An initial-boundary value problem for the generalized Sasa-Satsuma equation on the half-line

doi:10.3969/j.issn.1000-5641.2019.04.004

Abstract

Abstract: In this paper, we implement the Fokas unified transform method to study initial-boundary value problems of the generalized Sasa-Satsuma equation on the half-line. Assuming that the solution u(x, t) of the generalized Sasa-Satsuma equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ.

Key words: Riemann-Hilbert problem, generalized Sasa-Satsuma equation, initialboundary value problems, Fokas unified transform

CLC Number:

O175.29

DONG Feng-jiao, HU Bei-bei. An initial-boundary value problem for the generalized Sasa-Satsuma equation on the half-line[J]. Journal of East China Normal University(Natural Sc, 2019, 2019(4): 33-41.

References

[1] FOKAS A S. A unified transform method for solving linear and certain nonlinear PDEs[J]. Proc R Soc Lond A, 1997, 453:1411-1443.
[2] FOKAS A S. Integrable nonlinear evolution equations on the half-line[J]. Commun Math Phys, 2002, 230:1-39.
[3] FOKAS A S, ITS A R, SUNG L Y. The nonlinear Schrödinger equation on the half-line[J]. Nonlinearity, 2005, 18:1771-1822.
[4] XIA B Q, FOKAS A S. Initial-boundary value problems associated with the Ablowitz-Ladik system[J]. Physica D, 2018, 364:27-61.
[5] ZHANG N, XIA T C, HU B B. A Riemann-Hilbert approach to the complex Sharma-Tasso-Olver equation on the half line[J]. Commun Theor Phys, 2017, 68(2):580-594.
[6] LENELLS J. Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs[J]. Phys D, 2012, 241:857-875.
[7] LENELLS J. The Degasperis-Procesi equation on the half-line[J]. Nonlinear Anal, 2013, 76:122-139.
[8] DE MONVEL B A, SHEPELSKY D, ZIELINSKI L. A Riemann-Hilbert approach for the Novikov equation[J]. SIGMA, 2016, 12:095.
[9] XU J, FAN E G. The unified method for the Sasa-Satsuma equation on the half-line[J]. Proc R Soc A Math Phys Eng Sci, 2013, 469:1-25.
[10] GENG X G, LIU H, ZHU J Y. Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line[J]. Stud Appl Math, 2015, 135:310-346.
[11] TIAN S F. Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method[J]. J Differ Equations, 2017, 262:506-558.
[12] YAN Z Y. An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4×4 Lax pair on the half-line[J]. CHAOS, 2017, 27:053117.
[13] HU B B, XIA T C, ZHANG N, et al. Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line[J]. Int J Nonlin Sci Num, 2018, 19(1):83-92.
[14] HU B B, XIA T C, MA W X. Riemann-Hilbert approach for an initial-boundary value problem of the twocomponent modified Korteweg-de Vries equation on the half-line[J]. Appl Math Comput, 2018, 332:148-159.
[15] HU B B, XIA T C. A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line[J]. Math Method Appl Sci, 2018, 41(13):5112-5123.
[16] HU B B, XIA T C, MA W X. Riemann-Hilbert approach for an initial-boundary value problem of an integrable coherently coupled nonlinear Schrödinger system on the half-line[J]. E Asian J Appld Math, 2018, 8(3):531-548.
[17] SASA N, SATSUMA J. New-type of solutions for a higher-order nonlinear evolution equation[J]. J Phys Soc Japan, 1991, 60:409-417.
[18] GENG X G, WU J P. Riemann-Hilbert approach and N-soliton solutions for a generalized Sasa-Satsuma equation[J]. Wave Motion, 2016, 60:62-72.