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Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations. (English) Zbl 0896.35123

Summary: We review qualitative properties of solutions of critical nonlinear Schrödinger equation \[ iu_t= -\Delta u-| u|^{p-1}u, \quad u(0)= u_0, \] and Zakharov equations \[ iu_t= -\Delta u+nu, \quad n_t= -\nabla\cdot v, \quad \frac{1}{c_0^2} v_t= -\nabla(n+| u|^2), \] which develop a singularity in finite time.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
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