Linear versus nonlinear dissipation for critical NLS equation

• Published in: Physica. D Vol. 203; no. 3; pp. 167 - 184
• Main Authors: Passot, T., Sulem, C., Sulem, P.L.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.04.2005; Elsevier
• Subjects: Exact sciences and technology; Nonlinear Schrödinger equation; Physics; Wave collapse; Weak dissipation; 05.45. − a; 42.65.Jx; Weak dissipation; 52.35 Bj; 52.35 Mw; Nonlinear Schrödinger equation; Wave collapse; Collapse; Alfven waves; 42.65.Jx Nonlinear Schrödinger equation; Linear damping; Intensity; Photon absorption; Non linear phenomenon; 05.45.-a; Landau damping; Non linear damping; Energy dissipation; Differential equations; Filamentation instabilities; Schroedinger equation; Models; Nonlinear optics
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2005.03.011

The focusing nonlinear Schrödinger equation at critical dimension is considered in the presence of a weak damping, either linear or involving a power of the wave intensity. Special attention is paid to the case of a diffusive damping term, that describes Landau damping for a low-frequency Alfvén wave subject to filamentation instability. When compared with the nonlinear damping classically used to model multi-photon absorption in nonlinear optics, significant differences concerning the wave energy dissipation are numerically observed and interpreted by means of an asymptotic ODE model resulting from a modulation analysis of the solution near collapse. It is in particular shown that whereas for a nonlinear damping, dissipation is almost totally concentrated in the collapse events, it remains sizeable while the wave defocuses when the damping is linear.