Approximation of the thermodynamic limit of finite-gap solutions of the focusing NLS hierarchy by multisoliton solutions

TL;DR

This paper develops a method to approximate finite-gap solutions of the focusing NLS hierarchy by multisoliton solutions in the thermodynamic limit, using Riemann-Hilbert problems and generalizing primitive potentials.

Contribution

It introduces a novel approximation technique for finite-gap solutions of integrable equations via multisolitons, with error estimates in the thermodynamic limit.

Findings

Finite-gap and multisoliton solutions are approximated in the thermodynamic limit.

Error estimates are provided for the approximation on compact subsets.

The approach generalizes primitive potentials to the focusing NLS hierarchy.

Abstract

In this paper we approximate the thermodynamic limit of finite-gap solutions to any integrable equations in the focusing NLS hierarchy (fNLS, mKdV, ...) with an associated multisoliton solutions using the Riemann-Hilbert Problem approach. Moreover, we show that both the finite-gap and multisoliton solutions are approximated in the thermodynamic limit by a generalization of the primitive potentials introduced by V. Zakharov and his collaborators in the KdV context. Under certain assumptions on the spectral data for the finite gap potentials, we provide error estimates for the approximation on compact subsets of the $(x,t)$-plane.