ON ONE BLOW UP POINT SOLUTIONS TO THE CRITICAL NONLINEAR SCHRÖDINGER EQUATION

2005 ◽  
Vol 02 (04) ◽  
pp. 919-962 ◽  
Author(s):  
FRANK MERLE ◽  
PIERRE RAPHAEL
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Ground State ◽  
Finite Time ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Specific Class ◽  
Nonlinear Schrödinger ◽  
Existence And Stability

We consider the L2 critical nonlinear Schrödinger equation [Formula: see text] in the energy space H1. In the series of papers [11–15,18], we studied finite time blow up solutions for which lim t↑T < + ∞ |∇ u(t)|L2 = + ∞ and proved classification results of the blow up dynamics for the specific class of small super critical L2 mass initial data. We extend these results here to a wider class of finite time blow up solutions corresponding to the ones which accumulate at one point exactly the ground state mass. In particular, we prove the existence and stability of large L2 mass log-log type solutions which are believed to describe the generic blow up dynamics.

scholarly journals Blow-up solutions with minimal mass for nonlinear Schrödinger equation with variable potential

2021 ◽  
Vol 11 (1) ◽  
pp. 58-71
Author(s):  
Jingjing Pan ◽  
Jian Zhang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Uniqueness Result ◽  
Nonlinear Schrodinger Equation ◽  
Compactness Result ◽  
Nonlinear Schrödinger ◽  
Minimal Mass

Abstract This paper studies the mass-critical variable coefficient nonlinear Schrödinger equation. We first get the existence of the ground state by solving a minimization problem. Then we prove a compactness result by the variational characterization of the ground state solutions. In addition, we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up solutions which are pseudo-conformal transformation of the ground states.

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.

Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

2021 ◽  
Author(s):  
Annie Millet ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Random Variable ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Deterministic Setting

Abstract We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

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