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Nonlinear integro-differential models for intense waves in media like biological tissues and geostructures with complex internal relaxation-type dynamics

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Abstract

The paper discusses a universal scheme for constructing nonlinear integro-differential models to describe intense waves in media with a complex internal relaxation-type dynamics. Examples of such media are presented. Various forms of kernels are described. Situations are shown in which the models can be simplified by reducing them to differential or differential-difference equations with partial derivatives. Integral relations for the linear momentum and energy transferred by the wave are obtained. Exact solutions are found. The mapping method is used to obtain approximate solutions and analyze them in the form of difference relations.

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References

  1. L. D. Landau and E. M. Lifshits, Theoretical Physics, vol. 6, Fluid Mechanics (Pergamon, London, 1987; Fizmatlit, Moscow, 2003).

    Google Scholar 

  2. A. L. Polyakova, Akust. Zh. 6, 356 (1960).

    Google Scholar 

  3. A. L. Polyakova, S. I. Soluyan, and R. V. Khokhlov, Akust. Zh. 8, 107 (1962).

    MathSciNet  Google Scholar 

  4. S. I. Soluyan and R. V. Khokhlov, Akust. Zh. 8, 220 (1962).

    MathSciNet  Google Scholar 

  5. O. A. Vasil’eva, A. A. Karabutov, E. A. Lapshin, and O. V. Rudenko, Interaction of One-Dimensional Waves in Media without Dispersion (Mos. Gos. Univ., Moscow, 1983) [in Russian].

    MATH  Google Scholar 

  6. O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Plenum, Consultants Bureau, NewYork, 1977).

    Book  MATH  Google Scholar 

  7. O. V. Rudenko and S. I. Soluyan, Sov. Phys. Acoust. 18, 352 (1973).

    Google Scholar 

  8. O. V. Rudenko, S. I. Soluyan, and R. V. Khokhlov, Sov. Phys. Acoust. 20, 356 (1974).

    Google Scholar 

  9. O. V. Rudenko, in Nonlinear Waves-2012 (Inst. Appl. Phys. Russ. Acad. Sci., N. Novgorod, 2013), pp. 83–97.

    Google Scholar 

  10. O. V. Rudenko and C. M. Hedberg, Acoust. Phys. 59, 644 (2013).

    Article  ADS  Google Scholar 

  11. O. V. Rudenko, Acoust. Phys. 56, 457 (2010).

    Article  ADS  Google Scholar 

  12. A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Folwkes, and S. Y. Emelianov, Ultrasound in Med. Biol. 24, 1419 (1998).

    Article  Google Scholar 

  13. O. V. Rudenko and A. P. Sarvazyan, Acoust. Phys. 52, 720 (2006).

    Article  ADS  Google Scholar 

  14. P. V. Lebedev-Stepanov and O. V. Rudenko, Acoust. Phys. 55, 729 (2009).

    Article  ADS  Google Scholar 

  15. O. V. Rudenko and S. N. Gurbatov, Acoust. Phys. 58, 247 (2012).

    ADS  Google Scholar 

  16. O. V. Rudenko, A. L. Sobisevich, L. E. Sobisevich, C. M. Hedberg, and N. V. Shamaev, Acoust. Phys. 58, 99 (2012).

    Article  ADS  Google Scholar 

  17. O. V. Rudenko, S. N. Gurbatov, and I. Yu. Demin, Acoust. Phys. 59, 584 (2013).

    Article  ADS  Google Scholar 

  18. O. V. Rudenko, in Proc. 5th Int. Conf. “Frontiers of Nonlinear Physics,” N. Novgorod, 2013, pp. 63–64.

  19. S. N. Gurbatov and O. V. Rudenko, in Proc. 5th Int. Conf. “Frontiers of Nonlinear Physics,” N. Novgorod, 2013, pp. 56–57.

  20. N. H. Ibragimov, S. V. Meleshko, and O. V. Rudenko, J. Phys. A: Math. Theor. 44, 315201 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  21. O. V. Rudenko, S. N. Gurbatov, and C. M. Hedberg, Nonlinear Acoustics through Problems and Examples (Trafford, Victoria, Canada, 2010).

    Google Scholar 

  22. O. V. Rudenko, Phys.-Usp. 49, 69 (2006).

    Article  ADS  Google Scholar 

  23. M. A. Isakovich, Phys.-Usp. 22, 928 (1979).

    ADS  Google Scholar 

  24. I. G. Mikhailov, V. A. Solov’ev, and Yu. P. Syrnikov, Principles of Molecular Acoustics (Nauka, Moscow, 1964).

    Google Scholar 

  25. B. A. Larichev and G. A. Maksimov, Acoust. Phys. 44, 709 (1998).

    ADS  Google Scholar 

  26. S. A. Akhmanov and N. I. Koroteev, Methods of Nonlinear Optics in Light Dispersion Spectroscopy (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Physics Department, Moscow State University, Moscow, Russia

    O. V. Rudenko

  2. Nizhni Novgorod State University, Nizhni Novgorod, Russia

    O. V. Rudenko

  3. Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Russia

    O. V. Rudenko

  4. Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia

    O. V. Rudenko

  5. Blekinge Institute of Technology, Karlskrona, 37179, Sweden

    O. V. Rudenko

Authors
  1. O. V. Rudenko
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Correspondence to O. V. Rudenko.

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Original Russian Text © O.V. Rudenko, 2014, published in Akusticheskii Zhurnal, 2014, Vol. 60, No. 4, pp. 368–375.

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Rudenko, O.V. Nonlinear integro-differential models for intense waves in media like biological tissues and geostructures with complex internal relaxation-type dynamics. Acoust. Phys. 60, 398–404 (2014). https://doi.org/10.1134/S1063771014040162

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  • DOI: https://doi.org/10.1134/S1063771014040162

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