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## Theory of Random Fibre Laser

We develop the theory explaining the role of nonlinearity in generation radiation in a fiber laser that is pumped by external light. The pumping energy is converted into the generating signal due to the Raman scattering supplying an effective gain for the signal. The signal is generated with frequencies near the one corresponding to the maximum value of the gain. Generation conditions and spectral properties of the generated signal are examined. We focus mainly on the case of a random laser where reflection of the signal occurs on impurities of the fiber. From the theoretical standpoint, kinetics of a wave system close to an integrable one are investigated. We demonstrate that in this case, the perturbation expansion in the kinetic equation has to use the closeness to the integrable case.

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example—random fibre laser—we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

We present a proof of the monotonic entropy growth for a nonlinear discrete-time model of a random market. This model, based on binary collisions, also may be viewed as a particular case of Ulam’s redistribution of energy problem. We represent each step of this dynamics as a combination of two processes. The first one is a linear energy-conserving evolution of the two-particle distribution, for which the entropy growth can be easily verified. The original nonlinear process is actually a result of a specific “coarse graining” of this linear evolution, when after the collision one variable is integrated away. This coarse graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth. The combination of these two factors produces the required result which is obtained only by means of information theory inequalities.

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing ‘entropy’ functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist.

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.