Self-similar singular solutions for the nonlinear Schrödinger equation and the complex Ginzburg-Landau equation

In 2001, Plecháč and Sverak gave strong numerical evidence for the existence of branches of backwards self-similar singular solutions to the complex Ginzburg-Landau equation. We now present a rigorous proof of these branches' existence, which in particular includes the 3D cubic Schrödinger equation.

Our proof follows the same strategy as Plecháč and Sverak, which reduces the problem to proving the existence of a solution to a certain ODE with prescribed behaviour at zero and infinity. Near zero, the solution is constructed using a rigorous numerical ODE solver, and near infinity, by carefully analysing the asymptotic expansion. These two solutions are then glued together to form the full solution. To get the quantitative control required for the gluing, we make use of computer-assisted proof methods to bound the relevant terms.