# American Institute of Mathematical Sciences

March  2002, 1(1): 1-18. doi: 10.3934/cpaa.2002.1.1

## A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation

 1 Department of Mathematics, University of California, Santa Barbara California, 93106, United States

Received  March 2001 Published  December 2001

An efficient adaptive moving mesh method for investigation of the semi-classical limit of the focusing nonlinear schrödinger equation is presented. The method employs a dynamic mesh to resolve the sea of solitons observed for small dispersion parameters. A second order semi-implicit discretization is used in conjunction with a dynamic mesh generator to achieve a cost-efficient, accurate, and stable adaptive scheme. This method is used to investigate with highly resolved numerics the solution's behavior for small dispersion parameters. Convincing evidence is presented of striking regular space-time patterns for both analytic and non-analytic inital data.
Citation: Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1
 [1] Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic and Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893 [2] Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059 [3] Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic and Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505 [4] Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $n$. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030 [5] Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038 [6] Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial and Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521 [7] Zheng-Ru Zhang, Tao Tang. An adaptive mesh redistribution algorithm for convection-dominated problems. Communications on Pure and Applied Analysis, 2002, 1 (3) : 341-357. doi: 10.3934/cpaa.2002.1.341 [8] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [9] Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689 [10] Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098 [11] Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669 [12] Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869 [13] Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1133-1144. doi: 10.3934/jimo.2021012 [14] Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845 [15] Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic and Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011 [16] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [17] Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137 [18] Xiaoxue Gong, Ying Xu, Vinay Mahadeo, Tulin Kaman, Johan Larsson, James Glimm. Mesh convergence for turbulent combustion. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4383-4402. doi: 10.3934/dcds.2016.36.4383 [19] Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165 [20] Zecen He, Haihua Liang, Xiang Zhang. Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 421-441. doi: 10.3934/dcdsb.2021049

2020 Impact Factor: 1.916