Focusing singularity in a derivative nonlinear Schrödinger equation

Abstract

Abstract. We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, |ψ| 2σ ψ x . In the L 2 -supercritical regime, σ > 1, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.

Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equations

“…We show that for n = 4, the asymptotic description of theorem 1 still applies though the studied initial data are very far from the soliton. Note, however, that derivative NLS equations show a very similar behavior as gKdV equations, i.e., both dispersive shocks and blow-up as can be seen for instance in [28].…”

Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition inH12

“…Whether DNLS exhibits finite-time blowup solutions for initial data with large mass remains an important open question. A numerical study by Liu, Simpson and Sulem [28] indicates that there is finite time singularity for the L 2 -supercritical nonlinearity i|u| 2σ ∂ x u, σ > 1. We also refer to [5] for a further numerical investigation of the structure of the singular profile near blowup times.…”

Local Existence Theory for Derivative Nonlinear Schrödinger Equations with Noninteger Power Nonlinearities

“…We were, unfortunately, unable to show that the H 1 solutions belonged to C(0, T ; H 1 ). Our H 2 solutions are a step toward the justifying the time-dependent simulations appearing in [18,19].…”

“…Such a higher regularity result is of interest in light of the time dependent simulations in [18,19]. In those works, the pseudospectral method was used, which exactly corresponds to the Fourier cutoff mollifier of our analysis.…”

“…This equation, gDNLS, was recently studied in [18,19] for both the properties of its solitons and its potential for singularity formation. The original derivative nonlinear Schrödinger equation was written as…”