Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

Multidimensional Dirichlet series in the problem of the asymptotics of spectral series of nonlinear elliptic operators

Series of asymptotic solutions of nonlinear elliptic boundary-value problems in compact domains with a spectral parameter contained in the boundary condition are constructed, and the connection of these solutions with the trajectories of classical Hamiltonian systems defined on the boundary of the domains considered is established. The asymptotic solutions indicated are expressed in terms of multidimensional Dirichlet series, and a superposition law is established for them which, as it turns out, does not depend either on the number of independent variables in the original problem or on the form of the nonlinearity.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 23, pp. 137–222, 1983.     

Asymptotic Solution of the Helmholtz Equation in a Three-Dimensional Layer of Variable Thickness with a Localized Right-Hand Side

Computational Mathematics and Mathematical Physics - The asymptotics of the solution to the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side in...     

Quantum dynamics in a thin film. II. Stationary states

The spectrum of quantum waveguides simulating thin toroidal tubes and thin spherical surfaces is investigated. Asymptotic formulas are obtained and a geometric classification using the so-called Reeb graphs is carried out.     

Integrability of Truncated Hugoniot–Maslov Chains for Trajectories of Mesoscale Vortices on Shallow Water

The problem of trajectories of large (mesoscale) shallow-water vortices manifests integrability properties. The Maslov hypothesis states that such vortices can be generated using solutions with weak pointlike singularities of the type of the square root of a quadratic form; such square-root singular solutions may describe the propagation of mesoscale vortices in the atmosphere (typhoons and cyclones). Such solutions are necessarily described by infinite systems of ordinary differential equations (chains) in the Taylor coefficients of solutions in the vicinities of singularities. A proper truncation of the vortex chain for a shallow-water system is a system of 17 nonlinear equations. This system becomes the Hill equation when the Coriolis force is constant and almost becomes the physical pendulum equations when the Coriolis force depends on the latitude. In a rough approximation, we can then explicitly describe possible trajectories of mesoscale vortices, which are analogous to oscillations of a rotating solid body swinging on an elastic thread.     

Cauchy—Riemann conditions and point singularities of solutions to linearized shallow-water equations

Singular solutions with algebraic “square-root” type singularity of two-dimensional equations of shallow-water theory are propagated along the trajectories of the external velocity field on which the field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow is proportional to an orthogonal operator on such a trajectory. It turns out that, in the linear approximation, this fact is closely related to the effect of “blurring” of solutions of hydrodynamical equations; namely, a singular solution of the Cauchy problem for the linearized shallow-water equations preserves its shape exactly (i.e., is not blurred) if and only if the Cauchy-Riemann conditions are satisfied on the trajectory (of the external field) along which the perturbation is propagated.