Integrable Nonlocal Nonlinear Equations

By Mark J. Ablowitz, Ziad Musslimani. 2016

538 citations recorded by the source corpus.

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” 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 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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/69001f26da0bc3cb9367e04d7f1d98cf9f11fc1c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Dmitry Shepelsky
2. Yan Rybalko
3. Xu‐Dan Luo
4. Jingsong He
5. Jiguang Rao
6. Yi Cheng
7. B. Feng
8. Ziad Musslimani

In Statistical and Nonlinear Physics

1. Dmitry Shepelsky
2. Yan Rybalko
3. Xu‐Dan Luo
4. Jingsong He
5. Jiguang Rao
6. Yi Cheng
7. B. Feng
8. Ziad Musslimani

Citations in the corpus

• Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data (2021, methodology): “Typical examples are the reverse space-time nonlocal NLS equation and the reverse time nonlocal NLS equation, the complex/real space-time SineGordon equation, the complex/real reverse space-time mKdV equation [1, 6, 7], the nonlocal derivative NLS equation [39], and the multidimensional nonlocal Dav”
• Nonlocal Nonlinear Schr\"odinger Equations and Their Soliton Solutions (2017, background): “Recently, Ablowitz and Musslimani [2]-[4] found another integrable reduction.”
• General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions (2017, methodology): “led NLS, as well as their semi-discrete analogues [60]–[70]. Moreover, the IST to solve the initial-value problem with vanishing boundary condition for the nonlocal NLS equation has been developed in [17, 21, 22]. For pure soliton solutions corresponding to reflectionless potentials, one obtains one-”
• Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations (2017, methodology): “Finally, it is worthy to emphasize that the technique presented in this paper may also succeed to other nonlinear systems with different kinds of fully or partially PT symmetric potentials, or reverse space-time nonlocal equations [19,20].”