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

Khachai, O. Yu.

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.

MSC:

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

Keywords:

matched asymptotic expansions; numerical methods; formal asymptotic expansion; step-like contrast structure; shock layer

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