Asymptotics of a solution of a three-dimensional nonlinear wave equation near a butterfly catastrophe point. (English. Russian original) Zbl 1400.35019

Proc. Steklov Inst. Math. 301, Suppl. 1, S72-S87 (2018); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 23, No. 2, 250-265 (2017).
Summary: The solution of the three-dimensional nonlinear wave equation \(-U''_{TT} + U''_{XX} + U''_{YY} + U''_{ZZ} = f(\epsilon T, \epsilon X, \epsilon Y, \epsilon Z, U)\) by means of the method of matched asymptotic expansions is considered. Here \(\epsilon\) 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: \(u''_{xx} = u^5 - tu^3 - zu^2 - yu - x\). 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.


35B25 Singular perturbations in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
Full Text: DOI


[1] Khachay, O. Yu.; Nosov, P. A., On some numerical integration curves for PDE in neighborhood of “ butterfly” catastrophe point, Ural Math. J., 2, 127-140, (2016) · Zbl 1396.34039
[2] Vasil’eva, A. B.; Butuzov, V. F.; Nefedov, N. N., Contrast structures in singularly perturbed problems, Fundam. Prikl. Mat., 4, 799-851, (1998) · Zbl 0963.34043
[3] Suleimanov, B. I., Cusp catastrophe in slowly varying equilibriums, J. Exp. Theor. Phys., 95, 944-956, (2002)
[4] Il’in, A. M.; Suleimanov, B. I., On two special functions related to fold singularities, Dokl. Math., 66, 327-329, (2002) · Zbl 1223.34084
[5] Il’in, A. M.; Suleimanov, B. I., Birth of step-like contrast structures connected with a cusp catastrophe, Sb. Math., 195, 1727-1746, (2004) · Zbl 1129.35387
[6] V. P. Maslov, V. G. Danilov, and K. A. Volosov, Mathematical Modelling of Heat and Mass Transfer Processes (Nauka, Moscow, 1987; Kluwer Acad., Dordrecht, 1995). · Zbl 0839.35001
[7] A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Amer. Math. Soc., Providence, RI, 1992). · Zbl 0754.34002
[8] R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, New York, 1981; Mir, Moscow, 1984). · Zbl 0597.58001
[9] V. S. Vladimirov and V. V. Zharinov, Equations of Mathematical Physics (Fizmatlit, Moscow, 2004) [in Russian]. · Zbl 0954.35001
[10] V. A. Zorich, Mathematical Analysis I (MTsNMO, Moscow, 2002) [in Russian].
[11] E. Fehlberg, Low-Order Classical Runge-Kutta Formulas with Stepsize Control, NASA Technical Report R-315 (1969).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.