Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new.Comment: 15 page

Construction of coupled Harry Dym hierarchy and its solutions from St\"ackel systems

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical St\"ackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding St\"ackel systems.Comment: 16 page

Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page

Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation

We consider singular solutions of a system of two cross-coupled Camassa-Holm (CCCH) equations. This CCCH system admits peakon solutions, but it is not in the two-component CH integrable hierarchy. The system is a pair of coupled Hamiltonian partial differential equations for two types of solutions on the real line, each of which separately possesses exp(-|x|) peakon solutions with a discontinuity in the first derivative at the peak. However, there are no self-interactions, so each of the two types of peakon solutions moves only under the induced velocity of the other type. We analyse the `waltzing' solution behaviour of the cases with a single bound peakon pair (a peakon couple), as well as the over-taking collisions of peakon couples and the antisymmetric case of the head-on collision of a peakon couple and a peakon anti-couple. We then present numerical solutions of these collisions, which are inelastic because the waltzing peakon couples each possess an internal degree of freedom corresponding to their `tempo' -- that is, the period at which the two peakons of opposite type in the couple cycle around each other in phase space. Finally, we discuss compacton couple solutions of the cross-coupled Euler-Poincar\'e (CCEP) equations and illustrate the same types of collisions as for peakon couples, with triangular and parabolic compacton couples. We finish with a number of outstanding questions and challenges remaining for understanding couple dynamics of the CCCH and CCEP equations

Confluence of hypergeometric functions and integrable hydrodynamic type systems

It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details.Comment: 22 pages, PMNP 2015, added some comments and reference

From St\"{a}ckel systems to integrable hierarchies of PDE's: Benenti class of separation relations

We propose a general scheme of constructing of soliton hierarchies from finite dimensional St\"{a}ckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in momenta integrals of motion.Comment: 24 page

Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate

We provide a classification of the possible flow of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferro-magnet. We present the full set of one-phase periodic solutions. The corresponding Whitham modulation equations are obtained together with formulas connecting their solutions with the Riemann invariants of the modulation equations. The problem is not genuinely nonlinear, and this results in a non-single-valued mapping of the solutions of the Whitham equations with physical wave patterns as well as to the appearance of new elements --- contact dispersive shock waves --- that are absent in more standard, genuinely nonlinear situations. Our analytic results are confirmed by numerical simulations

String Field Theory Vertices, Integrability and Boundary States

We study Neumann coefficients of the various vertices in the Witten's open string field theory (SFT). We show that they are not independent, but satisfy an infinite set of algebraic relations. These relations are identified as so-called Hirota identities. Therefore, Neumann coefficients are equal to the second derivatives of tau-function of dispersionless Toda Lattice hierarchy (this tau-function is just the partition sum of normal matrix model). As a result, certain two-vertices of SFT are identified with the boundary states, corresponding to boundary conditions on an arbitrary curve. Such two-vertices can be obtained by the contraction of special surface states with Witten's three vertex. We analyze a class of SFT surface states,which give rise to boundary states under this procedure. We conjecture that these special states can be considered as describing D-branes and other non-perturbative objects as "solitons" in SFT. We consider some explicit examples, one of them is a surface states corresponding to orientifold.Comment: 28pages plus appendices, acknowledgments adde