A multi-D model for Raman amplification

Abstract

Abstract.In this paper, we continue the study of the Raman amplification in plasmas that we initiated in [Colin and Colin, Diff. Int. Eqs. 17 (2004) 297-330; Colin and Colin, J. Comput. Appl. Math. 193 (2006) 535-562]. We point out that the Raman instability gives rise to three components. The first one is collinear to the incident laser pulse and counter propagates. In 2-D, the two other ones make a non-zero angle with the initial pulse and propagate forward. Furthermore they are symmetric with respect to the direction of propagation of the incident pulse. We construct a non-linear system taking into account all these components and perform some 2-D numerical simulations.Mathematics Subject Classification. 35Q55, 35Q60, 78A60, 74S20.

A Stability Criterion for High-Frequency Oscillations

“…Proof of Lemma 3.1. 10. -By Lemma 3.1.9, the datum V (0) decomposes as a leading term and a remainder ε K+1/2Ṽ ε…”

“…With γ ij are associated an upper rate of growth γ + ij , defined as in (3.30), and, unless γ ij = 0, a lower rate of growth γ − ij , defined as in the statement of Lemma 3.1. 10. Both depend on how tightly we cut around the resonance.…”

“…In [41], three-wave interaction systems are rigorously derived from the gravity-capillary water-wave system. Another derivation from Euler-Maxwell is given in [10].…”

“…-The 2d model of Colin and Colin[10] is a refinement of system (5.32) for the description of Raman scattering.…”

Liouville type theorems and periodic solutions for $χ^{(2)}$ type systems with non-homogeneous nonlinearities

“…For the purpose of modeling the nonlinear effects, one replaces the linear terms λ i u i in (1.4) by µ i |u i | p−1 u i . For a complete description of the model (1.3) (as well as a precise description of the physical coefficients) and the χ (2) system (1.4), we refer the readers to [6,7,8,9,11,42] and the references therein. A system similar to (1.3) also appears as an optics model with quadratic nonlinearity, see [40,41].…”