scholarly journals Wellposedness for the fourth order nonlinear Schrödinger equations

2006 ◽  
Vol 320 (1) ◽  
pp. 246-265 ◽  
Author(s):  
Chengchun Hao ◽  
Ling Hsiao ◽  
Baoxiang Wang
Keyword(s):  
Fourth Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

scholarly journals Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 510-541 ◽  
Author(s):  
X. Liang ◽  
A. Q. M. Khaliq ◽  
Y. Xing
Keyword(s):  
Discontinuous Galerkin ◽  
Fourth Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Nonlinear Schrödinger ◽  
Local Discontinuous Galerkin ◽  
Nonlinear Schrodinger Equations

AbstractThis paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

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