Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinksy equation

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.     

Free-surface film flow over topography under electric fields

We study a gravity-driven flow of a liquid down an inclined uneven wall in the presence of a normal electric field. We derive an evolution equation for the free-surface, and analyse steady-state solutions for flows into a rectangular trench and over a rectangular mound. The results are corroborated by boundary-element calculations for Stokes flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory

We analyze interaction of nonlinear pulses in active–dispersive–dissipative nonlinear media. A particular example of such media is a viscous thin film coating a vertical fibre. Experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place. In such events, larger pulses catch up with smaller ones and annihilate them. However, for certain flow conditions and after a certain distance from the inlet, no more coalescence is observed and the flow is described by quasi-equilibrium solitary pulses interacting continuously with each other through attractions and repulsions, and, eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole. This experimental study represents the first evidence of formation of bound states in low-Reynolds-number interfacial hydrodynamics. To gain theoretical insight into the interaction of the pulses and formation of bound states, we derive a weakly nonlinear model for the flow, the generalized Kuramoto–Sivashinsky (gKS) equation, that retains the fundamental mechanisms of the wave evolution, namely, dominant nonlinearity, instability, stability and dispersion. Much like in the experiments, the spatio-temporal evolution of the gKS equation is dominated by quasi-stationary solitary pulses which continuously interact with each other through coalescence events or attractions/repulsions. To understand the latter case, we utilize a weak-interaction theory for the solitary pulses of the gKS equation. The theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses, or, alternatively, the evolution of the separation distances. By analyzing the fixed points of this system, we obtain bound states of interacting pulses. For two pulses, we provide a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The interaction theory and resulting bound states are corroborated by computations of the full equation. We also find qualitative agreement between the theory and the experiments.     

Interaction of Flows on Quadrics

We interpret the Rodrigues formula describing a torsion of a rigid body in terms of Lie derivatives. We consider a more general situation where vector fields on quadrics (ellipsoids and hyperboloids) are dragged by other vector fields. The question is about the adjoint representation of a Lie group which is more general than the usual torsion group of the threedimensional Euclidean space.