Numerical resolution of some nonlinear Schr�dinger-like equations in plasmas

Abstract

Schrödinger-like equations play an essential role in the modeling of plasmas created by laser beams, and taking into account of relativistic terms yields a new kind of nonlinearity in them, as compared to the classical case. A numerical scheme involving a new handling of absorbing conditions suitable for Schrödinger-like equations is given, and several general models are studied: the classical cubic case linked to the Kerr effect for an anharmonic plasma, where numerical experiments prove the efficiency of our code applied to the computation of solitons and explosive solutions. Furthermore, new kinds of explosive solutions are computed, and we numerically show the essential role of the ground state to get approximations of the cubic explosive solution by global ones. A multiphotonic ionization model is also studied including higher-order terms in the nonlinearity. In this case, we obtain stable structures physically explained by competing saturation processes. These structures have been experimentally detected. Finally, a relativistic model involving the Lorentz kinematic factor in the evolution equations is investigated numerically; filamentary structures are found in agreement with predicted relativistic phenomena. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 672–696, 1999