On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. (English) Zbl 1007.35087

Perelman, Galina

Ann. Henri Poincaré 2, No. 4, 605-673 (2001).

The author considers the nonlinear Schrödinger equation \[ i\psi_{t}=-\psi -\left|\psi\right|^{4}\psi \] with initial data \(\psi|_{t=0}=\varphi+\kappa\), where \(\kappa\) is small in suitable sense. It is shown that for a certain set of initial perturbations \(\kappa\) the solution \(\psi\) blows up in finite time. An asymptotic representation of the solution is obtained.

Reviewer: Vladislav Kravchenko (México)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A20	Analyticity in context of PDEs
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

nonlinear Schrödinger equation; initial perturbations; asymptotic representation