THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1846-1860.

THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

朱巧珍¹, 范恩贵¹, 徐建²

1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

收稿日期:2017-07-20 修回日期:2018-02-28 出版日期:2018-12-25 发布日期:2018-12-28
通讯作者: Engui FAN E-mail:faneg@fudan.edu.cn
作者简介:Qiaozhen ZHU,E-mail:qiaozhenzhu13@fudan.edu.cn;Jian XU,E-mail:jianxu@usst.edu.cn
基金资助:
This work was support by NSFC (11671095; 11501365). Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant NO.15YF1408100, Shanghai Youth Teacher Assistance program NO.ZZslg15056 and the Hujiang Foundation of China (B14005).

THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

Qiaozhen ZHU¹, Engui FAN¹, Jian XU²

1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received:2017-07-20 Revised:2018-02-28 Online:2018-12-25 Published:2018-12-28
Contact: Engui FAN E-mail:faneg@fudan.edu.cn
Supported by:
This work was support by NSFC (11671095; 11501365). Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant NO.15YF1408100, Shanghai Youth Teacher Assistance program NO.ZZslg15056 and the Hujiang Foundation of China (B14005).

摘要/Abstract

摘要： The Gelfand-Levitan-Marchenko representation is used to analyze the initial-boundary value problem of two-component nonlinear Schrödinger equation on the half-line. It has shown that the global relation can be effectively analyzed by the Gelfand-Levitan-Marchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.

关键词: Gelfand-Levitan-Marchenko representation, initial-boundary value problem, two-component nonlinear Schrödinger equation

Abstract: The Gelfand-Levitan-Marchenko representation is used to analyze the initial-boundary value problem of two-component nonlinear Schrödinger equation on the half-line. It has shown that the global relation can be effectively analyzed by the Gelfand-Levitan-Marchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.

Key words: Gelfand-Levitan-Marchenko representation, initial-boundary value problem, two-component nonlinear Schrödinger equation

朱巧珍, 范恩贵, 徐建. THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE[J]. 数学物理学报(英文版), 2018, 38(6): 1846-1860.

Qiaozhen ZHU, Engui FAN, Jian XU. THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1846-1860.

参考文献

[1] Lax P D. Integrals of nonlinear equations of evolution and solitary waves. Comm Pure Appl Math, 1968, 21:467-490
[2] Gardner C S, Greene J M, Kruskal M D, Miura R M. Method for the solving for the Korteweg-de Veries equation. Phys Rev Lett, 1967, 19:1095-1097
[3] Fokas A S. A unified transform method for solving linear and certain nonlinear PDEs. Proc R Soc Lond A, 1997, 453:1411-1443
[4] Fokas A S. Integrable nonlinear evolution equations on the half-line. Commun Math Phys, 2002, 230:1-39
[5] Fokas A S, Its A R. An initial-boundary value problem for the sine-Gordon eqution. Theoret Math Phys, 1992, 92:388-403
[6] Fokas A S, Its A R. An initial-boundary value problem for the Korteweg-de Vries eqution. Math Comput Simulation, 1994, 37:293-321
[7] Fokas A S, Its A R. The linearization of the initial-boundary value problem of the nonlinear Schröding eqution. SAIM J Math Anal, 1996, 27:738-764
[8] Fokas A S, Its A R. The nonlinear Schröding equation on the interval. J Phys A, 2004, 37:6091-6114
[9] Fokas A S, Its A R, Sung L Y. The nonlinear Schröding equation on the half line. Nonlinearity, 2005, 18:1771-1822
[10] Lenells J. Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs. Physica D, 2012, 241:857-875
[11] Lenells J. The Degasperis-Procesi equation on the half-line. Nonlinear Anal, 2013, 76:122-139
[12] Xu J, Fan E. The unified transform method for the Sasa-Satsuma equation on the half-line. Proc R Soc Ser A, 2013, 469:20130068
[13] Xu J, Fan E. The three wave equation on the half-line. Phys Lett A, 2014, 378:26-33
[14] Xu J, Fan E. Initial-boundary value problem for integrable nonlinear evolution equations with 3×3 Lax pairs on the interval. Stud Appl Math, 2016, 136:321-354
[15] Xu J, Fan E. The GLM representation of the global relation for the two-component nonlinear Schröding equation on the interval. J Math Phys, 2017, 58:023509
[16] Xu J, Fan E. Initial-boundary value problem for the two-component nonlinear schrödinger equation on the half-line. J Nonlinear Math Phys, 2016, 23:167-189
[17] Xu J, Fan E G. A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation. Acta Math Sci, 2014, 34B:973-994
[18] De Monvel A B, Fokas A S, Shepelsky D. Analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett Math Phys, 2003, 65:199-212
[19] Fokas A S. The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Commun Pure Appl Math, 2005, 58:639-670

THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

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