#### For citation:

Rudenko O. V. «Exotic» models of high-intensity wave physics: linearizing equations, exactly solvable problems and non-analytic nonlinearities. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 3, pp. 7-34. DOI: 10.18500/0869-6632-2018-26-3-7-34

# «Exotic» models of high-intensity wave physics: linearizing equations, exactly solvable problems and non-analytic nonlinearities

Topic and aim. A brief review of publications and discussion of some mathematical models are presented, which, in the author’s opinion, are well-known only to a few specialists. These models are not well studied, despite their universality and practical significance. Since the results were published at different times and in different journals, it is useful to summarize them in one article. The goal is to form a general idea of the subject for the readers and to interest them with mathematical, physical or applied details described in the cited references. Investigated models. Higher-order dissipative models are discussed. Precisely linearizable equations containing nonanalytic nonlinearities – quadratically-cubic (QC) and modular (M) – are considered. Equations like Burgers, KdV, KZ, Ostrovsky–Vakhnenko, inhomogeneous and nonlinear integro-differential equations are analyzed. Results. The appearance of dissipative oscillations near the shock front is explained. The formation in the QC-medium of compression and rarefaction shocks, which are stable only for certain parameters of the «jump», as well as the formation of periodic trapezoidal sawtooth waves and self-similar N-pulse signals are described. Collisions of single pulses in the M-medium are discussed, revealing new corpuscular properties (mutual absorption and annihilation). Collisions are similar to interactions of clusters of chemically reacting substances, for example, fuel and oxidizer. The features of the behavior of «modular» solitons are described. The phenomenon of nonlinear wave resonance in media with QC-, Q- and M-nonlinearities is studied. Precisely linearizable inhomogeneous equations with external sources are used. The shift of maximum of resonance curves relative to the linear position, which is determined by the equality of velocities of freely propagating and forced waves, is indicated. Simplified models for diffracting beams obtained by projecting 3D equations onto the beam axis are analyzed. Strongly nonlinear waves in systems with holonomic constraints are discussed. Integro-differential equations with relaxation type kernel, and the possibility of reducing them to differential and differentialdifference equations are considered. Discussion. The material is outlined on a popular level. Apparently, these studies can be continued if the readers find them interesting enough.

- Rabinovich M.I., Trubetskov D.I. Oscillations and Waves in Linear and Nonlinear Systems. New York: Springer, 1989.
- Rudenko O.V. The 40th anniversary of the Khokhlov–Zabolotskaya equation. Acoust. Phys., 2010, vol. 56, no. 4, pp. 457–466.
- Rudenko O.V., Robsman V.A. Equation of nonlinear waves in a scattering medium. Doklady-Physics., 2002, vol. 47, no. 6, pp. 443–446.
- Gueguen Y., Palciauskas V. Introduction to the Physics of Rocks. Princeton, NJ: Princeton University Press, 1994.
- Hill C.R., Bamber J.C., Ter Haar G.R. (Eds.). Physical Principles of Medical Ultrasonics. John Wiley & Sons, 2004.
- Boldea A.L. Generalized and potential symmetries of the Rudenko–Robsman equation. Cent. Eur. J. Phys., 2010, vol. 6, no. 6, pp. 995–1000.
- Аveriyanov M.V., Basova M.S., Khokhlova V.A. Stationary and quasi-stationary waves in even-order dissipative systems. Acoust. Phys., 2005, vol. 51, no. 5, pp. 495–501.
- Nakoryakov V.E., Pokusaev B.G., Shreiber I.R. Wave Dynamics of Gas- and VapourLiquid Media. New York: Begell House Publishers, 1992.
- Rudenko O.V. Equation admitting linearization and describing waves in dissipative media with modular, quadratic, and quadratically cubic nonlinearities. Doklady Mathematics, 2016, vol. 94, no. 3, pp. 703–707.
- Rudenko O.V. Nonlinear dynamics of quadratically cubic systems. Physics-Uspekhi, 2013, vol. 56, no. 7, pp. 683–690.
