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Title: Integrable Nonlocal Nonlinear Equations

A nonlocal nonlinear Schrödinger (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” issymmetric thus the nonlocal NLS equation is alsosymmetric. In this paper, newreverse space‐timeandreverse timenonlocal 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 NLS, modified Korteweg‐deVries (mKdV), sine‐Gordon, (1 + 1) and (2 + 1) dimensional three‐wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partiallysymmetric DS and partially reverse space‐time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space‐time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space‐time and reverse time nonlocal NLS equations.

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Author(s) / Creator(s):
Publisher / Repository:
Date Published:
Journal Name:
Studies in Applied Mathematics
Page Range / eLocation ID:
p. 7-59
Medium: X
Sponsoring Org:
National Science Foundation
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