Blowup in the Nonlinear Schrödinger Equation Near Critical Dimension

✍ Scribed by Vivi Rottschäfer; Tasso J. Kaper

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 235 KB
• Volume: 268
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7814

✦ Synopsis

This article contains an analysis of the cubic nonlinear Schrödinger equation and solutions that become singular in finite time. Numerical simulations show that in three dimensions the blowup is self-similar and symmetric. In two dimensions, the blowup still appears to be symmetric but is no longer self-similar. In the case that the dimension, d, is greater than and exponentially close to 2 in terms of a small parameter associated to the norm of the blow-up solution, a locally unique, monotonically decreasing in modulus, self-similar solution that satisfies the boundary and global conditions associated with the blow-up solution is constructed in Kopell and Landman , SIAM J. Appl., Math. 55, 1297-1323]. In this article, it is shown that this locally unique solution also exists for d > 2 and algebraically close to 2 in the same small parameter. The central idea of the proof involves constructing a pair of manifolds of solutions (to the nonautonomous ordinary differential equation satisfied by the self-similar solutions) that satisfy the conditions at r = 0 and the asymptotic conditions respectively and then showing that these intersect transversally. A key step involves tracking one of the manifolds over a midrange in which the ordinary differential equation has a turning point and hence obtaining good control over the solutions on the manifold.  2002 Elsevier Science (USA)