Integrable boundary conditions for the nonlinear Schr\"{o}dinger hierarchy

TL;DR

This paper explores integrable boundary conditions for the entire nonlinear Schrödinger hierarchy on the half-line, revealing differences between even and odd orders and introducing new boundary conditions involving time reversal.

Contribution

It introduces a novel class of integrable boundary conditions for odd order NLS equations involving time reversal, and proves the hierarchy's integrability with these conditions.

Findings

Odd order NLS equations admit new boundary conditions involving time reversal.

The hierarchy remains integrable with the new boundary conditions, possessing infinite conserved quantities.

Constructs soliton solutions for the boundary value problems using boundary dressing techniques.

Abstract

We study integrable boundary conditions associated with the whole hierarchy of nonlinear Schr\"{o}dinger (NLS) equations defined on the half-line. We find that the even order NLS equations and the odd order NLS equations admit rather different integrable boundary conditions. In particular, the odd order NLS equations permit a new class of integrable boundary conditions that involves the time reversal. We prove the integrability of the NLS hierarchy in the presence of our new boundary conditions in the sense that the models possess infinitely many integrals of the motion in involution. Moreover, we develop further the boundary dressing technique to construct soliton solutions for our new boundary value problems.