Geodesic
Inverse scattering transform for a nonlocal derivative nonlinear Schr\"odinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 1, pp. 38-53

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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.
Keywords: nonlocal derivative nonlinear Schrödinger equation, zero boundary conditions, symmetry properties, matrix Riemann–Hilbert problem, singularity. 
@article{TMF_2022_210_1_a2,
     author = {Xinxin Ma and Yonghui Kuang},
     title = {Inverse scattering transform for a nonlocal derivative nonlinear {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {38--53},
     publisher = {mathdoc},
     volume = {210},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_210_1_a2/}
}
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Xinxin Ma; Yonghui Kuang. Inverse scattering transform for a nonlocal derivative nonlinear Schr\"odinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 1, pp. 38-53. http://geodesic.mathdoc.fr/item/TMF_2022_210_1_a2/