Abstract
A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.
Similar content being viewed by others
References
O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Plenum, New York, 1977).
S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media (Springer, Berlin, 2011).
O. V. Rudenko, Phys.-Usp. 38, 965–989 (1995).
O. V. Rudenko, Phys.-Usp. 29, 620–641 (1986).
B. K. Novikov, O. V. Rudenko, and V. I. Timoshenko, Nonlinear Underwater Acoustics (Am. Inst. Physics, New York, 1987).
O. V. Rudenko, Phys.-Usp. 183 (7), 719–726 (2013).
O. V. Rudenko and C. M. Hedberg, Dokl. Math. 91 (2), 232–235 (2015).
O. V. Rudenko and C. M. Hedberg, Nonlin. Dyn. 85 (2), 767–776 (2016).
V. A. Gusev and O. V. Rudenko, Dokl. Math. 93 (1), 94–98 (2016).
V. E. Nazarov, S. B. Kiyashko, and A. V. Radostin, Radiophys. Quantum Electron. 59 (3), 246–256 (2016).
A. V. Radostin, V. E. Nazarov, and S. B. Kiyashko, Wave Motion 50 (2), 191–196 (2013).
V. E. Nazarov, S. B. Kiyashko, and A. V. Radostin, Nelin. Din. 11 (2), 209–218 (2015) [in Russian].
S. A. Ambartsumyan, Elasticity Theory of Different Moduli (Nauka, Moscow, 1982; China Rail. Publ. House, Beijing, 1986).
A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981; Mir, Moscow, 1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Doklady Akademii Nauk, 2016, Vol. 471, No. 1, pp. 23–27.
Rights and permissions
About this article
Cite this article
Rudenko, O.V. Equation admitting linearization and describing waves in dissipative media with modular, quadratic, and quadratically cubic nonlinearities. Dokl. Math. 94, 703–707 (2016). https://doi.org/10.1134/S1064562416060053
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562416060053