A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

Modern nanotechnology allows one to scale down various important devices (sensors, chips, fibers, etc.) and thus opens up new horizons for their applications. The efficiency of most of them is based on fundamental physical phenomena, such as transport of wave excitations and resonances. Short propagation distances make phase-coherent processes of waves important. Often the scattering of waves involves propagation along different paths and, as a consequence, results in interference phenomena, where constructive interference corresponds to resonant enhancement and destructive interference to resonant suppression of the transmission. Recently, a variety of experimental and theoretical work has revealed such patterns in different physical settings. The purpose of this review is to relate resonant scattering to Fano resonances, known from atomic physics. One of the main features of the Fano resonance is its asymmetric line profile. The asymmetry originates from a close coexistence of resonant transmission and resonant reflection and can be reduced to the interaction of a discrete (localized) state with a continuum of propagation modes. The basic concepts of Fano resonances are introduced, their geometrical and/or dynamical origin are explained, and theoretical and experimental studies of light propagation in photonic devices, charge transport through quantum dots, plasmon scattering in Josephson-junction networks, and matter-wave scattering in ultracold atom systems, among others are reviewed.

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Fano resonances in nanoscale structures

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

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Introduction to the Modern Theory of Dynamical Systems

Tables of integrals

Existence of 'breathers', that is, time-periodic, spatially localized solutions, is proved for a broad range of time-reversible or Hamiltonian networks of weakly coupled oscillators. Some of their properties are discussed, some generalizations suggested, and several open questions raised.

https://iopscience.iop.org/article/10.1088/0951-7715/7/6/006/pdf

Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators

Breathers in nonlinear lattices: existence, linear stability and quantization

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

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The twist map, the extended Frenkel-Kontorova model and the devil's staircase

Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi - Pasta - Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate `rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.

https://iopscience.iop.org/article/10.1088/0951-7715/9/6/007/pdf

Serge Aubry

Papers

Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators

Breathers in nonlinear lattices: existence, linear stability and quantization

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

The twist map, the extended Frenkel-Kontorova model and the devil's staircase

Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit