Abstract: In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov-Zabolotskaya-Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of the Khokhlov-Zabolotskaya-Kuznetsov type equations for the existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.

Keywords: KZK equation, global classical solution, fixed point, sum of~operators, initial value problem

For citation: Bouakaz, A., Bouhmila, F., Georgiev, S. G., Kheloufi, A. and Khoufache, S.
Existence of Classical Solutions for a Class of the Khokhlov-Zabolotskaya-Kuznetsov Type Equations, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 36-50. DOI 10.46698/n8469-5074-4131-b

1. Kuznetsov, V. P. Equations of Nonlinear Acoustics,
Soviet Physics Acoustics, 1971, vol. 16, pp. 467-470.

2. Zabolotskaya, E. A. and Khokhlov, R. V. Quasi-Plane Waves in the Nonlinear Acoustics
of Confined Beams, Soviet Physics Acoustics, 1969, vol. 15, pp. 35-40.

3. Chou, C.-S., Sun, W., Xing, Y. and Yang, H. Local Discontinuous Galerkin Methods
for the Khokhlov-Zabolotskaya-Kuznetzov Equation, Journal of Scientific
Computing, 2017, vol. 73, no. 2-3, pp. 593-616. DOI: 10.1007/s10915-017-0502-z.

4. Rozanova-Pierrat, A. Mathematical Analysis of Khokhlov-Zabolotskaya-Kuznetsov
(KZK) Equation, 2006, hal-00112147, 68 p.

5. Averkiou, M. A. and Cleveland, R. O. Modeling of an Electrohydraulic Lithotripter with the KZK Equation,
The Journal of the Acoustical Society of America, 1999, vol. 106, no. 1, pp. 102-112.
DOI: 10.1121/1.427039.

6. Destrade, M., Goriely, A. and Saccomandi, G. Scalar Evolution Equations for Shear
Waves in Incompressible Solids: a Simple Derivation of the Z, ZK, KZK and KP Equations,
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
2011, vol. 467, no. 2131, pp. 1823-1834. DOI: 10.1098/rspa.2010.0508.

7. Kostin, I. and Panasenko, G. Khokhlov-Zabolotskaya-Kuznetsov Type Equation:
Nonlinear Acoustics in Heterogeneous Media, SIAM Journal on Mathematical Analysis,
2008, vol. 40, no. 2, pp. 699-715. DOI: 10.1137/060674272.

8. Zhang, L., Ji, J., Jiang, J. and Zhang, C. The New Exact Analytical Solutions
and Numerical Simulation of (3 + 1)-Dimensional Time Fractional KZK Equation,
International Journal of Computing Science and Mathematics,
2019, vol. 10, no. 2, pp. 174-192. DOI: 10.1504/IJCSM.2019.098744.

9. Akcagil, S. and Aydemir, T. New Exact Solutions for the Khokhlov-Zabolotskaya-Kuznetsov,
the Newell-Whitehead-Segel and the Rabinovich Wave Equations by Using a New Modification of the Tanh-Coth Method,
Cogent Mathematics, 2016, vol. 3, art. ID 1193104, 12 p. DOI: 10.1080/23311835.2016.1193104.

10. Satapathy, P., Raja Sekhar, T. and Zeidan, D. Codimension Two Lie Invariant
Solutions of the Modified Khokhlov-Zabolotskaya-Kuznetsov Equation,
Mathematical Methods in the Applied Sciences, 2021, vol. 44, no. 6, pp. 4938-4951.
DOI: 10.1002/mma.7078.

11. Dontsov, E. V. and Guzina, B. B. On the KZK-Type Equation for Modulated Ultrasound Fields,
Wave Motion, 2013, vol. 50, no. 4, pp. 763-775. DOI: 10.1016/j.wavemoti.2013.02.008.

12. Georgiev, S. G. and Zennir, K.
Existence of Solutions for a Class of Nonlinear Impulsive Wave Equations,
Ricerche di Matematica, 2022, vol. 71, no. 1, pp. 211-225. DOI: 10.1007/s11587-021-00649-2.

13. Djebali, S. and Mebarki, K. Fixed Point Index Theory for Perturbation of Expansive
Mappings by \(k\)-Set Contractions, Topological Methods in Nonlinear Analysis,
2019, vol. 54, no. 2A, pp. 613-640. DOI: 10.12775/TMNA.2019.055.

14 Polyanin, A. and Manzhirov, A. Handbook of Integral Equations, CRC Press, 1998, 796 p.