- Rudenko O.V., Hedberg C.M. Quadratically cubic Burgers’ equation as exactly solvable model of mathematical physics. Doklady Mathematics, 2015, vol. 91, no. 2, pp. 232–235.
- Rudenko O.V., Hedberg C.M. The quadratically cubic Burgers equation: An exactly solvable nonlinear model for shocks, pulses and periodic waves. Nonlinear Dynamics, 2016, Vol. 85, no. 2, pp. 767–776.
- Rudenko O.V., Gusev V.A. Self-similar solutions of a Burgers-type equation with quadratically cubic nonlinearity. Doklady Mathematics, 2016, vol. 93, no. 1, pp. 94–98.
- Rudenko O.V., Soluyan S.I. Some nonstationary problems of the theory of finite amplitude waves in dissipative media. Dokl. Akad. Nauk SSSR, 1970, vol. 190, no. 4, pp. 815–818.
- PolyaninA.D., Vyaz’min A.V., Zhurov A.I., Kazenin D.A. Handbook of Exact Solutions to Heat and Mass transfer Equations. Moscow: Faktorial, 1998 [in Russian].
- Ingard U. Nonlinear distorsion of sound transmitted through an orifice. J. Acoust. Soc. Am., 1970, vol. 48, no. 1, pp. 32–33.
- Idelchik I.E. Flow Resistance: A Design Guide for Engineers. New York: Hemisphere, 1989.
- Rudenko O.V., Khirnykh K.L. Model of Helmholtz resonator for absorption of highintensity sound. Sov. Phys. Acoust., vol. 36, no. 3, pp. 527–534.
- Korobov A.I., Izosimova M.Yu. Nonlinear Lamb waves in a metal plate with defects. Acoust. Phys., 2006, vol. 52, no. 5, pp. 683–692.
- Nazarov V.E., Kiyashko S.B., Radostin A.V. The wave processes in micro-inhomogeneous media with different-modulus nonlinearity and relaxation. Radiophys. and Quant. Electr., 2016, vol. 59, no. 3, pp. 246–256.
- Nazarov V., Radostin A. Nonlinear Acoustic Waves in Micro-Inhomogeneous Solids. John Wiley & Sons, 2015.
- Radostin A.V., Nazarov V.E., Kiyashko S.B. Propagation of nonlinear acoustic waves in bimodular media with linear dissipation. Wave Motion, 2013, vol. 50, no. 2, pp. 191–196.
- Ambartsumyan S.A. Elasticity Theory of Different Moduli. Beijing: China Rail. Publ. House, 1986.
- Hedberg C.M., Rudenko O.V. Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium. Nonlinear Dynamics, 2017, vol. 90, no. 3, pp. 2083–2091.
- Rudenko O.V. Modular solitons. Doklady Mathematics, 2016, vol. 94, no. 3, pp. 708–711.
- Korobov A.I., Prokhorov V.M. Nonlinear acoustic properties of the B95 aluminum alloy and the B95/nanodiamond composite. Acoust. Phys., 2016, vol. 62, no. 6, pp. 661–667.
- Korobov A.I., Kokshaiskii A.I., Prokhorov V.M., Evdokimov I.A., Perfilov S.A., Volkov A.D. Mechanical and nonlinear elastic characteristics of polycrystalline aluminum alloy and nanocomposite. Phys.of Solid State, 2016, vol. 58, no. 12, pp. 2472–2480.
- Gray A.L., Rudenko O.V. High-intensity wave in defected media containing both quadratic and modular nonlinearities: Shocks, harmonics and nondestructive testing. Acoust. Phys., 2018, vol. 64, no. 4, pp. 402–407.
- Mikhailov S.G., Rudenko O.V. A Simple nonlinear element model. Acoust. Phys., 2017, vol. 63, no. 3, pp. 270–274.
- Karabutov А.А., Lapshin E.A., Rudenko O.V. Interaction between light waves and sound under acoustic nonlinearity conditions. J. Exp. Theor. Physics, 1976, vol. 44, no. 1, pp. 58–63.
