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Maslov’s Complex Germ in the Cauchy Problem for a Wave Equation with a Jumping Velocity

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Abstract

Using Maslov’s complex germ in the Cauchy problem for a wave equation, we consider the asymptotics of the solution of the Cauchy problem in which the velocity depends irregularly on a small parameter.

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References

  1. V. P. Maslov, Operator Methods, Izdat. MGU, Moscow, 1973.

    MATH  Google Scholar 

  2. V. P. Maslov, The Complex WKB Method for Nonlinear Equations. I, Birkhäuser, Basel, 1994.

    Book  Google Scholar 

  3. V. V. Belov and S. Yu. Dobrokhotov, “Semiclassical Maslov Asymptotics with Complex Phases. I. General Approach”, Theoret. and Math. Phys., 92:2 (1992), 843–868.

    Article  ADS  MathSciNet  Google Scholar 

  4. S. Yu. Dobrokhotov and A. I. Shafarevich, Semiclassical Quantization of Invariant Isotropic Manifolds of Hamiltonian Systems, in: Topological Methods in the Theory of Hamiltonian Systems (eds. A.T. Fomenko, A.B. Bolsinov, A.I. Shafarevich), Faktorial, Moscow, 1998.

    Google Scholar 

  5. A. I. Allilueva and A. I. Shafarevich, “Short-Wave Asymptotic Solutions of the Wave Equation with Localized Perturbations of the Velocity”, Russ. J. Math. Phys., 27:2 (2020), 145–154.

    Article  MathSciNet  Google Scholar 

  6. A. I. Allilueva and A. I. Shafarevich, “Reflection and Refraction of Lagrangian Manifolds Corresponding to Short-Wave Solutions of the Wave Equation with an Abruptly Varying Velocity”, Russ. J. Math. Phys., 28:2 (2021), 137–146.

    Article  MathSciNet  Google Scholar 

  7. S. Yu. Dobrokhotov, “Application of Maslov’s Theory to Two Problems with Operator-Valued Symbol”, Russ. Math. Surv., 39:4 (1984), 125.

    Article  Google Scholar 

  8. V. S. Buslaev, “Adiabatic Perturbation of a Periodic Potential”, Theoret. and Math. Phys., 58:2 (1984), 153–159.

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Yu. Dobrokhotov and A. I. Shafarevich, “Semiclassical Asymptotic Behavior of the Scattering of Wave Packets by the Rapidly Changing Potential Given by the Equation \(-2\vert \nabla\Phi\vert ^2/{\rm ch}^2(\Phi (x)/h)+V_0(x)\)”, Sov. Phys., Dokl., 32:8 (1987), 633–635.

    ADS  MATH  Google Scholar 

  10. A. I. Shafarevich, “Quasiclassical Scattering of Wave Packets on a Narrow Band in which the Potential Rapidly Changes”, Math. Notes, 45:1 (1989), 72–77.

    Article  MathSciNet  Google Scholar 

  11. V. P. Maslov and G. A. Omel’yanov, “Asymptotic Soliton-Form Solutions of Equations with Small Dispersion”, Russian Math. Surveys, 36:3 (1981), 73–149.

    Article  ADS  Google Scholar 

  12. F. Calogero and A. Degasperis, Spectral Transform and Solitons, North-Holland Publishing Company, 1982.

    MATH  Google Scholar 

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Funding

This work was supported financially by the Russian Science Foundation (grant no. 21-71-00050).

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Authors and Affiliations

  1. Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia

    A. I. Allilueva & A. I. Shafarevich

  2. Moscow State University, Moscow, Russia

    A. I. Shafarevich

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    A. I. Shafarevich

Authors
  1. A. I. Allilueva
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  2. A. I. Shafarevich
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Correspondence to A. I. Allilueva or A. I. Shafarevich.

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Allilueva, A.I., Shafarevich, A.I. Maslov’s Complex Germ in the Cauchy Problem for a Wave Equation with a Jumping Velocity. Russ. J. Math. Phys. 29, 1–10 (2022). https://doi.org/10.1134/S1061920822010010

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  • DOI: https://doi.org/10.1134/S1061920822010010

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