Maslov Canonical Operator Research Articles

Homogenization in the problem of long water waves over a bottom site with fast oscillations

The system of equations of gravity surface waves is considered in the case where the basin’s bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom’s oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors’ works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.

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 the caustics using a special class of coordinate systems ("eikonal coordinates") on Lagrangian manifolds. The specific features of the two-dimensional case are considered. The general case is treated in arXiv:1307.2292 [math-ph].

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.

Asymptotic solutions of the Cauchy problem with localized initial conditions for linearized two-dimensional Boussinesq-type equations with variable coefficients

In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.

Asymptotics of waves on the shallow water generated by spatially-localized sources and trapped by underwater ridges

We consider a set of examples describing the behavior of linear nonstationary waves on the shallow water generated by time-instantaneous spatially-localized sources and propagating over underwater banks and ridges. To this end, we use an asymptotic approach developed by the authors of the present paper and their collaborators. The approach uses the generalized Maslov canonical operator. We establish the occurrence and dynamics of wave trains travelling over underwater ridges (which are nonstationary waves trapped by ridges) with wave-front singularities, moving focal (turning) points, cascades of space-time caustics, etc.

Asymptotic solutions of the linear shallow-water equations with localized initial data

A new asymptotic method for solving Cauchy problems with localized initial data (perturbation) for the linearized shallow-water equation is suggested. The solution is decomposed into two parts: waves and vortices. Metamorphosis of the profile takes place for the wave part: it is localized in the neighborhood of the initial point and later on it is localized in the neighborhood of the front (1-D curve on the plane). Initially, the front is a smooth curve, but as the time increases turning (focal) and self-intersection points might appear. The vortical part is localized in the neighborhood of a point moving along the trajectory of the basic velocity vector field. Both parts are described by very simple formulae taking into account the form of the initial perturbation. These formulae are expressed by means of elementary functions for some special choice of the initial data. Due to the profile-metamorphosis phenomenon the method leading to the final formulae is not elementary. It makes use of semiclassical approximations and ray expansions. It is based on the generalization of the construction known as the Maslov canonical operator and boundary-layer expansions.

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).

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:

Description of tsunami propagation based on the Maslov canonical operator

Description of tsunami propagation based on the Maslov canonical operator

Asymptotics in the Number of Elements of a Statistical Ensemble for Maslov Equations in Quantum Thermodynamics

In this paper, we consider a simple model problem which provides us with an example showing that not only periodic trajectories of the classical Hamiltonian system are significant for solutions of asymptotic problems, but the trajectories filling invariant tori also should be taken into consideration. For this model problem, we find the invariant tori and prove that the corresponding Maslov canonical operator gives an asymptotic solution of the problem.

Blurring of the step for the Schroedinger equation

satisfying, for t = 0, the initial condition ~/(x, 0) = 0(-x)~bo(X), where 0 is the Heaviside function, and ~bo(X) = A(x)e is0(x)/h. Here V, S o, and A are infinitely smooth functions, V belonging to the Schwarz space, V and S O real-valued functions, and A a compactly supported function: supp A C (- a, a), a > 0 (at the end of the article we also consider the case of a semi-infinite step), and h > 0 is a small parameter. In the article we will be investigating what form is assumed by q~(x, t) when t = t* > 0. If we take fro(X) as the initial condition, the asymptote of the solution is well known and may be constructed using the Maslov canonical operator [3]. In the present situation this method cannot be applied directly, since the initial condition is represented by a discontinuous function. Suppose that ~b(x, t, k) is the solution of the initial-value problem for the Schroedinger equation with initial condition if(x, 0, k) = ~o(x)e ikx/h. To solve the problem we represent the function ~(x, 0) in the form of a superposition of waves ~(x, 0, k). By the linearity of the Schroedinger equation ,I~(x, t*) may be represented in the form of the same superposition of waves ~b(x, t*, k). Therefore, once we know the asymptote of the function ~b(x, T*,k) as h --, +0, we will be able to determine the asymptote of ff'(x, t*) by computing a corresponding rapidly oscillating integral. Moreover, we fix some k o > 0 and limit ourself to values k E [-ko, ko]. Finally, we will have to estimate the error of the inexact computation of ~b(x, t*, k) and the error due to ignoring k ~ [-k 0, ko]. Let us first recall the technique for constructing the asymptote of ~b(x, t*, k) using the canonical operator. We consider a Hamiltonian system with Hamiltonian H(q, p) = p2/2 + V(q); the corresponding Lagrangian K(q, r = dl2/2-V(q). Suppose that over time t* a point in the phase space (xo, P0), where Po = So'(Xo) + k, turns into the point (X, P)(x o, k), and J(x o, k) = Xxo(X o, k). We denote by I the closed interval with endpoints X(_a, 0). Let us start by limiting ourself to the case in which J(x o, 0) ~ 0 for all x0 E I-a, a].Then there exist k0 > 0 and some neighborhood U of the closed interval [-a, a] such that for all

