General procedure to formulate Hamiltonian theory of the completely integrable n onlinear equations and its application to the sine-Gordon equation

Abstract

For a completely integrable nonlinear equation, the Poisson bracket of monodramy matrix is known to be expressed in a form of integral with respect to x. The in tegrand is found to be an x-differential of a linear combination of direct produ ct of two pairs of Jost solutions definitely, and the coefficients can be determ ined by comparing the corresponding elements of direct product matrices on two s ides. Hence a general procedure for constructing Hamiltonian formalism is given for a completely integrable nonlinear equation. As an example, the Hamiltonian t heory of sine-Gordon equation is re-examined, which shows the essence of the lin ear combination method for its very complicated Poisson bracket. And the previou s works involve, as is known, some inappropriate violating simultaneity of varia bles in Poisson bracket, which is also revised now.