Geodesic
Riemann--Hilbert approach and $N$-soliton solutions of the~generalized mixed nonlinear Schr\"odinger equation with nonzero boundary conditions
Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 2, pp. 274-304

Voir la notice de l'article provenant de la source Math-Net.Ru

We apply the inverse scattering transformation to the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition at infinity. The scattering theories are investigated. In the direct problem, we analyze the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and the scattering matrix, and the properties of the discrete spectrum. In the inverse problem, an appropriate Riemann–Hilbert problem is formulated. By solving the problem, we obtain the reconstruction formula, the trace formula, and the “theta” condition. In the reflectionless case, a complicated integral factor is derived, which is a key ingredient of the explicit expression for $N$-soliton solutions. Using the $N$-soliton formula, we discuss the abundant dynamical features of the solution and its phases at different parameter values.
Keywords: Riemann–Hilbert problem, generalized mixed nonlinear Schrödinger equation
Mots-clés : soliton solution. 
@article{TMF_2021_209_2_a4,
     author = {DeQin Qiu and Cong Lv},
     title = {Riemann--Hilbert approach and $N$-soliton solutions of the~generalized mixed nonlinear {Schr\"odinger} equation with nonzero boundary conditions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {274--304},
     publisher = {mathdoc},
     volume = {209},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a4/}
}
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DeQin Qiu; Cong Lv. Riemann--Hilbert approach and $N$-soliton solutions of the~generalized mixed nonlinear Schr\"odinger equation with nonzero boundary conditions. Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 2, pp. 274-304. http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a4/