Doklady Mathematics

, Volume 94, Issue 3, pp 708–711 | Cite as

Modular solitons

  • O. V. RudenkoEmail author
Mathematical Physics


Solutions to a partial differential equation of the third order containing the modular nonlinearity are studied. The model describes, in particular, elastic waves in media with weak high-frequency dispersion and with different response to tensile and compressive stresses. This equation is linear for solutions preserving their sign. Nonlinear phenomena only manifest themselves to alternating solutions. Stationary solutions in the form of solitary waves or solitons are found. It is shown how the linear periodic wave becomes nonlinear after exceeding a certain critical value of the amplitude, and how it transforms into a soliton with further increase in the amplitude.


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  1. 1.
    O. V. Rudenko, Dokl. Math. 94 (3), 23–27 (2016).Google Scholar
  2. 2.
    S. A. Ambartsumyan, Elasticity Theory of Different Moduli (Nauka, Moscow, 1982; China Rail. Publ. House, Beijing, 1986).Google Scholar
  3. 3.
    V. E. Nazarov, S. B. Kiyashko, and A. V. Radostin, Radiophys. Quant. Electron. 58, 729–737 (2016).CrossRefGoogle Scholar
  4. 4.
    A. V. Radostin, V. E. Nazarov, and S. B. Kiyashko, Wave Motion 50 (2), 191–196 (2013).CrossRefMathSciNetGoogle Scholar
  5. 5.
    V. E. Nazarov, S. B. Kiyashko, and A. V. Radostin, Nelin. Din. 11 (2), 209–218 (2015).CrossRefGoogle Scholar
  6. 6.
    M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves, 3rd ed. (Lenand, Moscow, 2015) [in Russian].zbMATHGoogle Scholar
  7. 7.
    S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media (Springer, Berlin, 2011).zbMATHGoogle Scholar
  8. 8.
    O. V. Rudenko, Phys.-Usp. (Adv. Phys. Sci.) 56 (7), 683–690 (2013).CrossRefGoogle Scholar
  9. 9.
    O. V. Rudenko and C. M. Hedberg, Dokl. Math. 91 (2), 232–235 (2015).CrossRefMathSciNetGoogle Scholar
  10. 10.
    O. V. Rudenko and C. M. Hedberg, Nonlin. Dyn. 84 (2), 1–11 (2016). doi 10.1007/s11071-016-2721-5Google Scholar
  11. 11.
    V. A. Gusev and O. V. Rudenko, Dokl. Math. 93 (1), 94–98 (2016).CrossRefMathSciNetGoogle Scholar
  12. 12.
    V. E. Zakharov and L. D. Faddeev, Funct. Anal. Appl. 5 (4), 280–287 (1971).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Physics FacultyMoscow State UniversityMoscowRussia
  2. 2.Prokhorov General Physics InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Schmidt Institute of Physics of the EarthRussian Academy of SciencesMoscowRussia
  4. 4.Blekinge Institute of TechnologyKarlskronaSweden

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