N-SOLITON SOLUTION OF THE KUNDU-TYPE EQUATION VIA RIEMANN-HILBERT APPROACH

doi:10.1007/s10473-020-0108-x

Abstract

Abstract: In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, the N-soliton solution of the Kaup-Newell equation, Chen-Lee-Liu equation, and Gerjikov-Ivanov equation can be obtained respectively by choosing different parameters.

Key words: the Kundu-type equation, Lax pair, Riemann-Hilbert problem, soliton solution

CLC Number:

35Q51

Lili WEN, Ning ZHANG, Engui FAN. N-SOLITON SOLUTION OF THE KUNDU-TYPE EQUATION VIA RIEMANN-HILBERT APPROACH[J].Acta mathematica scientia,Series B, 2020, 40(1): 113-126.

Trendmd

References

[1] Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödingertype equations. J Math Phys, 1984, 25(12):3433-3438
[2] Kundu A. Exact solutions to higher-order nonlinear equations through gauge transformation. Physica D, 1987, 25(1/3):399-406
[3] Fan E G. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J Math Phys, 2001, 42(9):4327-4344
[4] Clarkson P A, Tuszynski J A. Exact-solutions of the multidimensional derivative nonlinear Schrödingerequation for many-body systems near criticality. J Phys A, 1990, 23(19):4269-4288
[5] Kodama Y. Optical solitons in a monomode fiber. J Stat Phys, 1985, 39(5/6):597-614
[6] Wang X, Yang B, Chen Y, Yang Y Q. Higher-order rogue wave solutions of the Kundu-Eckhaus equation. Phys Scr, 2014, 89(9):095210
[7] Wen X Y, Zhang G Q. Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation. Mod Phys Lett B, 2018, 32(1):1850005
[8] Xu S W, He J S, Wang L H. The Darboux transformation of the derivative nonlinear Schrödinger equation. J Phys A:Math Theor, 2011, 44(30):305203
[9] Zhang Y S, Guo L J, Xu S W, Wu Z W, He J S. The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation. Commun Nonlinear Sci Numer Simulat, 2014, 19(6):1706-1722
[10] Guo B L, Ling L M, Liu Q P. High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations. Stud Appl Math, 2013, 130(4):317-344
[11] Kaup D J, Newell A C. Exact solution for a derivative nonlinear Schrödinger equation. J Math Phys, 1978, 19(4):798-801
[12] Liu H, Geng X G. The vector derivative nonlinear Schrödinger equation on the half-line. IMA J Appl Math, 2018, 83(1):148-173
[13] Yang B, Zhang W G, Zhang H Q, Pei S B. Generalized Darboux transformation and rational soliton solutions for Chen-Lee-Liu equation. Appl Math Comput, 2014, 242:863-876
[14] Zhang N, Xia T C, Fan E G. A Riemann-Hilbert approach to the Chen-Lee-Liu equation on the half line. Act Math Appl Sin Engl Ser, 2018, 34(3):493-515
[15] Zhang Y S, Guo L J, He J S, Zhou Z X. Darboux transformation of the second type derivative nonlinear Schrödinger equation. Lett Math Phys, 2015, 105(6):853-891
[16] Xu S W, He J S. The rogue wave and breather solution of the Gerdjikov-Ivanov equation. J Math Phys, 2012, 53(6):063507
[17] Guo L J, Zhang Y S, Xu S W, Wu Z W, He J S. The higher order rogue wave solutions of the GerdjikovIvanov equation. Phys Scr, 2014, 89(3):035501
[18] Wen X Y, Yang Y Q, Yan Z Y. Generalized perturbation N-fold Darboux transformations and multi-roguewave structures for the modified self-steepening nonlinear Schrödinger equation. Phys Rev E, 2015, 92(1):012917
[19] Nie H, Zhu J Y, Geng X G. Trace formula and new form of N-soliton to the Gerdjikov-Ivanov equation. Anal Math Phys, 2018, 8(3):415-426
[20] Qiu D Q, He J S, Zhang Y S, Porsezian K. The Darboux transformation of the Kundu-Eckhaus equation. Proc R Sco A-Math Phys Eng, 2015, 471(2180):20150236
[21] Gardner C S, Greene J M, Kruskal M D, Miura R M. Method for solving the Korteweg-de Vries equation. Phys Rev Lett, 1967, 19(19):1095
[22] Yang J K. Nonlinear Waves in Intergrable and Nonintergrable Systems. Philadelphia:SIAM, 2010
[23] Ablowitz M J, Fokas A S. Complex Variables:Introduction and Applications. New York:Cambridge University Press, 2003
[24] Novikov S, Manakov S, Pitaevskii L, Zakharov V. Theory of Solitons:the Inverse Scattering Method. New York, London:Consultants Bureau, 1984
[25] Fokas A S. Two dimensional linear PDEs in a complex ploygon. Proc R Soc A-Math Phys Eng Sci, 2001, 457(2006):371-393
[26] Zhang Y S, Chen Y, He J S. Riemann-Hilbert method and N-soliton for two component Gerdjikov-Ivanov equation. J Nonlinear Math Phys, 2017, 24(2):210-223
[27] Wang D S, Zhang D J, Yang J K. Integrable properties of the general coupled nonlinear Schrödinger equations. J Math Phys, 2010, 51(2):023510
[28] Wu J, Geng X G. Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation. Commun Nonlinear Sci Numer Simul, 2017, 53:83-93
[29] 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. Appl Math Comput, 2018, 332:148-159
[30] Hu B B, Xia T C. A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line. Math Meth Appl Sci, 2018, 41(13):5112-5123
[31] Hu B B, Xia T C, Zhang N, Wang J B. Initial-boundary value problem for the coupled higher-order Nonlinear Schrödinger equation on the half-line. Int J Nonlinear Sci Numer Simul, 2018, 19(1):83-92
[32] Xiao Y, Fan E G, Xu J. The Fokas-Lenells equation on the finite interval. Acta Math Sci, 2017, 37B(3):852-876
[33] Xu J, Fan E G. A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation. Acta Math Sci, 201434B(4):973-994