Geodesic
Asymptotics of a solution of a three-dimensional nonlinear wave equation near a butterfly catastrophe point
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 250-265 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the framework of the method of matched asymptotic expansions, a solution of the three-dimensional nonlinear wave equation $-U''_{TT}+U''_{XX}+U''_{YY}+U''_{ZZ}=f({\varepsilon} T, {\varepsilon} X,{\varepsilon} Y,{\varepsilon} Z,U)$ is considered. Here $\varepsilon$ is a small positive parameter and the right-hand side is a smoothly changing source term of the equation. A formal asymptotic expansion of the solution of the equation is constructed in terms of the inner scale near a typical “butterfly” catastrophe point. It is assumed that there exists a standard outer asymptotic expansion of this solution suitable outside a small neighborhood of the catastrophe point. We study a nonlinear second-order ordinary differential equation (ODE) for the leading term of the inner asymptotic expansion depending on three parameters: $u''_{xx}=u^5-t u^3-z u^2-y u-x$. This equation describes the appearance of a step-like contrast structure near the catastrophe point. We briefly describe the procedure for deriving this ODE. For a bounded set of values of the parameters, we obtain a uniform asymptotics at infinity of a solution of the ODE that satisfies the matching conditions. We use numerical methods to show the possibility of locating a shock layer outside a neighborhood of zero in the inner scale. The integral curves found numerically are presented.
Keywords: matched asymptotic expansions, nonlinear ordinary differential equation, nonlinear equation of mathematical physics, butterfly catastrophe, numerical methods. 
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O. Yu. Khachay. Asymptotics of a solution of a three-dimensional nonlinear wave equation near a butterfly catastrophe point. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 250-265. http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a20/

[1] Khachay O.Y., Nosov P.A., “On some numerical integration curves for PDE in neighborhood of “butterfly” catastrophe point”, Ural Math. J., 2:2 (2016), 127–140 https://umjuran.ru/index.php/umj/article/view/66 | DOI

[2] Vasileva A.B., Butuzov V.F., Nefedov N.N., “Kontrastnye struktury v singulyarno vozmuschennykh zadachakh”, Fundament. i prikl. matematika, 4:3 (1998), 799–851 | MR | Zbl

[3] Suleimanov B.I., “Katastrofa sborki v medlenno menyayuschikhsya polozheniyakh ravnovesiya”, Zhurn. eksperiment. i teor. fiziki, 122:5 (11) (2002), 1093–1106

[4] Ilin A.M., Suleimanov B.I., “O dvukh spetsialnykh funktsiyakh, svyazannykh s osobennostyu sborki”, Dokl. RAN, 387:2 (2002), 156–158 | MR | Zbl

[5] Ilin A.M., Suleimanov B.I., “Zarozhdenie kontrastnykh struktur tipa stupenki, svyazannoe s katastrofoi sborki”, Mat. sb., 195:12 (2004), 27–46 | DOI | Zbl

[6] Maslov V.P., Danilov V.G., Volosov K.A., Matematicheskoe modelirovanie protsessov teplomassoperenosa. Evolyutsiya dissipativnykh struktur, Nauka, M., 1987, 352 pp. | MR

[7] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[8] Gilmor R., Prikladnaya teoriya katastrof, v. 1, Mir, M., 1984, 350 pp. | MR

[9] Vladimirov V.S., Zharinov V.V., Uravneniya matematicheskoi fiziki, Fizmatlit, M., 2004, 400 pp. | MR

[10] Zorich V.A., Matematicheskii analiz, Ch. I, 4-e, ispravlennoe izd., MTsNMO, M., 2002, 664 pp.

[11] Fehlberg E., Low-order classical Runge-Kutta formulas with stepsize control, NASA Technical. Report R-315, 1969, 43 pp.