Spectral conservation laws for periodic nonlinear equations of the Melnikov type
Аннотация
We dedicate this article to our teacher S.P. Novikov on the occasion of his 70th birthday In the seminal paper [24] in 1974 S.P. Novikov, in particular, established that the spectral curve of the one-dimensional periodic Schrödinger operator H = − d2 + u(x) dx2 is preserved when the real-valued potential u(x,t) evolves via the Korteweg– de Vries (KdV) equation and that for finite-zone (finite gap) potentials the classical conservation laws, i.e. the Kruskal–Miura integrals, are described in terms of branch points for this curve. The spectral curve Γ is a hyperelliptic where λ 2 = Q(E) Q(E) = (E − E0)... (E − E2N) is a polynomial of degree 2N +1 for N-zone potentials. It was proved in [24] that finite-zone potentials are exactly solutions of the Novikov equations, i.e., stationary points of higher KdV flows and their linear combinations, and that the KdV flow on the set of N-zone potentials reduces to a completely integrable finite-dimensional Hamiltonian system for which the ends of the stability zones, i.e., E0,...,E2N, supply the necessary family of first integrals. The article [24] was the starting point for the development of the finite gap integration theory in which the spectral curves play the main role. In this article we consider the deformation of the spectral curve via the periodic equations of the Melnikov type and we show that although the spectral curve is not preserved it is deformed in such a manner that it still gives many conservation laws for the system. 1.