Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation. (English. Russian original) Zbl 1486.81097
Theor. Math. Phys. 210, No. 1, 31-45 (2022); translation from Teor. Mat. Fiz. 210, No. 1, 38-53 (2022).
Summary: We give a detailed discussion of a nonlocal derivative nonlinear Schrödinger (NL-DNLS) equation with zero boundary conditions at infinity in terms of the inverse scattering transform. The direct scattering problem involves discussions of the analyticity, symmetries, and asymptotic behavior of the Jost solutions and scattering coefficients, and the distribution of the discrete spectrum points. Because of the symmetries of the NL-DNLS equation, the discrete spectrum is different from those for DNLS-type equations. The inverse scattering problem is solved by the method of a matrix Riemann-Hilbert problem. The reconstruction formula, the trace formula, and explicit solutions are presented. The soliton solutions with special parameters for the NL-DNLS equation with a reflectionless potential are obtained, which may have singularities.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 81U40 | Inverse scattering problems in quantum theory |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
Keywords:
nonlocal derivative nonlinear Schrödinger equation; zero boundary conditions; symmetry properties; matrix Riemann-Hilbert problem; singularityReferences:
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