RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

doi:10.1007/s10473-022-0106-z

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 127-140.doi: 10.1007/s10473-022-0106-z

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

马文秀^1,2,3,4

1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China;
2. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia;
3. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA;
4. School of Mathematical and Statistical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

收稿日期:2020-08-20 出版日期:2022-02-25 发布日期:2022-02-24
作者简介:Wenxiu MA,E-mail:mawx@cas.usf.edu
基金资助:
The work was supported in part by NSFC (11975145 and 11972291), and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020).

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

Wenxiu MA^1,2,3,4

1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China;
2. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia;
3. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA;
4. School of Mathematical and Statistical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

Received:2020-08-20 Online:2022-02-25 Published:2022-02-24
Supported by:
The work was supported in part by NSFC (11975145 and 11972291), and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020).

摘要/Abstract

摘要： The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrödinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

关键词: matrix spectral problem, nonlocal reverse-time integrable equation, integrable hierarchy, Riemann-Hilbert problem, inverse scattering transform, soliton solution

Abstract: The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrödinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

Key words: matrix spectral problem, nonlocal reverse-time integrable equation, integrable hierarchy, Riemann-Hilbert problem, inverse scattering transform, soliton solution

中图分类号:

37K15

马文秀. RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES[J]. 数学物理学报(英文版), 2022, 42(1): 127-140.

Wenxiu MA. RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES[J]. Acta mathematica scientia,Series B, 2022, 42(1): 127-140.

参考文献

[1] Ablowitz M J, Musslimani Z H. Integrable nonlocal nonlinear Schrödinger equation. Phys Rev Lett, 2013, 110(6):064105
[2] Ablowitz M J, Musslimani Z H. Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity, 2016, 29(3):915-946
[3] Ablowitz M J, Luo X D, Musslimani Z H. Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions. J Math Phys, 2018, 59(1):011501
[4] Ablowitz M J, Feng B F, Luo X D, Musslimani Z H. General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. Nonlinearity, 2018, 31(12):5385
[5] Yang J. General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys Lett A, 2019, 383(4):328-337
[6] Ma W X. Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations. Appl Math Lett, 2020, 102:106161
[7] Gürses M, Pekcan A. Nonlocal nonlinear Schrödinger equations and their soliton solutions. J Math Phys, 2018, 59(5):051501
[8] Ma L Y, Zhu Z N. Nonlocal nonlinear Schrödinger equation and its discrete version:soliton solutions and gauge equivalence. J Math Phys, 2016, 57(8):083507
[9] Vinayagam P S, Radha R, Al Khawaja U, Ling L. New classes of solutions in the coupled PT symmetric nonlocal nonlinear Schrödinger equations with four wave mixing. Commun Nonlinear Sci Numer Simul, 2018, 59:387-395
[10] Yang Y Q, Suzuki T, Cheng X P. Darbourx transformations and exact solutions for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation. Appl Math Lett, 2020, 99:105998
[11] Ablowitz M J, Musslimani Z H. Integrable nonlocal nonlinear equations. Stud Appl Math, 2017, 139(1):7-59
[12] Ji J L, Zhu Z N. Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform. J Math Anal Appl, 2017, 453(2):973-984
[13] Gürses M, Pekcan A. Nonlocal modified KdV equations and their soliton solutions by Hirota method. Commun Nonlinear Sci Numer Simul, 2019, 67:427-448
[14] Fokas A S. Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity, 2016, 29(2):319-324
[15] Song C Q, Xiao D M, Zhu Z N. Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation. Commun Nonlinear Sci Numer Simul, 2017, 45:13-28
[16] Ma W X. Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schrödinger equations. Proc Amer Math Soc, 2021, 149(1):251-263
[17] Novikov S P, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons:the Inverse Scattering Method. New York:Consultants Bureau, 1984
[18] Ma W X. The inverse scattering transform and soliton solutions of a combined modified Korteweg-de Vries equation. J Math Anal Appl, 2019, 471(1/2):796-811
[19] Ma W X, Zhou R G. Adjoint symmetry constraints of multicomponent AKNS equations. Chin Ann Math Ser B, 2002, 23(3):373-384
[20] Ablowitz M J, Kaup D J, Newell A C, Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud Appl Math, 1974, 53(4):249-315
[21] Tu G Z. On Liouville integrability of zero-curvature equations and the Yang hierarchy. J Phys A Math Gen, 1989, 22(13):2375-2392
[22] Ma W X, Chen M. Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. J Phys A Math Gen, 2006, 39(34):10787-10801
[23] Ma W X, Zhou R Z. Adjoint symmetry constraints leading to binary nonlinearization. J Nonlinear Math Phys, 2002, 9(Suppl. 1):106-126
[24] Ma W X. Riemann-Hilbert problems and soliton solutions of a multicomponent mKdV system and its reduction. Math Meth Appl Sci, 2019, 42(4):1099-1113
[25] Gakhov F D. Boundary Value Problems. London:Elsevier Science, 2014
[26] Kawata T. Riemann spectral method for the nonlinear evolution equation//Advances in Nonlinear Waves Vol I, Res Notes in Math 95. Boston, MA:Pitman, 1984:210-225
[27] Kamvissis S, McLaughlin K D T-R, Miller P D. Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation. Princeton:Princeton University Press, 2004
[28] Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264(4):2633-2659
[29] Ma W X. Lump and interaction solutions to linear PDEs in 2+1 dimensions via symbolic computation. Mod Phys Lett B, 2019, 33(36):1950457
[30] Ma W X, Zhang L Q. Lump solutions with higher-order rational dispersion relations. Pramana-J Phys, 2020, 94:43
[31] Zhang R G, Yang L G, Liu Q S, Yin X J. Dynamics of nonlinear Rossby waves in zonally varying flow with spatial-temporal varying topography. Appl Math Comput, 2019, 346:666-679
[32] Ma W X. Long-time asymptotics of a three-component coupled mKdV system. Mathematics, 2019, 7(7):573
[33] Rybalko Y, Shepelsky D. Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation. J Math Phys, 2019, 60(3):031504
[34] Ma W X. Long-time asymptotics of a three-component coupled nonlinear Schrödinger system. J Geom Phys, 2020, 153:103669
[35] Gesztesy F, Holden H. Soliton Equations and Their Algebro-geometric Solutions:(1+1)-Dimensional Continuous Models. Cambridge:Cambridge University Press, 2003
[36] Ma W X. Trigonal curves and algebro-geometric solutions to soliton hierarchies I, II. Proc Roy Soc A, 2017, 473(2203):20170232, 20170233
[37] Boiti M, Leon J J P, Martina L, Pempinelli F. Scattering of localized solitons in the plane. Phys Lett A, 1988, 132(8/9):432-439
[38] Hietarinta J. One-dromion solutions for generic classes of equations. Phys Lett A, 1990, 149(2/3):113-118

