On the Analytic Degenerate Dispersion Laws

Abstract

The concept of degenerate dispersion laws arose in the theory of Hamiltonian wave systems [1]. Their Hamiltonians in normal variables have the form

$$H = \sum\limits_{\alpha = 1}^Q {\int {{\omega ^\alpha }} } (k)a_k^\alpha a_k^{\alpha *2}dk + {H_{\operatorname{int} }}$$

The functions ${\omega ^\alpha }(\overrightarrow k )$, α = 1,…, Q represent dispersion laws of linear modes available in the system. Consider the simplest case Q = 1. The dynamics of the solution is controlled by usual Hamiltonian equation

$$i\partial ak/\partial t = \delta H/\delta a_k^*$$

(0.1)

It is shown in [1], (for review of further results see [2], [3]) that the following alternative is the necessary condition for system (0.1) to possess an additional motion invariant of the form

$$I = \int {{f_k}} {a_k}a_k^*dk + higherterms$$