A conservative spectral collocation method for the nonlinear Schrödinger equation in two dimensions

2017 ◽  
Vol 310 ◽  
pp. 194-203 ◽  
Author(s):  
Rongpei Zhang ◽  
Jiang Zhu ◽  
Xijun Yu ◽  
Mingjun Li ◽  
Abimael F.D. Loula
Keyword(s):  
Schrödinger Equation ◽  
Collocation Method ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Nonlinear Schrödinger

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.

scholarly journals Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

2021 ◽  
Vol 18 (01) ◽  
pp. 1-28
Author(s):  
Van Duong Dinh
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Nonlinear Schrödinger ◽  
Energy Scattering ◽  
The One ◽  
Inhomogeneous Nonlinear Schrödinger Equation

We consider a class of [Formula: see text]-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions [Formula: see text] where [Formula: see text] and [Formula: see text]. Using a new approach of Arora et al. [Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 148 (2020) 1653–1663], we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah and Guzmán [Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.) 51 (2020) 449–512] to the whole range of [Formula: see text] where the local well-posedness is available. In the defocusing case, our result extends the one in [V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 19(2) (2019) 411–434], where the energy scattering for non-radial initial data was established in dimensions [Formula: see text].

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