Why Patterns Appear Spontaneously in Dissipative Systems?

It is proposed that the spatial (and temporal) patterns spontaneously appearing in dissipative systems maximize the energy flow through the pattern forming interface. In other words - the patterns maximize the entropy growth rate in an extended conservative system (consisting of the pattern forming interface and the energy bathes). The proposal is supported by examples of the pattern formation in different systems. No example contradicting the proposal is known.Comment: 7 pages, 1 figur

Linear and Nonlinear Bullets of the Bogoliubov-de Gennes Excitations

We report on the focalization of Bogoliubov–de Gennes excitations of the nonlinear Schrödinger equation in the defocusing regime (Gross-Pitaevskii equation for repulsive Bose-Einstein condensates) with a spatially modulated periodic potential. Exploiting the modification of the dispersion relation induced by the modulation, we demonstrate the existence of localized structures of the Bogoliubov–de Gennes excitations, in both the linear and nonlinear regimes (linear and nonlinear “bullets”). These traveling Bogoliubov–de Gennes bullets, localized both spatially and temporally in the comoving reference frame, are robust and propagate remaining stable, without spreading or filamentation. The phenomena reported in this Letter could be observed in atomic Bose-Einstein condensates in the presence of a spatially periodic potential induced by an optical lattice.Peer ReviewedPostprint (published version

Transverse Patterns in Nonlinear Optical Resonators

The book is devoted to the formation and dynamics of localized structures (vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in nonlinear optical resonators such as lasers, optical parametric oscillators, and photorefractive oscillators. The theoretical analysis is performed by deriving order parameter equations, and also through numerical integration of microscopic models of the systems under investigation. Experimental observations, and possible technological implementations of transverse optical patterns are also discussed. A comparison with patterns found in other nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is given. This article contains the table of contents and the introductory chapter of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of the boo

Anisotropic Subdiffractive Solitons

We study solitons in the two-dimensional defocusing nonlinear Schroedinger equation with the spatio-temporal modulation of the external potential. The spatial modulation is due to a square lattice; the resulting macroscopic diffraction is rotationally symmetric in the long-wavelength limit but becomes anisotropic for shorter wavelengths. Anisotropic solitons -- solitons with the square (x,y)-geometry -- are obtained both in the original nonlinear Schroedinger model and in its averaged amplitude equation