Chin. Phys. B, 2022, Vol. 31(11): 110201    DOI: 10.1088/1674-1056/ac7a1b
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# Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

Guofei Zhang(张国飞)1, Jingsong He(贺劲松)2,†, and Yi Cheng(程艺)1
1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
2 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
Abstract  We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.
Keywords:  Schrödinger-type equation      Riemann-Hilbert problem      zero boundary conditions      N double-pole solitons
Received:  09 May 2022      Revised:  09 May 2022      Accepted manuscript online:  18 June 2022
 PACS: 02.30.Ik (Integrable systems) 05.45.Yv (Solitons)
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515012554).
Corresponding Authors:  Jingsong He     E-mail:  hejingsong@szu.edu.cn