title: Numerical study of fractional nonlinear Schrödinger equations.
abstract: Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
authors.affiliation: Institut de Mathématiques de Bourgogne , 9 avenue Alain Savary , BP 47870, 21078 Dijon, France.
authors.forename: Christian
authors.initials: C
authors.lastname: Klein
authors.affiliation: Department of Mathematics, Statistics, and Computer Science, M/C 249 , University of Illinois at Chicago , 851 S. Morgan St., Chicago, IL 60607, USA.
authors.forename: Christof
authors.initials: C
authors.lastname: Sparber
authors.affiliation: King Abdullah University of Science and Technology (KAUST) , MCSE Division , Thuwal 23955-6900, Saudi Arabia.
authors.forename: Peter
authors.initials: P
authors.lastname: Markowich
keywords: Fourier spectral method; dispersion; finite time blow-up; fractional Laplacian; nonlinear Schrödinger equations