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

Metrics

本文评价

推荐阅读 10

[1]	肖羽, 徐建, 范恩贵. THE INITIAL BOUNDARY VALUE PROBLEMS FOR A NONLINEAR INTEGRABLE EQUATION WITH 3×3 LAX PAIR ON THE FINITE INTERVAL[J]. 数学物理学报(英文版), 2021, 41(5): 1733-1748.
[2]	温丽丽, 张宁, 范恩贵. N-SOLITON SOLUTION OF THE KUNDU-TYPE EQUATION VIA RIEMANN-HILBERT APPROACH[J]. 数学物理学报(英文版), 2020, 40(1): 113-126.
[3]	马文秀. RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS[J]. 数学物理学报(英文版), 2019, 39(2): 509-523.
[4]	李传忠. MÖBIUS-TODA HIERARCHY AND ITS INTEGRABILITY[J]. 数学物理学报(英文版), 2017, 37(4): 1151-1161.
[5]	肖羽, 范恩贵, 徐建. THE FOKAS-LENELLS EQUATION ON THE FINITE INTERVAL[J]. 数学物理学报(英文版), 2017, 37(3): 852-876.
[6]	徐建, 范恩贵. A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION[J]. 数学物理学报(英文版), 2014, 34(4): 973-994.
[7]	Spyridon Kamvissis. A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE[J]. 数学物理学报(英文版), 2011, 31(6): 2233-2246.
[8]	蔡海涛. TWO BOUNDED HALF ORTHOTROPIC PLANE MATERIALS WITH CRACKS[J]. 数学物理学报(英文版), 1995, 15(2): 214-218.