### Article

## Conditions for modulation instability in higher order Korteweg-de Vries equations

The universal mechanism of modulation instability (MI) has been discovered first for the Nonlinear

Schr¨odinger equation (NLS) and is well studied in the frame of the higher order NLS equations.

Recent studies demonstrated by pure existence theorems that also the higher order Korteweg-de

Vries (KdV) equations might possess the MI. In this Letter we present explicit form of the conditions

for the MI to appear in the KdV-family of equations of the general form ut + sup ux + uxxx =0

where s = ±1 with p > 0 being an arbitrary integer.

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations ut + sup ux + uxxx (here s = 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.

The issue of a recurrence of the modulationally unstable water wave trains within the framework of the fully nonlinear potential Euler equations is addressed. It is examined, in particular, if a modulation which appears from nowhere (i.e., is infinitesimal initially) and generates a rogue wave which then disappears with no trace. If so, this wave solution would be a breather solution of the primitive hydrodynamic equations. It is shown with the help of the fully nonlinear numerical simulation that when a rogue wave occurs from a uniform Stokes wave train, it excites other waves which have different lengths, what prevents the complete recurrence and, eventually, results in a quasi-periodic breathing of the wave envelope. Meanwhile the discovered effects are rather small in magnitude, and the period of the modulation breathing may be thousands of the dominant wave periods. Thus, the obtained solution may be called a quasi-breather of the Euler equations.

The development of modulational instability involving dust acoustic perturbations in dusty ionospheric plasma and in dusty plasmas of meteor tails in Earth’s ionosphere was considered. The effect of collisions of electrons, ions, and dust grains with neutrals at different altitudes was estimated. It is shown that, in this case, the influence of collisions of electrons and ions with neutrals is usually less significant than the influence of collisions between dust grains and neutrals. It is demonstrated that the effect of the modulational instability on the propagation of electromagnetic waves in the dusty ionospheric plasma is the most significant at heights of 100-120 km. The values of the wave vectors of the electromagnetic pump wave at which inelastic collisions with neutrals are important for the development of modulational interaction are calculated. The modulational interaction in the dusty ionosphere is important for the explanation of different phenomena such as ground-based observations of low-frequency ionospheric radio noises with frequencies below 60 Hz. The absence of observations of low-frequency ionospheric radio noise during such phenomena as noctilucent clouds and polar mesosphere summer echoes caused by the presence of dusty plasmas at heights of 80-95 km is explained by suppression of the development of the modulational instability at these heights. The role of inelastic collisions with neutrals in meteor tails is also discussed. It is shown that for typical parameters of dusty plasmas of meteor tails such collisions do not influence the development of the modulation instability in meteor tails.

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.