The KZK type equation introduced in this Note differs from the traditional form of the KZK model known in acoustics by the assumptions on the nonlinear term. For this modified form, a global existence and uniqueness result is established for the case of non-constant coefficients. Afterwards the asymptotic behaviour of the solution of the KZK type equation with rapidly oscillating coefficients is studied.
L'équation de type KZK introduit dans cette Note est une version modifiée du modèle KZK connu en acoustique (ces modifications concernent les hypothèses sur le terme non linéaire). Pour cette forme modifiée, un résultat d'existence et unicité globales est établi dans le cas des coefficients variables. Ensuite le comportement asymptotique de la solution de l'équation de type KZK avec les coefficients rapidement oscillants est étudié.
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Mots-clés : Acoustique, Acoustique non linéaire, Équation KZK, Homogéneisation
Ilya Kostin 1; Grigory Panasenko 1
@article{CRMECA_2006__334_4_220_0, author = {Ilya Kostin and Grigory Panasenko}, title = {Khokhlov{\textendash}Zabolotskaya{\textendash}Kuznetsov type equation: nonlinear acoustics in heterogeneous media}, journal = {Comptes Rendus. M\'ecanique}, pages = {220--224}, publisher = {Elsevier}, volume = {334}, number = {4}, year = {2006}, doi = {10.1016/j.crme.2006.01.010}, language = {en}, }
TY - JOUR AU - Ilya Kostin AU - Grigory Panasenko TI - Khokhlov–Zabolotskaya–Kuznetsov type equation: nonlinear acoustics in heterogeneous media JO - Comptes Rendus. Mécanique PY - 2006 SP - 220 EP - 224 VL - 334 IS - 4 PB - Elsevier DO - 10.1016/j.crme.2006.01.010 LA - en ID - CRMECA_2006__334_4_220_0 ER -
Ilya Kostin; Grigory Panasenko. Khokhlov–Zabolotskaya–Kuznetsov type equation: nonlinear acoustics in heterogeneous media. Comptes Rendus. Mécanique, Volume 334 (2006) no. 4, pp. 220-224. doi : 10.1016/j.crme.2006.01.010. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.01.010/
[1] Nonlinear Underwater Acoustics, Amer. Inst. of Phys., New York, 1987
[2] Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics, Ultrasound Med. Biol., Volume 24 (1999) no. 9, pp. 1419-1435
[3] Nonlinear Theory of Sound Beams, Amer. Inst. of Phys., New York, 1987
[4] Nonlinear sawtooth-shaped waves, Physics Uspekhi, Volume 38 (1995) no. 9, pp. 965-989
[5] Theoretical Foundations of Nonlinear Acoustics, Plenum, New York, 1977
[6] Interaction of One-Dimensional Waves in Dispersion-Free Media, Moscow University Publ., Moscow, 1983
[7] Cauchy problem for Zakharov–Kuznetsov equation, Differential Equations, Volume 31 (1995) no. 6, pp. 1070-1081
[8] Cauchy problem for quasi-linear equations of odd order, Math. USSR Sbornik, Volume 180 (1989) no. 9, pp. 1183-1210
[9] Homogenization of the equations of high frequency nonlinear acoustics, C. R. Acad. Sci. Paris, Sér. 1, Volume 325 (1997), pp. 931-936
[10] Homogenization of high frequency nonlinear acoustics equations, Appl. Anal., Volume 74 (2000), pp. 311-331
[11] Equations of high frequency nonlinear acoustics of heterogeneous media, Acoustic J., Volume 40 (1994) no. 2, pp. 290-294
[12] Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht–London–Boston, 1989
[13] C. Bardos, A. Rozanova, KZK equation, in: International Crimean Autumn Mathematical School-Symposium, Book of Abstracts Laspi–Batiliman, Ukraine, 2004
[14] C. Bardos, A. Rozanova, Khokhlov–Zabolotskaya–Kuznetsov equation, in: The IV Internat. Conference on Differential and Functional Differential Equations, Book of Abstracts, Moscow, 2005
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