Blowup in the Nonlinear Schr¨odinger Equation Near Critical Dimension

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.