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

Abstract

The solution of the three-dimensional nonlinear wave equation −U″_TT + U″_XX + U″_YY + U″_ZZ = 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: u″_xx = u⁵ − tu³ − zu² − 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.