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## Волны перепада в расширенном нелинейном уравнении Шредингера при учете индуцированного рассеяния и нелинейной дисперсии

The stationary waves with nonlinear phase modulation in an extended nonlinear Schrödinger equation with nonlinear dispersion and stimulated Raman–scattering terms are considered. New class of a kink–waves are found. This waves exists as the result of balance of the stimulated Raman–scattering and nonlinear dispersion.

The stationary waves with nonlinear phase modulation in the frame of the extended nonlinear Schrodinger equation with taking into account both the nonlinear dispersion and stimulated Raman-scattering terms are considered. Two classes of a kink–waves are found: one class exists as the result of balance of the stimulated Raman--scattering and nonlinear dispersion, other class – as the result of balance of the stimulated Raman-scattering and second-order linear dispersion. Is show that kink-waves with pedestal exist only in present of the nonlinear dispersion.

Dynamics of solitons in the frame of the extended nonlinear Schr¨odinger equation (NSE) taking into account stimulated Raman scattering (SRS) and inhomogeneous second-order dispersion (SOD) is considered. Compensation of soliton Raman self-wave number downshift in media with increasing second-order linear dispersion is shown. Quasi-soliton solution with small wave number spectrum variation, amplitude and extension are found analytically in adiabatic approximation and numerically. The soliton is considered as the equilibrium of SRS and increasing SOD. For dominate SRS soliton wave number spectrum tends to long wave region. For dominate increasing SOD soliton wave number spectrum tends to shortwave region.

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.

Dynamics of short solitons envelope in the frame of the third-order nonlinear Schrodinger equations taking into account stimulated Raman-scattering and inhomogeneous second– and third-order linear dispersion, nonlinear dispersion and cubic nonlinearity is considered. Compensation of the stimulated Raman-scattering effect by the increasing of the second-order linear dispersion is shown. In adiabatic approximation stable soliton’s propagation regime is found. Third-order linear dispersion and nonlinear dispersion inhomogeneity effect to stimulated Raman-scattering compensation is analyzed.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.