Vol 188, No 3 (2016)

We consider the three-dimensional rdDym equation u_ty = u_xu_xy −u_yu_xx. Using the known Lax representation with a nonremovable parameter and two hierarchies of nonlocal conservation laws associated with it, we describe the algebras of nonlocal symmetries in the corresponding coverings.

We study extensions of N-wave systems with PT symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the P (spatial reflection) and T (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras sl(3,C) and so(5,C)), we discuss the properties of different types of one- and two-soliton solutions. We show that the PT-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.

We construct a new class of integrable hydrodynamic-type systems governing the dynamics of the critical points of confluent Lauricella-type functions defined on finite-dimensional Grassmannian Gr(2, n), i.e., on the set of 2×n matrices of rank two. These confluent functions satisfy certain degenerate Euler–Poisson–Darboux equations. We show that in the general case, a hydrodynamic-type system associated with the confluent Lauricella function is an integrable and nondiagonalizable quasilinear system of a Jordan matrix form. We consider the cases of the Grassmannians Gr(2, 5) for two-component systems and Gr(2, 6) for three-component systems in detail.

We review some of our recent work devoted to the problem of quantization with preservation of Noether symmetries, finding hidden linearity in superintegrable systems, and showing that nonlocal symmetries are in fact local. In particular, we derive the Schrödinger equation for the isochronous Calogero goldfish model using its relation to Darwin equation. We prove the linearity of a classical superintegrable system on a plane of nonconstant curvature. We find the Lie point symmetries that correspond to the nonlocal symmetries (also reinterpreted as λ-symmetries) of the Riccati chain.

For an arbitrary number of species, we derive a Hamiltonian fluid model for strongly magnetized plasmas describing the evolution of the density, velocity, and electromagnetic fluctuations and also of the temperature and heat flux fluctuations associated with motions parallel and perpendicular to the direction of a background magnetic field. We derive the model as a reduction of the infinite hierarchy of equations obtained by taking moments of a Hamiltonian drift-kinetic system with respect to Hermite–Laguerre polynomials in velocity–magnetic-moment coordinates. We show that a closure relation directly coupling the heat flux fluctuations in the directions parallel and perpendicular to the background magnetic field provides a fluid reduction that preserves the Hamiltonian character of the parent drift-kinetic model. We find an alternative set of dynamical variables in terms of which the Poisson bracket of the fluid model takes a structure of a simple direct sum and permits an easy identification of the Casimir invariants. Such invariants in the limit of translational symmetry with respect to the direction of the background magnetic field turn out to be associated with Lagrangian invariants of the fluid model. We show that the coupling between the parallel and perpendicular heat flux evolutions introduced by the closure is necessary for ensuring the existence of a Hamiltonian structure with a Poisson bracket obtained as an extension of a Lie–Poisson bracket.