Spectral conservation laws for periodic nonlinear equations of the Melnikov type

TL;DR

This paper investigates how adding self-consistent sources to soliton equations, like the KP equation, affects their spectral properties and conservation laws, revealing that certain spectral invariants are preserved despite deformations.

Contribution

It demonstrates that for nonlinear equations with self-consistent sources, spectral curves deform but Floquet multipliers remain invariant, preserving associated conservation laws.

Findings

Spectral curves deform under self-consistent sources.

Floquet multipliers are preserved during deformation.

Conservation laws are maintained despite spectral curve changes.

Abstract

We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral. They deform the spectral curve but preserve the multipliers of the Floquet functions. The latter property implies that the conservation laws, for soliton equations, which may be described in terms of the Floquet multipliers give rise to conservation laws for the corresponding equations with self-consistent sources. Such a property was first observed by us for some geometrical flow which appears in the conformal geometry of tori in three- and four-dimensional Euclidean spaces (math/0611215).