Integrable nonlocal nonlinear equations
Mark J. Ablowitz, Ziad H. Musslimani

TL;DR
This paper introduces new classes of integrable nonlocal nonlinear equations, including reverse space-time and reverse time variants, expanding the understanding of PT-symmetric and nonlocal integrable systems with explicit solutions and conservation laws.
Contribution
It presents novel reverse space-time and reverse time nonlocal integrable equations derived from symmetry reductions, along with their Lax pairs, conservation laws, and explicit soliton solutions.
Findings
Introduced reverse space-time and reverse time nonlocal integrable equations.
Constructed Lax pairs and found explicit one-soliton solutions.
Established conservation laws and inverse scattering transforms for these systems.
Abstract
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is symmetric thus the nonlocal NLS equation is also symmetric. In this paper, new {\it reverse space-time} and {\it reverse time} nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal nonlinear Schr\"odinger, modified Korteweg-deVries (mKdV), sine-Gordon, and dimensional three-wave interaction, derivative NLS, "loop soliton",…
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Taxonomy
TopicsQuantum Mechanics and Non-Hermitian Physics · Nonlinear Waves and Solitons · Nonlinear Photonic Systems
