Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u_t=V(u)u_x, by the generalized hodograph method requires the diagonalizability of the mxm matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -- infinite-component systems of hydrodynamic type for which the infinite matrix V(u) is `sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.Comment: 36 pages, the classification results and proofs are refined. A section on generating functions is adde

Wigner distribution function formalism for superconductors and collisionless dynamics of the superconducting order parameter

A technique to study collisionless dynamics of a homogeneous superconducting system is developed, which is based on Riccati parametrization of Wigner distribution function. The quantum evolution of the superconductiung order parameter, initially deviated from the equilibrium value, is calculated using this technique. The effect of a time-dependent BCS paring interaction on the dynamics of the order parameter is also studied.Comment: 14 pages, 5 figure

D5D5-brane type I superstring background fields in terms of type IIB ones by canonical method and T-duality approach

We consider type IIB superstring theory with embedded $D5$-brane and choose boundary conditions which preserve half of the initial supersymmetry. In the canonical approach that we use, boundary conditions are treated as canonical constraints. The effective theory, obtained from the initial one on the solution of boundary conditions, has the form of the type I superstring theory with embedded $D5$-brane. We obtain the expressions for $D5$-brane background fields of type I theory in terms of the $D5$-brane background fields of type IIB theory. We show that beside known $\Omega$ even fields, they contain squares of $\Omega$ odd ones, where $\Omega$ is world-sheet parity transformation, $\Omega:\sigma\to -\sigma$. We relate result of this paper and the results of [1] using T-dualities along four directions orthogonal to $D5$-brane

Lagrangian and Hamiltonian Formalism on a Quantum Plane

We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino, we construct two different differential calculi on $TQ_{q,p}$. These two differential calculi can in principle give rise to two different particle dynamics, starting from a single Lagrangian. For Hamiltonian mechanics, we define a phase space $T^*Q_{q,p}$ spanned by noncommuting particle coordinates and momenta. The commutation relations for the momenta can be determined only after knowing their functional dependence on coordinates and velocities. Thus these commutation relations, as well as the differential calculus on $T^*Q_{q,p}$, depend on the initial choice of Lagrangian. We obtain the deformed Hamilton's equations of motion and the deformed Poisson brackets, and their definitions also depend on our initial choice of Lagrangian. We illustrate these ideas for two sample Lagrangians. The first system we examine corresponds to that of a nonrelativistic particle in a scalar potential. The other Lagrangian we consider is first order in time derivative

Quadratic invariants of elastic moduli

A quadratic invariant is defined as a quadratic form in the elements of a tensor that remains invariant under a group of coordinate transformations. It is proved that there are 7 quadratic invariants of the 21-element elastic modulus tensor under SO(3) and 35 under SO(2). This answers some open questions raised by Ting (1987) and Ahmad (2002).Comment: 20 page