We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis main...We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.展开更多
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,suc...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.展开更多
基金the Fundamental Research Funds for the Central Universities(Grant No.2024MS126).
文摘We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12071304 and 11871446)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2022A1515012554)。
文摘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.
