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Darboux transformation and analytic solutions of the discrete \(\mathcal{PT}\)-symmetric nonlocal nonlinear Schrödinger equation. (English) Zbl 1351.35195

Summary: In this letter, for the discrete parity-time-symmetric nonlocal nonlinear Schrödinger equation, we construct the Darboux transformation, which provides an algebraic iterative algorithm to obtain a series of analytic solutions from a known one. To illustrate, the breathing-soliton solutions, periodic-wave solutions and localized rational soliton solutions are derived with the zero and plane-wave solutions as the seeds. The properties of those solutions are also discussed, and particularly the asymptotic analysis reveals all possible cases of the interaction between the discrete rational dark and antidark solitons.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
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