulate its multisymplectic structure in both continuous and discrete versions which provides an important approach of investigating the corresponding differential equation theoretically and numerically.

The Lagrangian density for the “good” Boussinesq equation was proposed in Ref. [4], but the derivation was not given there. It is interesting to note that the Lagrangian density of the “good” Boussinesq equation in the form of (1.1) does not exist. To find a Lagrangian density, we need to cast (1.1) into a system of equations. Therefore, it will be helpful to give a detailed derivation of the Lagrangian density for the Boussinesq equation since this derivation will provide some guiding principles in finding Lagrangian density for other evolutionary equations.

We also consider a multisymplectic Fourier pseudospectral discretization for (1.1) and demonstrate its convergence by simulating the evolution of the soliton.

In Section 2, we will derive the Lagrangian density for the “good” Boussinesq equation. Its multisymplectic Hamiltonian formulation and discretization are given in Section 3. We end this paper with some numerical experiments in Section 4.

Section snippets

Lagrangian density for the “good” Boussinesq equation

To facilitate our discussions, we first present some notations and concepts introduced in Ref. [11]. Let x=(x1,,xp) and u=(u1,,uq) denote the independent variables and dependent variables, respectively. We denote the kth partial derivatives of a function u(x) with respect to x byJu(x)=ku(x)xj1xj2xjk,where J=(j1,,jk) is a k-tuple of integers with 1jip,(i=1,,k) indicating which derivatives are being taken. Notice that we allow equal integers amongst the integers ji,(i=1,,k).

Let A

Multisymplectic Hamiltonian formulation

In Section 2, we see that the system (2.8) is the Euler–Lagrange equation for the Lagrangian density (2.10). Introducing a pair of conjugate momenta p=ux,q=vx (Legendre transformation), the system (2.8) can be reformulated as the following first-order system:vt+px=u2+u,-ut+qx=0,-ux=-p,-vx=-q,which can be written as a multisymplectic Hamiltonian systemMzt+Kzx=zS(z),wherez=(u,v,p,q)T,S(z)=12u2+23u3-p2-q2,andM=0100-100000000000,K=00100001-10000-100.The multisymplectic system (3.2) was first

Numerical experiments

In this section, we perform some numerical experiments to demonstrate the convergence of the integrator (3.3). The “good” Boussinesq Eq. (1.1) possesses soliton solutions of the following formu(x,t)=-Asech2A/6(x+1-2A/3t+x0),where A is the amplitude of the pulse and x0 indicates the initial position of the pulse.

We first simulate the motion of the soliton. The initial conditions are:u(x,0)=-Asech2A/6(x+x0),ut(x,0)=2A1-2A/3sech2A/6(x+x0)tanhA/6(x+x0).In our experiments, we take A=0.5, x0=0, Δx=0.5

Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant No. 40774069 and partially by National Hi-Tech Research and Development Program of China (No. 2006AA09A102-08) and National Key Development Planning Project for the Basic Research (No. 2007CB209603).

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