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.

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

“…There the L 2 norm is invariant under the rescaling (15), and the blow-up profile has essentially the mass of the initial data. In the case n = 5 the L 2 norm of the part of the solution blowing up vanishes in the limit t → t * as follows from (21). This can be recognized already in Fig.…”

“…In the case of very large mass of the initial data, the direct integration of the dynamically rescaled equation (16) suffers from numerical instabilities at the boundaries of the computational domains. As in [16,20,19,21] the quantities of the dynamical rescaling are obtained in these cases by postprocessing the data from the direct integration of the gKdV equation (1). Our numerical simulations are compatible with the following conjectures about the analytic behavior of the solution in the critical and supercritical case.…”

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

“…The latter will be identified by tracing the norm of the solution and matching it to what would be expected from a dynamical rescaling of the DS II equation. Assuming that vanishes at the blow‐up, we can study the type of the blow‐up for DS II in a similar way as it has been done for generalized KdV equations in , for generalized KP equations in , and for fractional NLS equations in : we integrate DS II directly, as described above, and then we use some postprocessing to characterize the type of blow‐up via the rescaling . Because the L 2 norm of ψ is invariant under this rescaling, we consider the norm of ψ.…”

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

“…Along the numerical front, different reliable and efficient numerical methods have been developed, such as finite difference methods [8][9][10][11][12][13][14][15][16][17], spectral or collocation methods [18][19][20][21][22][23][24][25][26] and finite element methods [27][28][29]. It is well known that for differential equations with geometric structures, the structure-preserving schemes always perform better than the general-purpose ones [30][31][32][33].…”