New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-hamiltonian structure

TL;DR

This paper develops an algebraic inverse scattering approach for a restricted three-wave interaction system, revealing its bi-Hamiltonian structure and expressing solutions via theta functions, including special cases with Weierstrass functions.

Contribution

It introduces a new Lax pair and algebraic framework for the system, providing explicit quasiperiodic solutions and elucidating its integrable structure.

Findings

Derived a new Lax pair for the system

Expressed solutions using theta and Weierstrass functions

Established the bi-Hamiltonian structure and classical r-matrix algebra

Abstract

We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical $r$-matrix and give an interpretation of the Poisson brackets as linear $r$-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for $n=1$ in terms of Weierstrass functions.