- Rudenko O.V. Wave excitation in a dissipative medium with a double quadraticallymodular nonlinearity: A generalization of the inhomogeneous Burgers equation. Doklady Mathematics, 2018, vol. 97, no. 3, pp. 721–724.
- Sinai Ya.G. Asymptotic behavior of solutions of 1D-Burgers equation with quasiperiodic forcing. Topol. Methods Nonlinear Anal., 1998, vol. 11, no. 2, pp. 219–226.
- Kudryavtsev A. G., Sapozhnikov O. A. Determination of the exact solutions to the inhomogeneous Burgers equation with the use of the Darboux transformation. Acoust. Phys., 2011, vol. 57, no. 3, pp. 311–319.
- Pasmanter R.A. Stability and Backlund transform of the forced Burgers equation. J. Math. Phys., 1986, vol. 29, pp. 2744–2746.
- Rudenko O.V., Hedberg C.M. Wave resonance in media with modular, quadratic and quadratically-Cubic nonlinearities described by inhomogeneous Burgers-type equations. Acoust. Phys., 2018, vol. 64, no. 4, pp. 422–431.
- Karabutov A.A., Rudenko O.V. Modified Khokhlov’s method for nonstationary trans-sonic flows of compressible gas. Dokl. Akad. Nauk SSSR, 1979, vol. 248, no. 5, pp. 1092–1085.
- Rudenko O.V. Nonlinear standing waves, resonant phenomena and frequency characteristics of distributed systems. Acoust. Phys., 2009, vol. 55, no. 1, pp. 27–54.
- Gurbatov S.N., Rudenko O.V., Saichev A.I. Waves and Structures in Nonlinear Nondispersive Media. Berlin and Beijing: Springer and Higher Education Press, 2011.
- Rudenko O.V. Giant nonlinearities in structurally inhomogeneous media and the fundamentals of nonlinear acoustic diagnostic techniques. Physics Uspekhi, 2006, vol. 49, no. 1, pp. 69–87.
- Frish U„ Bec J. In: New Trends in Turbulence. Pp. 341–384. Berlin, Heidelberg, Springer, 2001.
- Zabolotskaya E.A., Khokhlov R.V. Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust., 1969, vol. 15, no. 1, pp. 35–40.
- Bakhvalov N. S., Zhileikin Ya. M., Zabolotskaya E.A. Nonlinear Theory of Sound Beams. New York: AIP, 1987.
- Ostrovskii L.A., Sutin A.M. Focusing of acoustic waves of finite amplitude. Sov. Phys. Doklady, 1975, vol. 221, no. 6, pp. 1300–1303.
- Rudenko O.V., Hedberg C.M. Diffraction of high-intensity field in focal region as dynamics of nonlinear system with low-frequency dispersion. Acoust. Phys., 2015, vol. 61, no. 1, pp. 28–36.
- Brunelli J.C., Sakovich S. Hamiltonian structures for the Ostrovsky–Vakhnenko equation. Commun. Nonlinear Sci. Numer. Simulat., 2013, vol. 18, pp. 56–62.
- Ostrovsky L.A. Nonlinear internal waves in a rotating ocean. Okeanologia, 1978, vol. 18, no. 2, pp. 181–191.
- Vakhnenko V.A. Solitons in a nonlinear model medium. J. Phys. A, 1992, vol. 25A, pp. 4181–4187.
- Naugol’nykh K.A., Romanenko E.V. Amplification factor of a focusing system as a function of sound intensity. Sov. Phys. Acoustics, 1959, vol. 5, no. 2, pp. 191–195.
- Bessonova O.V., Khokhlova V.A., Bailey M.R., Canney M.R., Crum L.A. Focusing of high power ultrasound beams and limiting values of shock wave parameters. Acoust. Phys., 2009, vol. 55, no. 4–5, pp. 463–473.
- Wu F., Wang Z.B., Chen W.Z., et al. Extracorporeal focused ultrasound surgery for treatment of human solid carcinomas: Early Chinese clinical experience. Ultrasound Med. Biol., 2004, vol. 30, no. 2, pp. 245-260.
