Semiclassical Asymptotics of the Solution to the Cauchy Problem for the Schrödinger Equation with a Delta Potential Localized on a Codimension 1 Surface

2020 ◽  
Vol 310 (1) ◽  
pp. 304-313
Author(s):  
A. I. Shafarevich ◽  
O. A. Shchegortsova
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Asymptotics ◽  
Delta Potential ◽  
The Cauchy Problem

Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

2002 ◽  
Vol 54 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Space Dimension ◽  
Nonlinear Schrodinger Equation ◽  
The Cauchy Problem ◽  
Cubic Nonlinear Schrödinger Equation ◽  
One Space Dimension

AbstractWe consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimensionCubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data are small and such that for some n ∈ Z, and , then the solution has an additional logarithmic timedecay in the short range region . In the far region the asymptotics have a quasilinear character.

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