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

Weinstein, Michael I.

Commun. Partial Differ. Equations 11, 545-565 (1986).

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

Keywords:

critical case; laser propagation; self-trapping; intense focusing; of beams; solution blow up; nonlinear Schrödinger equation; generalized Korteweg-de Vries equation

References:

[1]	Berestycki H, Arch. Rat. Mech. Anal 8 pp 313– (1983)
[2]	Berestycki H, C.R. Acad 293 pp 489– (1983)
[3]	DOI: 10.1007/BF01217349 · Zbl 0579.35025 · doi:10.1007/BF01217349
[4]	DOI: 10.1007/BF01403504 · Zbl 0513.35007 · doi:10.1007/BF01403504
[5]	Chiao R.Y, Phys. Rev. Lett 3 pp 479– (1984)
[6]	DOI: 10.1007/BF00250684 · Zbl 0249.35029 · doi:10.1007/BF00250684
[7]	DOI: 10.1063/1.523491 · Zbl 0372.35009 · doi:10.1063/1.523491
[8]	Ginibre J, J. Func. Anal 46 pp 81– (1972)
[9]	Ginibre J, in Nonlinear P.D.E. and Applications, College de France, Seminar (1981)
[10]	Kato T, in Studies in Appl. Math., Advanced in Math. Supplementary Studies
[11]	DOI: 10.1103/PhysRevLett.15.1005 · doi:10.1103/PhysRevLett.15.1005
[12]	Lions P.L, the locally compact case 5 (1984)
[13]	Lions P.L, the limit case, parts 1,2,” Rev. Math. Iberomericana 5 (1984)
[14]	McLeod P.L, Proc. Natl. Acad. Sci pp 6592–
[15]	Shatah ,J. and Strauss, W.A. ”Instability of nonlinear bound states,” preprint · Zbl 0603.35007
[16]	DOI: 10.1007/BF01626517 · Zbl 0356.35028 · doi:10.1007/BF01626517
[17]	DOI: 10.1016/0021-9991(83)90045-1 · Zbl 0519.76002 · doi:10.1016/0021-9991(83)90045-1
[18]	DOI: 10.1002/cpa.3160370603 · Zbl 0543.65081 · doi:10.1002/cpa.3160370603
[19]	Tsutsumi,M.” nonexistance of global solutions to nonlinear Schrödinger equations,”(unpuplished manuscript).
[20]	Vlasov V.N, Izv. Rad. U 37 pp 1353– (1971)
[21]	DOI: 10.1007/BF01208265 · Zbl 0527.35023 · doi:10.1007/BF01208265
[22]	DOI: 10.1137/0516034 · Zbl 0583.35028 · doi:10.1137/0516034
[23]	weinstein, M.I ”Lyapunov stability of ground of nonlinear dispersive evolution equations,”Commun,Pure Appl.Math.,in Press · Zbl 0594.35005
[24]	Weinstein M.I, in Proceedings of AMS/SIAM Research Seminar on Nonlinear Systems of PDE in Applied Mathematics Santa Fe
[25]	Zakharov V.E, Sov. Phys. JETP 41 pp 465– (1976)
[26]	DOI: 10.1007/BF01394245 · Zbl 0538.35058 · doi:10.1007/BF01394245