Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition

1990 ◽  
pp. 236-251 ◽  
Author(s):  
Takayoshi Ogawa ◽  
Yoshio Tsutsumi
Keyword(s):  
Boundary Condition ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Quartic Potential

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.

Lpc-concentration of blow-up solutions for the biharmonic nonlinear Schrödinger equation

2010 ◽  
Vol 89 (12) ◽  
pp. 1827-1835 ◽  
Author(s):  
Shihui Zhu ◽  
Jian Zhang ◽  
Han Yang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

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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