S. Yu. Dobrokhotov

Finite-zone, almost-periodic solutions in WKB approximations

It is shown that the recently discovered finite-zone, almost-periodic solutions may, on the one hand, serve as the foundation for the development of the multiphase WKB method in nonlinear equations (the method of Whitham) and, on the other hand, serve to define Lagrangian manifolds with complex germs which can be (second) quantized in the quasiclassical approximation.

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients.

Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I

where x is the spatial coordinate, t is time, D = D(x) is the basin depth (which we assume to be a smooth function of x), and g is the acceleration due to gravity. Assume that the point x = 0 corresponds to the shoreline. More precisely, D(x) > 0 for x > 0, D(0) = 0, D(x) = γx + O(x) for small x, where γ = 0, and Eqs. (0.1) are regarded in the domain x > 0 only. Let us pose the following Cauchy problem with initial data localized in a neighborhood of some point x = a, a > 0, for Eq. (0.1):

Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008).

Semiclassical maslov asymptotics with complex phases. I. General approach

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk<n (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of “intermediate” quantum numbers. The construction includes, in particular, new quantization conditions of Bohr—Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.

Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. I. The Cauchy problem

We present a systematic study of the construction of localized asymptotic solutions of the one-dimensional wave equation with variable velocity. In part I, we discuss the solution of the Cauchy problem with localized initial data and zero right-hand side in detail. Our aim is to give a description of various representations of the solution, their geometric interpretation, computer visualization, and illustration of various general approaches (such as the WKB and Whitham methods) concerning asymptotic expansions. We discuss ideas that can be used in more complicated cases (and will be considered in subsequent parts of this paper) such as inhomogeneous wave equations, the linear surge problem, the small dispersion case, etc. and can eventually be generalized to the 2-(and n-) dimensional cases.

Representations of rapidly decaying functions by the Maslov canonical operator

The function V (x/μ) rapidly decreases as μ → +0 in the exterior of a small neighborhood of the point x = 0. In order to study asymptotic solutions of the Cauchy problem with initial condition of the form (2) for partial differential equations, the expression on the right-hand side of Eq. (2) can be rewritten as the Maslov canonical operator [1] K Λδ on the LagrangianmanifoldΛδ = {p = α, x = 0, α ∈ R n} (a plane), acting on the function V (α) defined on Λδ:

Splitting amplitudes of the lowest energy levels of the Schrödinger operator with double-well potential

An analytic method is proposed for calculating the asymptotic splitting of the lowest energy levels of the Schrödinger operator with a symmetric double-well potential. The potential describing a chain of pairwise interacting quantum particles in a common double-well potential is considered as an example. The limit of a large number of particles is investigated.

New formulas for Maslov’s canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics

We suggest a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds. We present the results in the two-dimensional case and illustrate them with examples.

“Momentum” Tunneling between Tori and the Splitting of Eigenvalues of the Laplace–Beltrami Operator on Liouville Surfaces

Tunneling in the spectral problem for the Laplace–Beltrami operator on a torus with Liouville metric is considered. The formula for exponential splitting of eigenvalues is obtained. The splitting can be expressed in terms of gomology and cohomology classes of a complex Lagrangian manifold. These classes are constructed with the help of the phase flow of a certain gradient-Hamiltonian vector field on the manifold.