Abstract :
This article contains an analysis of the cubic nonlinear Schr¨odinger 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 [1995, 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.
