Spatial Structure of the Focusing Singularity of the Nonlinear Schrödinger Equation: A Geometrical Analysis

Abstract

The cubic nonlinear Schrödinger equation has solutions that become singular in finite time for spatial dimension $d \geq 2$. Numerical simulations demonstrate that in three dimensions, the blowup is self-similar and symmetric, with the structure described by a nonlinear, nonautonomous “profile equation.” In two dimensions, the blowup is again believed to be symmetric, but the self-similarity is weakly broken, with structure described by the profile equation in the limit as d tends to 2 from above. This paper gives a proof of the existence of a locally unique solution to the profile equation, for $d > 2$ and sufficiently close to 2, satisfying the boundary and global conditions associated with the blowup solution. Dynamical systems methods are used to transform previously derived asymptotic analysis into constructions of manifolds of solutions satisfying the relevant boundary conditions, and to follow these manifolds to show that they have a transverse intersection.