On the Initial Value Problem for the One Dimensional Quasi-Linear Schrödinger Equations

2002 ◽  
Vol 34 (2) ◽  
pp. 435-459 ◽  
Author(s):  
Wee Keong Lim ◽  
Gustavo Ponce
Keyword(s):  
Initial Value Problem ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
One Dimensional ◽  
The One

scholarly journals Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

2011 ◽  
Vol 10 (5) ◽  
pp. 1280-1304 ◽  
Author(s):  
Pauline Klein ◽  
Xavier Antoine ◽  
Christophe Besse ◽  
Matthias Ehrhardt
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Absorbing Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
Nonlinear Potential ◽  
Linear And Nonlinear ◽  
The One

AbstractWe propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

scholarly journals A SYSTEM OF COUPLED SCHRÖDINGER EQUATIONS WITH TIME-OSCILLATING NONLINEARITY

2012 ◽  
Vol 23 (11) ◽  
pp. 1250119 ◽  
Author(s):  
X. CARVAJAL ◽  
P. GAMBOA ◽  
M. PANTHEE
Keyword(s):  
Nonlinear Optics ◽  
Initial Data ◽  
Periodic Function ◽  
Initial Value Problem ◽  
Coupled System ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Periodic Functions ◽  
Initial Value ◽  
Physical Problems

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear Schrödinger equations [Formula: see text] where θ1 and θ2 are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data φ, ψ ∈ H1(ℝn), as |ω| → ∞, the solution (uω, vω) of the above IVP converges to the solution (U, V) of the IVP associated to [Formula: see text] with the same initial data, where I(g) is the average of the periodic function g. Moreover, if the solution (U, V) is global and bounded, then we prove that the solution (uω, vω) is also global provided |ω| ≫ 1.

scholarly journals On the one dimensional cubic NLS in a critical space

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marco Bravin ◽  
Luis Vega
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Three Dimensions ◽  
Critical Space ◽  
Initial Value ◽  
One Dimensional ◽  
Geometrical Problem ◽  
The One

<p style='text-indent:20px;'>In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.</p>

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