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Breather collapse for a dispersive effective equation: Asymptotics on the well-posedness boundary

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Abstract

For the solution of the Cauchy problem for the equation

$$ u_{tt} = u_{xx} + i2u_{ttx} + u_{ttxx} $$

with discontinuous initial data, asymptotic formulas as t → ∞ are derived, which agree well with numerical results. The stability of the numerical methods used is analyzed. Other results are presented concerning nonstandard linear equations produced by homogenizing the equations describing wave processes in periodic stratified media.

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References

  1. N. S. Bakhvalov and M. E. Eglit, “Effective Dispersive Equations for Wave Propagation in Periodic Media,” Dokl. Akad. Nauk 370, 7–11 (2000) [Dokl. Math. 61, 1–4 (2000)].

    MathSciNet  Google Scholar 

  2. N. S. Bakhvalov and M. E. Eglit, “Analysis of the Effective Dispersive Equations Describing Wave Propagation in Stratified Media and Thin Plates,” Dokl. Akad. Nauk 383, 742–746 (2002) [Dokl. Math. 65, 301–305 (2002)].

    MathSciNet  Google Scholar 

  3. S. I. Serdyukova, “Exotic Asymptotics for a Linear Hyperbolic Equation,” Dokl. Akad. Nauk 389, 1–5 (2003) [Dokl. Math. 67, 203–207 (2003)].

    MathSciNet  Google Scholar 

  4. S. I. Serdyukova, “Hard Transition from a Stationary State to Oscillations for a Linear Differential Equation,” Dokl. Akad. Nauk 415, 310–314 (2007) [Dokl. Math. 76, 554–558 (2007)].

    MathSciNet  Google Scholar 

  5. S. I. Serdyukova, “Deformation of a Breather-Type Solution after Adding a Lower Order Term with a Complex Coefficient,” Dokl. Akad. Nauk 427, 17–23 (2009) [Dokl. Math. 80, 463–469 (2009)].

    MathSciNet  Google Scholar 

  6. V. S. Vladimirov, Equations of Mathematical Physics (Marcel Dekker, New York, 1971; Nauka, Moscow, 1988).

    Google Scholar 

  7. I. G. Petrovskii, “On the Cauchy Problem for Systems of Linear Partial Differential Equations in the Domain of Nonanalytic Functions,” Byull. Mosk. Gos. Univ. A, No. 1 (1938).

  8. M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  9. REDUCE User’s Guide for UNIX Systems: Version 3.8, Ed. by Neun Winfried (Berlin-Dahlem, ZiB D-14195, Sept. 2004).

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980, Russia

    S. I. Serdyukova

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  1. S. I. Serdyukova
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Correspondence to S. I. Serdyukova.

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Dedicated to the memory of Academician N.S. Bakhvalov

Original Russian Text © S.I. Serdyukova, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 7, pp. 1276–1284.

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Serdyukova, S.I. Breather collapse for a dispersive effective equation: Asymptotics on the well-posedness boundary. Comput. Math. and Math. Phys. 50, 1212–1220 (2010). https://doi.org/10.1134/S0965542510070109

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  • DOI: https://doi.org/10.1134/S0965542510070109

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