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

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.

### 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
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