Initial value problem of nonlinear KdV-Schrödinger system

Pei Yi-Tong; Wang Jin-Kun; Guo Bo-Ling; Liu Wu-Ming

doi:10.7498/aps.72.20230241

Abstract

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501), and also gives the existence space of the local solutions.

Keywords:

Authors and contacts

1.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
3.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Corresponding author: Liu Wu-Ming, wmliu@iphy.ac.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2021YFA1400900, 2021YFA0718300, 2021YFA1402100), the National Natural Science Foundation of China (Grant Nos. 61835013, 12234012), and the Space Application System of China Manned Space Program

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References

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Cited By

[1]	Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095 Google Scholar
[2]	Shabat A, Zakharov V 1972 Sov. Phys. JETP 34 62
[3]	Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501 Google Scholar
[4]	Albert J, Bhattarai S 2013 Adv. Differ. Equ. 18 112964 Google Scholar
[5]	Angulo Pava J 2006 Electron. J. Differ. Equ. 72 1 Google Scholar
[6]	Angulo Pava J, Matheus C, Pilod D 2009 Commun. Pure Appl. Anal. 8 81544 Google Scholar
[7]	Chen L 1999 J. Partial Differ. Equ. 12 11
[8]	Corcho A J, Linares F 2007 Trans. Am. Math. Soc. 359 4089106 Google Scholar
[9]	Dias J P, Figueira M, Oliveira F 2010 Nonlinear Anal. 73 268698 Google Scholar
[10]	Appert K, Vaclavik J 1977 Phys. Fluids 20 18459 Google Scholar
[11]	Ikezi H, Nishikawa K, Hojo H, Mima K 1974 Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res. (Tokyo) 23948
[12]	Nishikawa K, Hojo H, Mima K, Ikezi H 1974 Phys. Rev. Lett. 33 14851 Google Scholar
[13]	Bubnov B A 1979 Differ. Equ. 15 17
[14]	Bubnov B A 1980 Differ. Equ. 16 24
[15]	Zhang B Y 1994 Int. Ser. Numer. Math. 118 371
[16]	Pei Y, Guo B, Liu W 2022 Math. Methods Appl Sci. 45 4853 Google Scholar
[17]	Capistrano-Filho R A, Sun S M, Zhang B Y 2018 Math. Control Relat. Fields 8 583 Google Scholar
[18]	Bona J L, Dougalis V A 1980 J. Math. Anal. Appl. 75 503 Google Scholar
[19]	Kumar V, Alqahtani A 2017 Nonlinear Dyn. 90 2903 Google Scholar
[20]	Kumar V, Alqahtani A 2019 Adv. Differ. Equ. 1 1 Google Scholar

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Vol. 72, Issue 10 - 2023-05-20

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