Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations

Abstract

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrödinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only $\dn(x,m)$ and $\cn(x,m)$ but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lamé polynomials of order 1, but it also admits solutions in terms of Lamé polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.

39 pages, no figures

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• Singular Soliton Molecules of the Nonlinear Schrodinger Equation
• Superposition principle and composite solutions to coupled nonlinear Schrödinger equations
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• Nonlocal Modified KdV Equations and Their Soliton Solutions
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• Symmetries and reductions of integrable nonlocal partial differential equations
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• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• Soliton solutions of the shifted nonlocal NLS and MKdV equations
• Families of rational and semi-rational solutions of the partial reverse space-time nonlocal Mel'nikov equation
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