In this talk, we study the Cauchy problem for the nonlinear Schodinger equation with a delta potential. We show that under certain conditions, the supremum norm of the solution tends to infinity in finite time. In order to prove this, we study the associated Lagrangian and Hamiltonian, and derive an estimate of the associated variance, which extends the usual Virial identity for the nonlinear Schrodinger equation without a potential. We also derive several conservation laws which a classical solution of the Cauchy problem must satisfy.
Finite Time Blowup for the Nonlinear Schrödinger Equation with a delta potential
Sarah Raynor, Wake Forest UniversityAuthors: Brandon Hauser, John Holmes, Eoghan O’Keefe, Sarah Raynor, and Chuanyang Yu
2023 AWM Research Symposium
Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]