## On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations.(English)Zbl 0596.35022

The initial value problem for the nonlinear Schrödinger equation (NLS) $i\phi_ t+\Delta \phi +| \phi |^{2\sigma}=0,\quad \phi: R^ N_ x\times R^+_ t\to C,\quad \phi (x,0)=\phi_ 0(x)\in H^ 1,$ in the limit case $$\sigma =2/N$$, is considered. The problem has solutions that ”blow up” in finite time, namely: there exist initial data $$\phi_ 0\in H^ 1$$ and a positive and finite constant $$T(\phi_ 0)$$, such that $(1.1)\quad \lim_{t\to T}\int | \nabla \phi (x,t)|^ 2 dx=\infty.$ The author considers initial data $$\phi_ 0$$ for which $$\| \phi_ 0\|_ 2=\| R\|_ 2$$ where R is the ”ground state solitary wave” of (NLS). It is proved that if (1.1) holds for $$T\in (0,\infty)$$, then, up to translations in space and phase, $\phi (x,t)\to (1/[\lambda (t)]^{N/2})R[x/\lambda (t)]\quad as\quad t\to T,$ strongly in $$H^ 1$$, where $$\lambda (t)=\| \nabla R\|_ 2/\| \nabla \phi (t)\|_ 2$$. Also a family of explicit solutions with such behavior is presented.
In the critical case $$(\sigma =2)$$ of the generalized Korteweg-de Vries equation (GKdV) $w_ t+(2\sigma +1)w^{2\sigma} w_ x+w_{xxx}=0,\quad w(x,0)=w_ 0(x)\in H^ 1,$ finite time blow up is believed to occur. An analogous result on convergence for solutions to (GKdV) with the asymptotics (1.1) is proved.
Reviewer: I.Onciulescu

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35Q99 Partial differential equations of mathematical physics and other areas of application 35K55 Nonlinear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
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