Skip to main content
SpringerLink
Log in

Nonstandard Lagrangian Singularities and Asymptotic Eigenfunctions of the Degenerating Operator \( - {d \over {dx}}D\left( x \right){d \over {dx}}\)

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We express the asymptotic eigenfunctions of the operator \( - {d \over {dx}}D\left( x \right){d \over {dx}}\) that degenerates at the endpoints of an interval in terms of the modified Maslov canonical operator introduced in our previous studies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Yu. Anikin, S. Yu. Dobrokhotov, and V. E. Nazaikinskii, “Simple asymptotics for a generalized wave equation with degenerating velocity and their applications in the linear long wave run-up problem,” Mat. Notes 104(4), 471–488 (2018) [transl. from Mat. Zametki 104 (4), 483–504 (2018)].

    Article  MathSciNet  MATH  Google Scholar 

  2. V. I. Arnol’d, “Characteristic class entering in quantization conditions,” Funct. Anal. Appl. 1(1), 1–13 (1967) [transl. from Funkts. Anal. Prilozh. 1 (1), 1–14 (1967)].

    Article  MATH  Google Scholar 

  3. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989). Engl. transl. of the 1st ed.: V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), Grad. Texts Math. 60.

    Book  Google Scholar 

  4. V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: The Classification of Critical Points, Caustics and Wave Fronts (Nauka, Moscow, 1982). Engl. transl.: V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1: The Classification of Critical Points, Caustics and Wave Fronts (Birkhäuser, Boston, 1985), Monogr. Math. 82.

    MATH  Google Scholar 

  5. M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningr. Gos. Univ., Leningrad, 1980; Kluwer, Dordrecht, 1987).

    Google Scholar 

  6. G. F. Carrier and H. P. Greenspan, “Water waves of finite amplitude on a sloping beach,” J. Fluid Mech. 4(1), 97–109 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Yu. Dobrokhotov and V. E. Nazaikinskii, “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation,” Math. Notes 100(5), 695–713 (2016) [transl. from Mat. Zametki 100 (5), 710–731 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Yu. Dobrokhotov and V. E. Nazaikinskii, “On the asymptotics of a Bessel-type integral having applications in wave run-up theory,” Math. Notes 102(6), 756–762 (2017) [transl. from Mat. Zametki 102 (6), 828–835 (2017)].

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I,” Russ. J. Math. Phys. 17(4), 434–447 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data,” St. Petersburg Math. J. 22(6), 895–911 (2011) [transl. from Algebra Anal. 22 (6), 67–90 (2010)].

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data,” Russ. J. Math. Phys. 20(4), 389–401 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Yu. Dobrokhotov, V. E. Nazaikinskii, and A. A. Tolchennikov, “Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach,” Math. Notes 101(5), 802–814 (2017) [transl. from Mat. Zametki 101 (5), 700–715 (2017)].

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms. Based, in Part, on Notes Left by Harry Bateman (McGraw-Hill, New York, 1954), Vol. 2, Bateman Manuscript Project, Calif. Inst. Technol.

    MATH  Google Scholar 

  14. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (Nauka, Moscow, 1983). Engl. transl.: M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Springer, Berlin, 1993).

    MATH  Google Scholar 

  15. V. A. Fock, “On the canonical transformation in classical and quantum mechanics,” Vestn. Leningr. Univ., Fiz., Khim., No. 16, 67–70 (1959). Rev. Engl. transl. in Acta Phys. Acad. Sci. Hung. 27, 219–224 (1969).

    Google Scholar 

  16. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989). Engl. transl.: A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Am. Math. Soc., Providence, RI, 1992), Transl. Math. Monogr. 102.

    MATH  Google Scholar 

  17. V. P. Maslov, Perturbation Theory and Asymptotic Methods (Mosk. Gos. Univ., Moscow, 1965) [in Russian].

    Google Scholar 

  18. V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation for Equations of Quantum Mechanics (Nauka, Moscow, 1976). Engl. transl.: V. P. Maslov and M. V. Fedoryuk, Semi-classical Approximation in Quantum Mechanics (D. Reidel, Dordrecht, 1981), Math. Phys. Appl. Math. 7.