Numerical realization of the maslov canonical operator method for problems of shortwave radio propagation in the ionosphere of the earth

Numerical realization of the maslov canonical operator method for problems of shortwave radio propagation in the ionosphere of the earth

Maslov canonical operator in problems of numerical simulation of diffraction and propagation of waves in inhomogeneous media

Article Maslov canonical operator in problems of numerical simulation of diffraction and propagation of waves in inhomogeneous media was published on January 1, 1990 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 5, issue 6).

THE LIMITS OF APPLICABILITY OF THE CANONICAL OPERATOR METHOD FOR NONSTRICTLY HYPERBOLIC EQUATIONS WITH NONSMOOTH CHARACTERISTICS

The Cauchy problem with rapidly oscillating initial conditions is investigated for the class of nonstrictly hyperbolic equations with nonsmooth characteristics. The domain of applicability of the Maslov canonical operator method is determined. Bibliography: 12 titles.

Numerical comparison of two asymptotic methods for solving wave diffraction problems in smooth inhomogeneous media

UDC 534.1;621.371 The methods of Maslov's canonical operator (MCO) and Gaussian beam summation (GBS), which are used to describe the diffraction structures of wave fields in complex inhomogeneous media, are numerically compared. The analysis is performed on a sample calculation of the amplitude of an acoustic wave field propagating in an inhomogeneous sound channel. It is shown that, on the axis of the sound channel, the results of calculation by the MCO and GBS methods agree within graphical accuracy. However, far from the axis, application of GBS method encounters definite difficulties. On the basis of the results of the comparison, it is concluded that the MCO method better reveals the mechanism of the diffraction field formation in the focal region. In the short wave approximation of wave diffraction and propagation problems in complex inhomogenous media, the task often arises to describe the field in the focal region, where the traditional ray (geometric) approach is incorrect. As the practice of such calculations shows, focal regions of dissimilar form are a similar phenomenon. They really exist in the ionosphere and tropospheric channels of radio communication, in underwater acoustic channels, in fiber optics, etc. This makes the task of the detailed description of the field in them very topical. At the present time, the field in the focal region is most fully described in the framework of methods for plotting asymptotic solutions, using rapidly oscillating integrals, and by the methods of parabolic equations (PE). Among the first group of such methods, we shall examine the means of plotting solutions based on Maslov's canonical operstor (MCO) [i, 2] including the topological classification of the focal regions and the local description of the field in them [3-7], since the practice of solving similar problems recommends just this method to us. We will compare our numerical solution with a solution plotted by a new and promising method -- the method of Gaussian beam summation (GBS) [8-11]. The present work compares the methods of MCO and GBS with respect to the final result of their application in a concrete problem. This will allow a more complete answer to the question of how effective the examined methods are in the applied problems, since the theoretical evaluations of their effectiveness in the literature often have a rather abstract character and are uncorrelated. We made use of the results of [8, 9] as a completed and fully documented GBS solution of the problem, which studied the propagation of monochromatic sound waves, excited by a point source located on the axis of a symmetric acoustic wave guide in a plane-laminar medium. i. Statement of the Problem. Asymptotic Solution in the MCO Form. We shall examine Helmholtz's equation, which we shall write in the symbology of [8, 9]:

ON FOURIER INTEGRAL OPERATORS

On the basis of the stationary phase method for oscillatory integrals with complex phase function, the authors prove the coincidence of Fourier integral operators and Maslov's canonical operator. Bibliography: 17 titles.

Asymptotic behavior of the fundamental solution of an elliptic equation with respect to a complex parameter

Maslov's canonical operator method is used for constructing the asymptotic behavior with respect to a complex parameter of the fundamental solution of a secondorder elliptic equation with smooth finite coefficients. The asymptotic form is constructed on the assumption that all trajectories of the corresponding Hamiltonian system depart to infinity. The asymptotic form is used for investigating the analytic properties of the fundamental solution.

On Maslov's canonical operator

On Maslov's canonical operator