Short Time Behavior of Solutions to Nonlinear Schrödinger Equations in One and Two Space Dimensions

2006 ◽  
Vol 31 (6) ◽  
pp. 945-957 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Two Space Dimensions ◽  
Short Time

Short Time Behavior of Solutions to Linear and Nonlinear Schrödinger Equations

2008 ◽  
Vol 60 (5) ◽  
pp. 1168-1200 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Broad Class ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Pinsky Phenomenon ◽  
Nonlinear Schrodinger Equations ◽  
Linear And Nonlinear ◽  
Short Time

AbstractWe examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations ut = iΔu+q(u) on I×ℝn, with initial data u(0, x) = f (x). Particular attention is paid to cases where f is piecewise smooth, with jump across an (n−1)-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general n in the linear case. We also have detailed analyses for a broad class of nonlinear equations when n = 1 and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.

scholarly journals SHARP ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS WITH REPULSIVE INTERACTIONS

2005 ◽  
Vol 07 (02) ◽  
pp. 167-176 ◽  
Author(s):  
NAOYASU KITA ◽  
TOHRU OZAWA
Keyword(s):  
Large Time Behavior ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Repulsive Interactions ◽  
The Cauchy Problem

A detailed description is given on the large time behavior of scattering solutions to the Cauchy problem for nonlinear Schrödinger equations with repulsive interactions in the short-range case without smallness condition on the data.

WAVE OPERATORS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH A NONLINEARITY OF LOW DEGREE IN ONE OR TWO SPACE DIMENSIONS

2003 ◽  
Vol 05 (06) ◽  
pp. 983-996 ◽  
Author(s):  
KAZUNORI MORIYAMA ◽  
SATOSHI TONEGAWA ◽  
YOSHIO TSUTSUMI
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Final States ◽  
Schrödinger Equations ◽  
Wave Operators ◽  
Nonlinear Schrödinger ◽  
Low Degree ◽  
Nonlinear Schrodinger Equations ◽  
Two Space Dimensions

In this paper, we study the asymptotic behavior of solutions to the cubic and the quadratic nonlinear Schrödinger equations in one and two space dimensions, respectively. When the nonlinearity is of a form f = |u|p-1u, it is known that there exist scattered states if p > 1+2/n and there does not otherwise. Therefore we may consider the nonlinearities treated in the present paper to be of critical order for the existence of scattered states though their forms differ slightly from that given above. We prove, however, that there exist scattered states for these critical nonlinear Schrödinger equations, in other words, that the wave operators exist on a certain set of final states. Our proof is mainly based on the construction of suitable approximate functions that approach to solutions of nonlinear Schrödinger equations at t = ∞.

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