Multiplicative operators in the spectral problem of integrable systems

TL;DR

This paper develops a method to analyze the spectral problem of integrable PDEs using polynomial multiplicative operators, deriving conservation laws, recursion formulas, and hyperelliptic curves for equations like KdV and NLS.

Contribution

It introduces a general approach for handling polynomial spectral problems in integrable systems, providing new tools for deriving conservation laws and spectral data.

Findings

Derived conservation laws for KdV and NLS equations

Established recursion formulas for integrable PDEs

Explicitly constructed hyperelliptic curves for NLS

Abstract

We consider the spectral problem of the Lax pair associated to periodic integrable partial differential equations. We assume this spectral problem to be a polynomial of degree $d$ in the spectral parameter $\lambda$. From this assumption, we find the conservation laws as well as the hyperelliptic curve required to solve the periodic inverse problem. A recursion formula is developed, as well as $d$ additional conditions which give additional information to integrate the equations under consideration. We also include two examples to show how the techniques developed work. For the Korteweg-deVries (KdV) equation, the degree of the multiplicative equation is $d=1$. Hence, we only have one condition and one recursion formula. The condition gives in each degree of the recursion the conserved densities for KdV equation, recovering the Lax hierarchy. For the Nonlinear Schr\"odinger (NLS) equation, the degree of the multiplicative operator is $d=2$. Hence, we have a couple of conditions that we use to deduce conserved density constants and the Lax hierarchy for such equation. Additionally, we explicitely write down the hyper-elliptic curve associted to the NLS equation. Our approach can be use for other completely integrable differential equations, as long as we have a polynomial multiplicative operator associated to them.