Integrable Nonlocal Nonlinear Equations
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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.read more
Citations
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Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions
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Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions
TL;DR: In this article, an integrable nonlocal nonlinear Schrodinger (NLS) equation with clear physical motivations is proposed, which is obtained from a special reduction of the Manakov system, and it describes Manakov solutions whose two components are related by a parity symmetry.
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Breather, lump and X soliton solutions to nonlocal KP equation
TL;DR: Breather, lump and X soliton solutions are presented via the Hirota bilinear method, to the nonlocal (2+1)-dimensional KP equation, derived from the Alice–Bob system.
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Nonlocal modified KdV equations and their soliton solutions by Hirota Method
Metin Gürses,Aslı Pekcan +1 more
TL;DR: In this article, the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions were studied.
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Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation
TL;DR: In this article, the inverse scattering transform (IST) was proposed for the reverse space-time NLS equation with nonzero boundary conditions (NZBCs) at infinity, which is more complicated because the branching structure of the associated linear eigenfunctions is complicated.
References
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Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
TL;DR: The condition of self-adjointness as discussed by the authors ensures that the eigenvalues of a Hamiltonian are real and bounded below, replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive.
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The Inverse scattering transform fourier analysis for nonlinear problems
TL;DR: In this article, a systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering, where the form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator.
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Beam dynamics in PT symmetric optical lattices.
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An exact solution for a derivative nonlinear Schrödinger equation
David J. Kaup,Alan C. Newell +1 more
TL;DR: In this paper, a method of solution for the derivative nonlinear Schrodinger equation is presented, where the appropriate inverse scattering problem is solved and the one-soliton solution is obtained, as well as the infinity of conservation laws.