Modulation Equations of Weakly Nonlinear Geometrical Optics in Media Exhibiting Mixed Nonlinearity

Abstract

The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propagating in materials exhibiting mixed nonlinearity is derived with the resonant wave interactions inherent in the system. Our analysis differs from the results of Hunter et al., in that we have employed a different scaling, keeping in view the delayed effects of nonlinearity in certain thermodynamic systems exhibiting mixed nonlinearity. The result is to modify the transport equations obtained by Hunter et al. by the addition of certain cubic nonlinear terms. Through the method of averaging, the secular terms are eliminated. However, the averaging process is carried out in two steps; first, along manifolds of codimension two giving an advection equation, the solution of which is then averaged in a direction transverse to the above‐mentioned manifold.