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
Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities,which originates from the parity-symmetric reduction of the Manakov system and has applications in some physica...Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities,which originates from the parity-symmetric reduction of the Manakov system and has applications in some physical systems with the parity symmetry constraint between two fields/components.Via the Riemann–Hilbert method,the theory of inverse scattering transform with the presence of double poles is extended for this equation under nonzero boundary conditions(NZBCs).Also,the double-pole soliton solutions with NZBCs are derived in the reflectionless case.It is shown that the quasi-periodic beating solitons can be obtained when the double pole lies off the circleΓcentered at the origin with radius√2q_(0)(where q_(0) is the modulus of NZBCs)on the spectrum plane.Moreover,using the improved asymptotic analysis method,the asymptotic solitons are found to be located in some logarithmic curves of the xt plane.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.
基金supported by the Beijing Natural Science Foundations(Grant Nos.1232022 and 1252016)the National Natural Science Foundations of China(Grant Nos.12475003 and 11705284)the Hebei Province Natural Science Foundation(Grant No.A2025502019)。
文摘Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities,which originates from the parity-symmetric reduction of the Manakov system and has applications in some physical systems with the parity symmetry constraint between two fields/components.Via the Riemann–Hilbert method,the theory of inverse scattering transform with the presence of double poles is extended for this equation under nonzero boundary conditions(NZBCs).Also,the double-pole soliton solutions with NZBCs are derived in the reflectionless case.It is shown that the quasi-periodic beating solitons can be obtained when the double pole lies off the circleΓcentered at the origin with radius√2q_(0)(where q_(0) is the modulus of NZBCs)on the spectrum plane.Moreover,using the improved asymptotic analysis method,the asymptotic solitons are found to be located in some logarithmic curves of the xt plane.
