Numerical simulations of the energy- supercritical Nonlinear Schrödinger equation

Details Numerical simulations of the energy- supercritical Nonlinear Schrödinger equation Numerical simulations of the energy- supercritical Nonlinear Schrödinger equation

2009-07-17

arxiv.org/abs/0907.3130v2

Mathematics

Analysis of PDEs

17 pages with 9 figures

Scientific paper

We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy supercritical wave equations and proved that if the solution is {a priori} bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the $H^2$-critical Sobolev norm for solutions of the NLS equation in dimension five with quintic nonlinearity. We find that for a class of initial conditions, this norm remains bounded, the solution exists for long time, and scatters.

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