偏微分方程式に対する指数漸近解析 (非線形波動現象の数理とその応用)

Abstract

A notable feature of ship waves at small Froude numbers (F ≪ 1) is that the wave amplitudes are exponentially small with respect to F. Moreover, as F→0 the assumption of linearization is not appropriate. Thus, the role of nonlinear effects in ship wave patterns at small F hinges on exponential (`beyond-all-orders') asymptotics. This outstanding theoretical issue is addressed here in the context of a simple partial differential equation with a quadratic nonlinear term. To understand how nonlinearity affects this ship wave pattern, we develop an exponential asymptotics technique applicable to a partial differential equation. The asymptotic predictions generally are supported by direct numerical computations.