Relativistic Toda Chain with Boundary Interaction at Root of Unity

We apply the Separation of Variables method to obtain eigenvectors of commuting Hamiltonians in the quantum relativistic Toda chain at a root of unity with boundary interaction

Bosons in a Lattice: Exciton-Phonon Condensate in Cu2O

We explore a nonlinear field model to describe the interplay between the ability of excitons to be Bose-condensed and their interaction with other modes of a crystal. We apply our consideration to the long-living para-excitons in Cu2O. Taking into account the exciton-phonon interaction and introducing a coherent phonon part of the moving condensate, we derive the dynamic equations for the exciton-phonon condensate. These equations can support localized solutions, and we discuss the conditions for the moving inhomogeneous condensate to appear in the crystal. We calculate the condensate wave function and energy, and a collective excitation spectrum in the semiclassical approximation; the inside-excitations were found to follow the asymptotic behavior of the macroscopic wave function exactly. The stability conditions of the moving condensate are analyzed by use of Landau arguments, and Landau critical parameters appear in the theory. Finally, we apply our model to describe the recently observed interference and strong nonlinear interaction between two coherent exciton-phonon packets in Cu2O.Comment: 34 pages, LaTeX, four figures (.ps) are incorporated by epsf. Submitted to Phys. Rev.

Holomorphic structures in hydrodynamical models of nearly . . .

We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. In many of these models an elliptic Monge-Ampere equation defines the relationship between a`balanced' velocity field, defined by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge-Ampere operators define an almost-complex structure, and in this paper we show that a natural extension of the so-called geostrophic momentum transformation of semi-geostrophic theory, which has a special importance in theoretical meteorology, defines Kähler and special Kähler structures on phase space. Furthermore, analogues of the`geostrophic momentum coordinates' are shown to be special Lagrangian coordinates under conditions which depend upon the physical approximations under consideration. Certain duality properties of the operators are studied within the framework of the Kähler geometry

1 Kähler Geometry and the Navier-Stokes Equations

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge–Ampère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kähler geometry is described, which is associated with the Monge–Ampère problem. This Kähler structure is then generalised to ‘two-and-a-half dimensional ’ flows, of which Burgers ’ vortex is one example. In three dimensions, we show how a generalized Calabi–Yau structure emerges in a special case. 1 Equations for an Incompressible Fluid Flow visualization methods, allied to large-scale computations of the three-dimensional incompressible Navier-Stokes equations, vividly illustrate the fact that vorticity has a tendency to accumulate on ‘thin sets ’ whose morphology is characterized by quasi onedimensional tubes or filaments and quasi two-dimensional sheets. This description is in itself approximate as these thin structures undergo dramatic morphological changes in time and space. The topology is highly complicated; sheets tend to roll-up into tubelike structures, while tubes tangle and knot like spaghetti boiling in a pan (Vincent & Meneguzzi 1994). Moreover, vortex tubes usually have short lifetimes, vanishing at one place and reforming at another. The behaviour of Navier-Stokes flows diverge in behaviour from Euler flows once viscosity has taken effect in reconnection processes. Nevertheless, the creation and early/intermediate evolution of their vortical sets appear to be similar. No adequate mathematical theory has been forthcoming explaining why thin sets tend to be favoured. The purpose of this paper is to investigate this enduring question in the light of the recent advances made in in the geometry of Kähler and other complex manifolds. While many difficult questions remain to be solved and explored, we believe that sufficient evidence exists that suggests that three-dimensional turbulent vortical dynamics may be governed by geometric principles. The incompressible Navier-Stokes equations, in two or three dimensions, are ∂u ∂t + u · ∇u

A geometric interpretation of coherent structures in Navier-Stokes Flows

The pressure in the incompressible three-dimensional Navier-Stokes and Euler equations is governed by a Poisson equation: this equation is studied using the geometry of three forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ 3 W ∗ where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ 3 W ∗ to the sign of the Laplacian of the pressure — and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, ∆p, satisfies ∆p> 0, the three-form associated with the Poisson equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When ∆p < 0 a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence. 1 Equations for an Incompressible Fluid It is rare in fluid dynamics for highly technical abstract geometrical criteria to have a direct correspondence with experimental observations. In a seminal paper, Douady et al

Quaternions and particle dynamics in the Euler fluid equations

Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation similarly considered