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By Catherine Sulem, Pierre-Louis Sulem

This monograph goals to fill the space among the mathematical literature which considerably contributed over the past decade to the certainty of the cave in phenomenon, and purposes to domain names like plasma physics and nonlinear optics the place this technique presents a primary mechanism for small scale formation and wave dissipation. This leads to a localized heating of the medium and with regards to propagation in a dielectric to attainable degradation of the fabric. For this objective, the authors have selected to deal with the matter of wave cave in by way of a number of tools starting from rigorous mathematical research to formal asymptotic expansions and numerical simulations.

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A localized solution is then associated to w = (2α)−1/2 sinh(α1/2 ξ), leading to a singularity at ξ = 0 for the amplitude v = 1/w, which is not acceptable. 22 1. The Physical Context waveguides was pointed out by Hasegawa and Tappert (1973). The resulting “optical temporal solitons ”5 were observed by Mollenauer, Stollen & Gordon (1980) (see also the review of Segev & Stegeman 1998). Shape-preserving localized solutions can also be constructed in higher dimensions (or for higher nonlinearities), and are usually referred to as standing or solitary waves.

22) where +∞ P0 (t) = 0 −∞ By Fourier transform in time, we have 1 0 P0 (Ω) = χ1 (Ω)E0 (Ω) = 1 ikz Ω−ω e χ1 (Ω)E( ). 24) where we expand χ1 (ω +εν) = χ1 (ω)+ χ1 (ω)εν + 12 χ1 (ω) 2 ν 2 +· · ·, where the primes hold for derivatives with respect to the frequency. 25) as a formal Taylor expansion, we are led to write symbolically 1 0 ∂ )E0 (T ) ∂T = ei(kz−ωt) χ1 (i∂t + ε∂T )E0 . 29) 16 1. 30) and n2 (ω) = 1 + χ1 (ω). The associated dispersion relation reduces to 2 2 k = ω nc2(ω) . 31) where in the expansion of the operator k 2 (ω + εi∂T ), the prime denotes derivatives and the corresponding functions are evaluated for the frequency ω.

Its asymptotes κ1 = ±κ2 are unstable. Note that this domain includes arbitrarily large wave numbers. Experimentally, the modulational instability has been observed in various physical contexts, including surface waves on deep water (Benjamin 1967a, Yuen and Lake 1975, 1980, Remoissenet 1996) and light waves in optical fibers (Tai, Hasegawa, and Tomita 1986). The nonlinear development of this instability depends on the space dimension. Numerical simulations of the focusing one-dimensional cubic NLS equation in a periodic domain has revealed a Fermi–Pasta–Ulam (1955) recurrent behavior of the solution (Yuen and Ferguson 1978a, Hafizi 1981).

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