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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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