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Xu, Tao; An, Li-Cong; Li, Min; Xu, Chuan-Xin

\(N\)-fold Darboux transformation of the discrete PT-symmetric nonlinear Schrödinger equation and new soliton solutions over the nonzero background. (English) Zbl 1543.35224

Stud. Appl. Math. 152, No. 4, 1338-1364 (2024).
Summary: For the discrete PT-symmetric nonlinear Schrödinger (dPTNLS) equation, this paper gives a rigorous proof of the \(N\)-fold Darboux transformation (DT) and especially verifies the PT-symmetric relation between transformed potentials in the Lax pair. Meanwhile, some determinant identities are developed in completing the proof. When the tanh-function solution is directly selected as a seed for the focusing case, the onefold DT yields a three-soliton solution that exhibits the solitonic behavior with a wide range of parameter regimes. Moreover, it is shown that the solution contains three pairs of asymptotic solitons, and that each asymptotic soliton can display both the dark and antidark soliton profiles or vanish as \(t\rightarrow\pm\infty\). It indicates that the focusing dPTNLS equation admits a rich variety of soliton interactions over the nonzero background, behaving like those in the continuous counterpart.
© 2024 Wiley Periodicals LLC.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35C08 Soliton solutions
35B40 Asymptotic behavior of solutions to PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K40 Second-order parabolic systems

Keywords:

asymptotic analysis; Darboux transformation; discrete PT-symmetric nonlinear Schrödinger equation; soliton solutions
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References:

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