Short Pulses Approximations in Dispersive Media

Abstract

Abstract. We derive various approximations for the solutions of nonlinear hyperbolic systems with fastly oscillating initial data. We first provide error estimates for the so-called slowly varying envelope, full dispersion, and Schrödinger approximations in a Wiener algebra; this functional framework allows us to give precise conditions on the validity of these models; we give in particular a rigorous proof of the "practical rule" which serves as a criterion for the use of the slowly varying envelope approximation (SVEA). We also discuss the extension of these models to short pulses and more generally to large spectrum waves, such as chirped pulses. We then derive and justify rigorously a modified Schrödinger equation with improved frequency dispersion. Numerical computations are then presented, which confirm the theoretical predictions.

Global Well-Posedness of the Short-Pulse and Sine–Gordon Equations in Energy Space

“…Short-pulse approximations of nonlinear wave packets in dispersive media were considered recently with various analytical techniques (see, e.g., [5] for a review of results). The previously known model for small-amplitude quasi-harmonic pulses, the nonlinear Schrödinger equation, is replaced in the short-pulse approximation by a new set of nonlinear evolution equations.…”

Intermediate Model for Spatial Evolution in Nonlinear Optics

“…See Ref. 7 for a deeper discussion on this topic and Refs. 16 and 1 when one takes the limit of ultra-short pulses.…”

A full dispersion asymptotic model to the water wave equations with surface tension

“…It seems however that an analytic proof of this conjecture is hard to obtain. 3 Numerical experiences conducted by Colin and Lannes [3] show that full dispersion models stay valid when modeling the evolution of rapidly oscillating initial data (short and chirped pulses), whereas the classical Schrödinger approximation breaks down in this case.…”

“…Let s ∈ N 0 , t 0 > d/2, T > 0, n 0 as defined in Proposition 3.1 and (c, d, p 0 ) ∈ C([0, T ]; H (s∨t 0 )+n 0 +2 (R d )3 ) be a solution to (60). Then the approximate solutionŨ = (ζ,ψ) constructed in Section 3 is consistent with the water wave equations(11) in the sense thatL(Ũ ) + N (Ũ ) = 3 N n=−N R 3n ( t, X)e inθ+ r R 4 (t, X) for some 2 < r 3 and some N 4.…”

“…The slight differences in the coefficients are explained by the fact that (2) is derived from a nondimensionalized version of the water wave equations 3. Even the rigorous justification the Benney-Roskes approximation remains an open question.…”