### Article

## Shedding and interaction of solitons in weakly disordered optical fibers

Dynamics of Langmuir solitons is considered in plasmas with spatially inhomogeneous electron temperature. An underlying Zakharov-type system of two unidirectional equations for the Langmuir and ion-sound fields is reduced to an inhomogeneous nonlinear Schrödinger equation (NLSE) with spatial variation of the second-order dispersion (SOD) and self-phase modulation (SPM) coefficients, induced by the spatially inhomogeneous profile of electron temperature. Analytical trajectories of the motion of a soliton in the plasma with an electron-temperature hole, barrier, or cavity between two barriers are found, using the method of integral moments. The possibility of the soliton to pass a high-temperature barrier is shown too. Analytical results are well corroborated by numerical simulations.

Dynamics of Langmuir solitons is considered in the framework of the extended nonlinear Schrödinger equation (NLSE), including a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, caused by stimulated scattering on damping ion-sound waves. Also included are spatially decreasing second-order dispersion (SOD) and increasing self-phase modulation (SPM), caused by spatial decreasing electron temperature of plasma. It is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, may be compensated by an upshift provided by the decreasing SOD and increasing SPM coefficients. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well.

Propagation of the short vector envelope solitons in a inhomogeneous medium with linear potential in coupled third–order nonlinear Shrodinger equations frame is considered. Explicit vector soliton solution is obtained. The explicit solution for the solitons trajectories is studied. In particular cases this solitons solution can be reduced as to the short scalar soliton solution on linear inhomogeneity profile, as to well – known Chen soliton solution.

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.