Mechanisms of Unstable Blowup in a Quadratic Nonlinear Schrödinger Equation

In the work Cho et al. (Jpn J Ind Appl Math 33:145–166, 2016) the authors conjecture that the quadratic nonlinear Schrödinger equation (NLS) iut=uxx+u2 for x∈T is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation ut=uxx+u2. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture.