Manchester Applied Mathematics and Numerical Analysis Seminars

Spring 1999

March 24, 1999, 4.00 pm

Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Dissipative effects in a nonlinear wave system with an unstable linear spectrum

Prof. Boris Malomed, Faculty of Engineering, Tel Aviv University

We consider a system describing the interaction of two resonantly coupled waves, which includes linear coupling terms, different group velocties, cubic nonlinear terms and linear losses (either symmetric or nonsymmetric). The system's dynamics are driven by an instability induced by the linear coupling. The collapse-free situation is considered. First, modulational instability of uniform continuous-wave solutions is investigated analytically. Then, numerical simulations demonstrate that, in most cases, development of an instability in the model with symmetric friction leads, after an initial ``latent'' stage, followed by a ``turbulent'' transient, to establishment of quasiperiodic or chaotic stationary spatial structures(the spatially chaotic structure is only quasi-stationary, demonstrating persistent residual ``breathings''). However, when the dissipative constant gets close to its limiting value, beyond which the system's dynamics are trivial, the duration of the transient turbulent stage diverges, so that persistent spatio-temporal chaos is observed. Qualitative arguments helping to interprete these features are presented. In the case when the dissipation is present only in one mode, stationary states do not exist. A specific nonstationary regime, with the undamped mode indefinitely growing and the damped one gradually vanishing but still playing an essential role, is predicted analytically and found numerically in this case. The simulations produce a nontrivial value of the scaling index controling the growth of the undamped mode. A general inference is that, in this system with an unstable linear spectrum, which generates spatio-temporal chaos in the absence of dissipation, adding dissipative terms leads, in most cases, to suppression of the temporal chaos, and generation instead of spatially chaotic quasi-stationary structures. But in some cases spatio-temporal chaos survives in the presence of the dissipation.

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