Two-dimensional wave propagation in a generalized elastic solid

Abstract

The authors analyze the propagation of two-dimensional waves in a weakly nonlinear and weakly dispersive elastic medium. The study is motivated by the importance to investigate elastic media with microlocal structure, such as crystals and polymers. It is known that the presence of dispersion and nonlinearity can produce solitary-like waves. For the one-dimensional case, their motion is described by Korteweg-de Vries equations. Up to now, the full theories for two- and three-dimensional cases are not available. Here the authors investigate two-dimensional waves by using a simplified theoretical model. The study is based on Euler-Lagrange equations derived from the Lagrangian of the system, and on an approximate form obtained by retaining the second-order terms which describe the nonlinearity. It appears that the dispersive character stems from the presence of high-order displacement gradients in constitutive relations. The main conclusion of the paper lies in showing that long-wave behaviour of elastic media with microlocal structure can be described by coupled modified Kadomtsev-Petviashvili equations.