scholarly journals Global solutions of the nonlinear Schrödinger equation in exterior domains

1983 ◽  
Vol 59 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Yoshio Tsutsumi
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Exterior Domains ◽  
Nonlinear Schrödinger

scholarly journals The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

1991 ◽  
Vol 117 (3-4) ◽  
pp. 251-273 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Schrödinger Equation ◽  
Global Solution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Structure Of Solutions ◽  
Nonlinear Schrödinger ◽  
Conformally Invariant

SynopsisWe study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

scholarly journals CONVERGENCE TO SCATTERING STATES IN THE NONLINEAR SCHRÖDINGER EQUATION

2001 ◽  
Vol 03 (03) ◽  
pp. 403-418 ◽  
Author(s):  
PASCAL BÉGOUT
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Scattering States

In this paper, we consider global solutions of the following nonlinear Schrödinger equation iut+Δ u+λ|u|αu=0, in ℝN, with λ∈ℝ, [Formula: see text] (α∈ (0,∞) if N=1) and u(0)∈X≡ H1(ℝN)∩ L2(|x|2;dx). We show that, under suitable conditions, if the solution u satisfies e-itΔu(t)-u±→0 in X as t→±∞ then u(t)-eitΔu±→0 in X as t→±∞. We also study the converse. Finally, we estimate |‖u(t)‖X- ‖eitΔu±‖X| under some less restrictive assumptions.

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