Integrable Nonlocal Nonlinear Equations

TLDR: In this article, the reverse space-time and reverse time nonlinear integrable equations are introduced, which arise from symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone.

Abstract: A nonlocal nonlinear Schrodinger (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 PT symmetric thus the nonlocal NLS equation is also PT symmetric. In this paper, new reverse space-time and 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 NLS, modified Korteweg-deVries (mKdV), sine-Gordon, (1 + 1) and (2 + 1) dimensional three-wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partially PT symmetric 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 Schrodinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painleve type equations are derived from the reverse space-time and reverse time nonlocal NLS equations.