    MATH  Google Scholar 

  19. A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, Lagrangian Manifolds and the Canonical Operator Method (Nauka, Moscow, 1978). Engl. transl.: A. S. Mishchenko, V. E. Shatalov, and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator (Springer, Berlin, 1990), Springer Ser. Sov. Math.

    Google Scholar 

  20. V. E. Nazaikinskii, “Phase space geometry for a wave equation degenerating on the boundary of the domain,” Math. Notes 92(1), 144–148 (2012) [transl. from Mat. Zametki 92 (1), 153–156 (2012)].

    Article  MathSciNet  MATH  Google Scholar 

  21. V. E. Nazaikinskii, “The Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary,” Math. Notes 96(2), 248–260 (2014) [transl. from Mat. Zametki 96 (2), 261–276 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  22. O. A. Oleinik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form (VINITI, Moscow, 1971), Itogi Nauki, Ser. Mat., Mat. Anal. 1969. Engl. transl.: O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form (Plenum, New York, 1973).

    MATH  Google Scholar 

  23. E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (Inst. Prikl. Fiz., Nizhni Novgorod, 1996) [in Russian].

    Google Scholar 

  24. T. Ratiu, T. A. Filatova, and A. I. Shafarevich, “Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrödinger operator with delta-potential on a surface of revolution,” Dokl. Math. 86(2), 694–696 (2012) [transl. from Dokl. Akad. Nauk 446 (6), 618–620 (2012)].

    Article  MathSciNet  MATH  Google Scholar 

  25. T. S. Ratiu, A. A. Suleimanova, and A. I. Shafarevich, “Spectral series of the Schröodinger operator with delta-potential on a three-dimensional spherically symmetric manifold,” Russ. J. Math. Phys. 20(3), 326–335 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Yu. Slavyanov, Asymptotics of Solutions of the One-Dimensional Schrödinger Equation (Leningrad. Gos. Univ., Leningrad, 1990). Engl. transl.: S. Yu. Slavyanov, Asymptotic Solutions of the One-Dimensional Schrödinger Equation (Am. Math. Soc., Providence, RI, 1996), Transl. Math. Monogr. 151.

    MATH  Google Scholar 

  27. S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, 1986).

    Google Scholar 

  28. J. J. Stoker, Water Waves: The Mathematical Theory with Applications (Interscience, New York, 1957).

    MATH  Google Scholar 

  29. C. E. Synolakis, “On the roots of f(z) = J 0(z) − iJ 1(z),” Q. Appl. Math. 46(1), 105–107 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. S. Vladimirov, Equations of Mathematical Physics, 4th ed. (Nauka, Moscow, 1981; Mir, Moscow, 1984).

    Google Scholar 

  31. T. Vukašinac and P. Zhevandrov, “Geometric asymptotics for a degenerate hyperbolic equation,” Russ. J. Math. Phys. 9(3), 371–381 (2002).

    MathSciNet  MATH  Google Scholar 

  32. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1944).

    MATH  Google Scholar 

  33. P. Zhevandrov, “Edge waves on a gently sloping beach: Uniform asymptotics,” J. Fluid Mech. 233, 483–493 (1991).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We are grateful to A. I. Shafarevich for useful discussions.

Funding

This work is supported by the Russian Science Foundation under grant 16-11-10282.

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia

    S. Yu. Dobrokhotov & V. E. Nazaikinskii

  2. Moscow Institute of Physics and Technology, State University, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701, Russia

    S. Yu. Dobrokhotov & V. E. Nazaikinskii

Authors
  1. S. Yu. Dobrokhotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. E. Nazaikinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Yu. Dobrokhotov or V. E. Nazaikinskii.

Additional information

In memory of Vasilii Sergeevich Vladimirov on the occasion of his 95-th birthday

Russian Text © The Author(s), 2019, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 306, pp. 83–99.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dobrokhotov, S.Y., Nazaikinskii, V.E. Nonstandard Lagrangian Singularities and Asymptotic Eigenfunctions of the Degenerating Operator \( - {d \over {dx}}D\left( x \right){d \over {dx}}\). Proc. Steklov Inst. Math. 306, 74–89 (2019). https://doi.org/10.1134/S0081543819050080

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819050080

Search

Navigation