New Singular Solutions of the Biharmonic NLS Equation

New Singular Solutions of the Biharmonic NLS Equation

Guy Baruch
Department of Applied Mathematics
Tel Aviv University

We consider blowup-type singular solutions of the fourth-order (biharmonic) nonlinear Schrodinger (BNLS) equation

iψ_t(t,x)-Δ²ψ+|ψ|^2σψ=0, x∈R^d.

Our formal and informal analysis, and numerical evidence, indcate that the blowup formation greatly resembles that of the standard (harmonic) NLS. However, the lack of some key analytic tools still makes BNLS theory challenging.

In the L²-critical case σ⋅d=4, we show that the collapsing core of the solution converges to a self-similar profile, and bounds the blowup rate.

In the super-critical case σ⋅d>4, we use asymptotic analysis to find and characterize new peak-type and ring-type singular solutions of the BNLS, which also converge to a self-similar profile. We determine whether the ring-type solutions blowup at either a sphere ("standing ring") or a point ("shrinking ring"), as determined by σ and d. These findings are verified numerically, using an adaptive mesh, at focusing factors of up to 10⁸.

Joint work with Gadi Fibich and Elad Mandelbaum.