scholarly journals On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Andrey Melnikov
Keyword(s):  
Inverse Scattering ◽  
Initial Value Problem ◽  
Scattering Theory ◽  
Schrodinger Equation ◽  
Classical Approach ◽  
Initial Value ◽  
Nls Equation ◽  
New Techniques ◽  
Nonlinear Schrödinger ◽  
Inverse Scattering Theory

In this work we present an application of a theory of vessels to a solution of the evolutionary nonlinear Schrödinger (NLS) equation. The classes of functions for which the initial value problem is solvable rely on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of Zakharov-Shabath for solving evolutionary NLS equation but has an advantage of simpler formulas and new techniques and notions to understand the solutions.

L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 309-327 ◽  
Author(s):  
Nakao Hayashi ◽  
Masayoshi Tsutsumi
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Classical Solutions ◽  
Initial Value ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Sign Conditions ◽  
Image Position

SynopsisWe study the initial value problem for the nonlinear Schrödinger equationUnder suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

Energy critical Schrödinger equation with weighted exponential nonlinearity: Local and global well-posedness

2018 ◽  
Vol 15 (04) ◽  
pp. 599-621
Author(s):  
Abdelwahab Bensouilah ◽  
Dhouha Draouil ◽  
Mohamed Majdoub
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Nonlinear Schrödinger

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity [Formula: see text] where [Formula: see text] and [Formula: see text]. We establish local and global well-posedness in the subcritical and critical regimes.

On smooth solutions to the initial value problem for the mixed nonlinear Schrödinger equations

1991 ◽  
Vol 119 (1-2) ◽  
pp. 31-45 ◽  
Author(s):  
Guo Boling ◽  
Tan Shaobin
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Smooth Solutions ◽  
Initial Value ◽  
Unique Existence ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Image Position

SynopsisSolutions to the initial value problem for the mixed nonlinear Schrödinger equationare considered. Conditions on the constants α,β, γ, function g(·) and initial data u(x, 0) are given so that, for this problem, the unique existence of smooth solutions is proved. In addition, the decay behaviours of the smooth solutions as |x|→+∞ are discussed.

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