Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point


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The solution of the three-dimensional nonlinear wave equation −UTT + UXX + UYY + UZZ = f(εT, εX, εY, εZ, U) by means of the method of matched asymptotic expansions is considered. Here ε 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: uxx = u5tu3zu2yux. 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.

Sobre autores

O. Khachai

Ural Federal University

Autor responsável pela correspondência
Email: khachay@yandex.ru
Rússia, Yekaterinburg, 620002

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