A Neumann System of the Third Order Differential Operator Associated with the Boussinesq Equation

Jianli Cao,Jingfang Han

doi:10.4236/jamp.2020.812211

Abstract

Finite-dimensional integrable Hamiltonian system, obtained through the nonlinearization of the 3 × 3 spectral problem associated with the Boussinesq equation, is investigated. A generating function method starting from the Lax-Moser matrix is used to give an effective way to prove the involutivity of integrals. Finite-parameter solution of the Boussinesq equation is calculated based on the commutative system of ordinary differential equations with these integrals as Hamiltonians. The problem of the third order differential operator associated with the Boussinesq Neumann system put forward by H. Flaschka in 1983 is solved.

Highlights

Boussinesq equation ( ) vtt = − 13 vxx − 4v2 xx (1)is one of the difficult soliton equations, which has been paid common attention in physical and mathematical fields [1]-[6]
The problem of the third order differential operator associated with the Boussinesq Neumann system put forward by H
Han order differential operator is obtained in this paper, which is the extension of the famous KdV Neumann system associated with the second order differential operator

Summary

Introduction

Is one of the difficult soliton equations, which has been paid common attention in physical and mathematical fields [1]-[6]. Flaschka put forward a problem of the third order differential operator associated with the Boussinesq Neumann system [7]. Some works focus on the decomposition and the structures of the Modified Boussinesq equation [8] [9] [10] [11]. The decomposition of the Boussinesq Neumann system has not been done thoroughly for a long time. A Neumann system of the Boussinesq equation associated with the third. There are many methods to deal with the integrability and involutivity [12] [13] [14]. A generating function method starting from the Lax-Moser matrix [15]-[20] is used to give an effective way to prove the involutivity of integrals

The Generating Function of Integrals

The Involutivity of Integrals

Hamiltonian Systems

Conclusion