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

@article{Dobrokhotov2013NewFF,
  title={New formulas for Maslov’s canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics},
  author={S. Yu. Dobrokhotov and George N. Makrakis and Vladimir Evgen'evich Nazaikinskii and T. Ya. Tudorovskii},
  journal={Theoretical and Mathematical Physics},
  year={2013},
  volume={177},
  pages={1579-1605}
}
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. 
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44 Citations
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44 Citations

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References

SHOWING 1-10 OF 31 REFERENCES

Fourier integrals and a new representation of Maslov's canonical operator near caustics

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.

Lagrangian Manifolds and the Maslov Operator

This book presents the topological and analytical foundations of the theory of Maslov's canonical operator for finding asymptotic solutions of a wide class of pseudodifferential equations. The…

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…

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…

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…

The semi-classical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory

If H (p,x) is completely integrable, or has merely has invariant diohantine torus $\Lambda$ in energy surface E, then a family of quasi-modes for H(x,hD_x) at the corresponding energy $E$ is constructed.

The semiclassical Maupertuis-Jacobi correspondence and applications to linear water wave theory

T d = R/2πZ, J = (J1, . . . , Jd) ∈ M . Here M is a domain in Rd, and φ = (φ1, . . . , φd), φj ∈ S1, are angle coordinates on Λ(J) defining the half-density |dφ1 ∧ · · · ∧ dφd|. Let γj , j = 1, . . .…

Evaluation of the Field near a Cusp of a Caustic

The field in the neighbourhood of a cusp of a caustic can be expressed in terms of a function of two variables, P(X,Y), known as Pearcey's integral or the Pearcey function. In this paper we develop…

Singularities of Caustics and Wave Fronts

1 Symplectic geometry.- 1.1 Symplectic manifolds.- 1.2 Submanifolds of symplectic manifolds.- 1.3 Lagrangian manifolds, fibrations, mappings, and singularities.- 2 Applications of the theory of…

Mathematical Methods of Classical Mechanics

Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid…