Well-posed Cauchy problem and the Hamiltonian form of (2+1) nonlinear equations integrable by inverse scattering transform

TL;DR

This paper establishes the Hamiltonian structure and well-posedness of the Cauchy problem for certain (2+1)-dimensional integrable nonlinear equations, including the Schrödinger and mKdV analogues, using inverse scattering methods.

Contribution

It proves the Hamiltonian form and well-posedness of the Cauchy problem for (2+1) nonlinear integrable equations, extending inverse scattering techniques.

Findings

Hamiltonian form of (2+1) nonlinear equations established

Well-posedness of the Cauchy problem proved for the (2+1) mKdV analogue

Solvability demonstrated using inverse scattering transform

Abstract

The Hamiltonian form of the (2+1) nonlinear integrable Schr\"odinger equation and the system of two (2+1) nonlinear analogue of the mKdV equation is proved. A well--posed Cauchy problem is formulated and the solvability of such a problem for the (2+1) nonlinear analogue of the mKdV equation is proved.