- Vasiljeva О.А., Lapshin Е.А., Rudenko О.V. Projection of the Khokhlov–Zabolotskaya equation on the axis of wave beam as a model of nonlinear diffraction. Doklady Mathematics, 2017, vol. 96, no. 3, pp. 646–649.
- Ibragimov N.H., Rudenko O.V. Principle of an A Priori use of symmetries in the theory of nonlinear waves. Acoust. Phys., 2004, vol. 50, no. 4, pp. 406-419.
- Rudenko O.V., Hedberg C.M. A new equation and exact solutions describing focal fields in media with modular nonlinearity. Nonlinear Dynamics, 2017, vol. 89, no. 3, pp. 1905–1913.
- Rudenko O.V. One-dimensional model of KZ-type equations for waves in the focal region of cubic and quadratically-cubic nonlinear media. Doklady Mathematics, 2017, vol. 96, no. 1, pp. 399–402.
- Rudenko O.V. Nonlinear sawtooth-shaped waves. Physics Uspekhi, 1995, vol. 38, no. 9, pp. 91-98.
- Panasenko G.P., Lapshin E.A. Homogenization of high frequency nonlinear acousticts equations. Applicable Analysis, 2013, vol. 74, no. 3, pp. 311–331.
- Landau L.D., Lifschitz E.M. Fluid Mechanics. New York: Pergamon Press, 1987.
- Broman G.I., Rudenko O.V. Submerged Landau jet: Exact solutions, their meaning and application. Physics Uspekhi, 2010, vol. 53, no. 1, pp. 91–98.
- Rudenko O.V. Оn strongly nonlinear waves and waves with strongly displayed weak nonlinearity. In: Nonlinear Waves – 2012 / Ed. A.V. Gaponov-Grekhov and V.I. Nekorkin, pp. 83–97. Nizhny Novgorod: IAP RAS publishing House, 2013.
- Rudenko O.V., Solodov E.V. Strongly nonlinear shear perturbations in discrete and continuous cubic nonlinear systems. Acoust. Phys., 2011, vol. 57, no. 1, pp. 51–58.
- Nikitenkova S.P., Pelinovskii E.N. Analysis of the Rudenko–Solodov equation in the theory of highly nonlinear shear vibrations. Acoust. Phys., 2014, vol. 60, no. 3, pp. 258–260.
- Heisenberg W. Zur Quantisierung nichtlinearer Gleichungen. Nachr. Acad. Wiss. Goettingen. IIa., 1953, pp. 111–127.
- Rudenko O.V., Tsyuryupa S.N., Sarvazyan A.P. Velocity and attenuation of shear waves in the phantom of a muscle-soft tissue matrix with embedded stretched fibers. Acoust. Phys., 2016, vol. 62, no. 5, pp. 608–614.
- Sarvazyan A.P., Rudenko O.V. United States Patent: 5, 810, 731. Date of Patent: Sep. 22, 1998. Method and apparatus for elasticity imaging using remotely induced shear wave.
- Rudenko O.V., Soluyan S.I., Khokhlov R.V. Problems of the theory of nonlinear acoustics. Sov. Phys. Acoust., vol. 20, no. 4, pp. 356–359.
- Ibragimov N.H., Meleshko S.V., Rudenko O.V. Group analysis of evolutionary integro-differential equations describing nonlinear waves: The general model. J. Physics A, vol. 44, no. 315201.
- Polyakova A.L., Soluyan S.I., Khokhlov R.V. Propagation of finite disturbances in a relaxing medium. Sov. Phys.Acoustics, 1962, vol. 8, no. 1, pp. 78–82.
- Rudenko O.V., Soluyan S.I. The scattering of sound by sound. Sov. Phys. Acoustics, 1973, vol. 18, no. 3, pp. 352–355.
- 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., 2014, vol. 60, no. 4, pp. 398–404.
- Rudenko O.V. Exact solutions of an integro-differential equation with quadratically cubic nonlinearity. Doklady Mathematics, 2016, vol. 94, no. 1, pp. 468–471.

- 2297 reads