Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method

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Abstract

Some generalized variational principles are obtained for ion-acoustic plasma waves by He's semi-inverse method. The obtained variational principle has profound implications in physical understandings, explaining the interaction between various variables in an energy view and the existence of conservation law.

Introduction

The ion acoustic plasma equations are [1]nt+x(nv)=0,vt+x12v2=0,2φx2eφ+n=0,where φ, n, and v are respectively the electric potential, ion density and ion velocity.

Although the partial differential model for ion acoustic plasma equations has been studied for a long time, yet the general variational principles for the discussed problem have not been dealt with. Variational model gives us another approach to the physical understanding of the problem in an energy view. Variational methods will become a popular tool for nonlinear analysis of ion acoustic waves.

This paper is a preliminary report on the construction of variational formulations for the discussed problem by the semi-inverse method proposed by He [2], [3], and it shows a great success.

Section snippets

Variational formulations by the semi-inverse method

From Eq. (1), we can introduce a special function, Ψ, defined asΨx=n,Ψt=−nv.Similarly from Eq. (2), another special function, Π, can be introduced, which satisfies the following relations:Πt=v,Πt=−12v2.Our aim in this paper is to construct some variational principles for the discussed problem by He's semi-inverse method [2], [3].

To proceed, we consider the following functional:J1(v,φ,Ψ)=∫t1t2dt∫x1x2Ldx,where the trial-Lagrangian, L, can be written in the form:L=vΨt+12v2Ψx+F,

Conclusions

To conclude, we obtain some variational principle by the semi-inverse method, which is a powerful mathematical tool to the search for variational formulations directly from the field equations.

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