A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions

2019 ◽  
Vol 62 (10) ◽  
pp. 1997-2014 ◽  
Author(s):  
Rongpei Zhang ◽  
Yong-Tao Zhang ◽  
Zhen Wang ◽  
Bo Chen ◽  
Yi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Numerical Method ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Nonlinear Schrödinger ◽  
Conservative Numerical Method ◽  
Fractional Nonlinear Schrödinger Equation

scholarly journals Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions

2019 ◽  
Vol 372 (1) ◽  
pp. 1-69 ◽  
Author(s):  
Maximilian Jeblick ◽  
Nikolai Leopold ◽  
Peter Pickl
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Particle System ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Pitaevskii Equation ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Reduced Density

Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by $$W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)$$Wβ(x)=N-1+2βW(Nβx), for any $$\beta >0$$β>0, or to be given by $$V_N(x)=e^{2N} V(e^N x)$$VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported $$W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})$$W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential $$V_N$$VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

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