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.

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 ◽  
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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.

scholarly journals CASCADE OF PHASE SHIFTS FOR NONLINEAR SCHRÖDINGER EQUATIONS

2007 ◽  
Vol 04 (02) ◽  
pp. 207-231 ◽  
Author(s):  
RÉMI CARLES
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Schrodinger Equation ◽  
Critical Time ◽  
Phase Shifts ◽  
Nonlinear Schrodinger Equation ◽  
Linear Case ◽  
Rigorous Proof ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

We consider a semi-classical nonlinear Schrödinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open.

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.

scholarly journals On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials

2019 ◽  
Vol 150 (6) ◽  
pp. 3187-3215
Author(s):  
Jianfu Yang ◽  
Jinge Yang
Keyword(s):  
Excited State ◽  
Lagrange Multiplier ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Limiting Behaviour ◽  
Normalized Solutions ◽  
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AbstractIn this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.

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