On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D (Q2752777)

The author considers the initial value problem with unknown \(\psi:= \psi(x,t)\), NEWLINE\[NEWLINE\begin{gathered} i\psi_t= -\psi_{xx}- |\psi|^{2p}\psi,\qquad x\in\mathbb{R},\\ \psi(x,0)= \psi_0(x),\quad \psi_0\in H^1.\end{gathered}\tag{1}NEWLINE\]NEWLINE It is pointed out that for \(p\geq 2\) the problem has solutions that blow-up in finite time and the case \(p= 2\) marks the transition between global existence and blow up. The author studies the influence of nonlinear bound states on singularity formation in the case \(p= 2\).NEWLINENEWLINE The nonlinear Schrödinger equation (1) has a soliton type solution of the form \(e^{it}\varphi_0(x)\) where \(\varphi_0\) is the ground state solitary wave. In this paper a Cauchy problem for (1) is studied when the initial data is close to a soliton in the sense that \(\psi(x,0)= \varphi_0(x)+ \chi_0(x)\) where \(\chi_0\) is, in an appropriate sense, a small quantity. It is shown that for a certain class of initial perturbations the solution \(\psi\) blows-up in finite time. Associated asymptotic representations are obtained which indicate that, up to a phase factor, the formation of a singularity is self-similar with a profile given by the ground state.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00038].