Maslov’s canonical operator, Hörmander’s formula, and localization of the Berry-Balazs solution in the theory of wave beams

The spectral problem − ⟨ ∇ , D ( x ) ∇ ψ ⟩ = λ ψ -\langle \nabla ,D(x)\nabla \psi \rangle = \lambda \psi in a bounded two-dimensional domain Ω \Omega is considered, where D ( x ) D(x) is a smooth function positive inside the domain and zero on the boundary whose gradient is different from zero on the boundary. This problem arises in the study of long waves trapped by the shore and by bottom irregularities. For its asymptotic solutions as λ → ∞ \lambda \rightarrow \infty , explicit formulas are given when D ( x ) D(x) has a special form that guarantees the complete integrability of the Hamiltonian system corresponding to the Hamiltonian H ( x , p ) = D ( x ) p 2 H(x,p)=D(x)p^2 . Since the problem is degenerate, the relevant Liouville tori are not in the standard phase space T ∗ Ω T^*\Omega , but in the “extended” phase space Φ ⊃ T ∗ Ω \mathbf {\Phi }\supset T^*\Omega , while their restrictions to T ∗ Ω T^*\Omega are not compact and “go to infinity” with respect to momenta near the boundary of Ω \Omega . As a result, nonstandard caustics emerge, formed by the boundary or its part, near which asymptotic eigenfunctions are expressed in terms of a Bessel function of composite argument. Standard caustics (within the domain) may also appear, which yield Airy functions in the asymptotics.

show abstract

Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp

Dobrokhotov

Назайкинский

2020

Math Notes

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Maslov’s canonical operator, Hörmander’s formula, and localization of the Berry-Balazs solution in the theory of wave beams

Cited by 12 publications

References 8 publications

New integral representations of the Maslov canonical operator in singular charts

New integral representations of the Maslov canonical operator in singular charts

Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for 2D standing coastal waves

Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp

Contact Info

Product

Resources

About