Stationary thermomagnetic waves in superconductors

The propagation of a nonlinear stationary thermomagnetic wave in superconducting media is qualitatively analyzed in view of the effect of an external magnetic field. The velocity and the thickness of the wave front are estimated. It is demonstrated that the inclusion of an external magnetic field affects the thermomagnetic wave profile only slightly.     

Multi-dimensional quasi-simple waves in weakly dissipative flows

A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to multi-orders analysis which is different from the standard perturbation method. Detailed solving up to the second order is presented for a 2D sound simple wave. A new 2D Burgers equation is derived for the wave phase. It essentially differs from the known 2D generalizations of the Burgers equation, e.g., from the Zabolotskaya-Khokhlov equation. The derived equation is investigated in the framework of group analysis of differential equations. Multi-parameter families of its exact solutions are constructed. Simplest solutions are chosen for analysis of their physical relevance in the initial variables.     

Stationary one-dimensional dispersive shock waves

We address shock waves generated upon the interaction of tilted plane waves with negative refractive index defects in defocusing media with linear gain and two-photon absorption. We found that, in contrast to conservative media where one-dimensional dispersive shock waves usually exist only as nonstationary objects expanding away from a defect or generating beam, the competition between gain and two-photon absorption in a dissipative medium results in the formation of localized stationary dispersive shock waves, whose transverse extent may considerably exceed that of the refractive index defect. One-dimensional dispersive shock waves are stable if the defect strength does not exceed a certain critical value.     

Geometric phases in dissipative systems

It is shown that a phenomenon analogous to the geometric phase shifts of Berry and Hannay occurs for dissipative oscillatory systems and can be detected in numerical simulations of chemical oscillators. The approach herein to the theory of geometric phases begins with a study of simple first-order differential equations on the circle (circle dynamics). It is shown how more complicated systems exhibit geometric phases through reduction to a circle dynamics. In this way, the various manifestations of the phenomenon are seen from a single unified perspective. The results are illustrated in numerical experiments on several model systems ranging from analytically solvable, but contrived, to realistic models of chemical oscillators.     

Intermittency in dissipative dynamical systems

We show that the intermittency displayed by a differential system proposed by Yamada and Fujisaka can be interpreted in the general framework of intermittent transitions to turbulence studied by Pomeau and the author.     

Stochastic oscillations in dissipative systems

The recent discovery of simple deterministic systems, the behaviour of which is intrinsically stochastic, has led to a new approach to the problem of turbulence. This paper is devoted to the discussion of some physical mechanisms for the appearance of chaos in simple dissipative systems. The authors show that stochastic behaviour is the result of the self-disorganization of such systems. Examples from electronics, chemistry, theory of nonlinear oscillations and waves are given as illustrations to the theoretical findings.     

Lorentz-covariant dissipative lagrangian systems

The concept of dissipative hamiltonian system is converted to Lorentz-covariant form, with evolution generated jointly by two scalar functionals, the lagrangian action and the global entropy. A bracket formulation yields the local covariant laws of energy-momentum conservation and of entropy production. The formalism is illustrated by a derivation of the covariant Landau kinetic equation.     

Energy propagation for dispersive waves in dissipative media

University of Bologna, Italy. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 650–664, July, 1993.