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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GBDT and algebro-geometric approaches to explicit solutions and wave functions for nonlocal NLS
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Vector dark solitons with oscillating background density
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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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TL;DR: In this paper, a method for solving the initial value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as
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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.
TL;DR: In this paper, parity-time symmetric periodic potentials are investigated in detail for both one-and two-dimensional lattice geometries, and it is shown that PT periodic structures can exhibit unique characteristics stemming from the nonorthogonality of the associated Floquet-Bloch modes.
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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.