The Zero-Dispersion Limit for the Odd Flows in the Focusing Zakharov-Shabat Hierarchy

TL;DR

This paper investigates the zero-dispersion limit of odd flows in the focusing Zakharov-Shabat hierarchy, including NLS and mKdV, establishing convergence of conserved densities and fluxes for real initial data using a modified Lax-Levermore approach.

Contribution

It extends the zero-dispersion analysis to all odd flows in the hierarchy, including nonselfadjoint spectral problems, and shows the limiting dynamics of mKdV matches KdV after transformation.

Findings

Zero-dispersion limit established for all odd flows.

Conserved densities and fluxes converge for real initial data.

Limiting dynamics of mKdV are identical to KdV after algebraic transformation.

Abstract

We present a numerical and theoretical study of the zero-dispersion limit of the focusing Zakharov-Shabat hierarchy, which includes NLS and mKdV flows as its second and third members. All the odd flows in the hierachy are shown to preserve real-valued data. We establish the zero-dispersion limit of all the nontrivial conserved densities and associated fluxes for these odd flows for a large class of real-valued initial data which includes all ``single hump'' initial data. In particular, it is done for the ``focusing'' mKdV flow. The method is based on the Lax-Levermore KdV strategy, but here it is carried out in the context of a nonselfadjoint spectral problem. We find that after an algebraic transformation the limiting dynamics of the mKdV equation is identical to that of the KdV equation.