Exponential time differencing Crank–Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations

2014 ◽  
Vol 235 ◽  
pp. 235-252 ◽  
Author(s):  
Xiao Liang ◽  
Abdul Q.M. Khaliq ◽  
Qin Sheng
Keyword(s):  
Spline Approximation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Crank Nicolson

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.

scholarly journals Crank–Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials

2017 ◽  
Vol 27 (11) ◽  
pp. 2147-2184 ◽  
Author(s):  
Patrick Henning ◽  
Daniel Peterseim
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Step Size ◽  
Time Step ◽  
Coupling Condition ◽  
Time Step Size ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Crank Nicolson

This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank–Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in [Formula: see text] under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

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