scholarly journals Schrödinger equations: pointwise convergence to the initial data

1988 ◽  
Vol 102 (4) ◽  
pp. 874-874 ◽  
Author(s):  
Luis Vega
Keyword(s):  
Initial Data ◽  
Pointwise Convergence ◽  
Schrodinger Equations ◽  
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.

Oscillatory Integrals with Nonhomogeneous Phase Functions Related to Schrödinger Equations

1998 ◽  
Vol 41 (3) ◽  
pp. 306-317
Author(s):  
Lawrence A. Kolasa
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Smooth Function ◽  
Schrodinger Equation ◽  
Oscillatory Integrals ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Phase Functions

AbstractIn this paper we consider solutions to the free Schrödinger equation in n + 1 dimensions. When we restrict the last variable to be a smooth function of the first n variables we find that the solution, so restricted, is locally in L2, when the initial data is in an appropriate Sobolev space.

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