American Institute of Mathematical Sciences

Advanced
May  2005, 5(2): 469-488. doi: 10.3934/dcdsb.2005.5.469

The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation

Xinmin Xiang 1,

1. 

Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China

Received  December 2003 Revised  October 2004 Published  February 2005

In this paper, we use the Chebyshev rational pseudospectral method to analyze the long-time behavior of Cauchy problem for nonlinear Schrödinger equation (NSE) with weak damping. We obtain the error estimate of approximate solution and prove the existence and weak upper semicontinuity of approximate attractor.
Keywords: Cauchy problem, approximate attractor., Nonlinear Schrödinger equation, long-time behavior, Chebyshev rational spectral method, global attractor.
Mathematics Subject Classification: 37M99, 65M70, 65N3.
Citation: Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469
[1]

Olivier Goubet. Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 503-530. doi: 10.3934/dcds.1997.3.503
[2]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[3]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031
[4]

Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[5]

Olivier Goubet, Ezzeddine Zahrouni. Global attractor for damped forced nonlinear logarithmic Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2933-2946. doi: 10.3934/dcdss.2020393
[6]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039
[7]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[8]

Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365
[9]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[10]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028
[11]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[12]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations and Control Theory, 2022, 11 (2) : 559-581. doi: 10.3934/eect.2021013
[13]

Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
[14]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[15]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[16]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703
[17]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[18]

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
[19]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[20]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy