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This is a Quantitative Study study.

September 17, 2025

Integrable Boundary Conditions for the Nonlinear Schrödinger Hierarchy Via the Fokas Method

Key Points

New integrable boundary conditions enhance the study of the nonlinear Schrödinger hierarchy, showing significant differences between even-order and odd-order equations.
The introduction of a new class of boundary conditions linked to time reversal expands the understanding of integrable systems, potentially leading to novel soliton solutions.
Demonstrating the existence of infinitely many integrals of motion reinforces the integrability of the nonlinear Schrödinger hierarchy under these conditions.
A boundary dressing technique is developed to construct soliton solutions, highlighting the practical application of these integrable boundary conditions.

Abstract

ABSTRACT We study integrable boundary conditions for the whole hierarchy of nonlinear Schrödinger (NLS) equations defined on the half‐line. We find that the even‐order and odd‐order NLS equations admit rather different integrable boundary conditions. In particular, the odd‐order NLS equations admit a new class of integrable boundary conditions that involves time reversal. We establish the integrability of the NLS hierarchy with our new boundary conditions by demonstrating the existence of infinitely many integrals of motion in involution. Moreover, we develop the boundary dressing technique to construct soliton solutions satisfying these new boundary conditions.

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