American Institute of Mathematical Sciences

Advanced
November  2003, 9(6): 1571-1586. doi: 10.3934/dcds.2003.9.1571

Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions

Akihiro Shimomura 1,

1. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

Received  October 2002 Revised  May 2003 Published  September 2003

We study the scattering theory for the coupled Wave-Schrödinger equation with the Yukawa type interaction, which is certain quadratic interaction, in three space dimensions. This equation belongs to the borderline between the short range case and the long range one. We construct modified wave operators for that equation for small scattered states with no restriction on the support of the Fourier transform of them.
Keywords: modified wave operators., long range scattering, asymptotic behavior of solutions, Wave-Schrödinger equations.
Mathematics Subject Classification: 35B40, 35P25, 35Q4.
Citation: Akihiro Shimomura. Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1571-1586. doi: 10.3934/dcds.2003.9.1571
[1]

Takafumi Akahori. Global solutions of the wave-Schrödinger system with rough data. Communications on Pure and Applied Analysis, 2005, 4 (2) : 209-240. doi: 10.3934/cpaa.2005.4.209
[2]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[3]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[4]

Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure and Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351
[5]

Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602
[6]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
[7]

Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
[8]

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275
[9]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[10]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure and Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
[11]

Bixiang Wang. Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021223
[12]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
[13]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
[14]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919
[15]

Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759
[16]

P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029
[17]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825
[18]

Amna Dabaa, O. Goubet. Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1743-1756. doi: 10.3934/cpaa.2016011
[19]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110
[20]

Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (145)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy