Modified elastic wave equations on Riemannian and Kähler manifolds (Q2459938)

The author gives a generalization of the elastic wave equation \( \rho (\partial /\partial _{t}) ^{2}u - (\lambda + 2\mu )\,\, \text{grad}\, \, \text{div} \, u + \mu\,\, \text{rot \, rot}\, u=0\) to the case of arbitrary Riemann manifolds and of Kähler manifolds. In the case of standard elasticity theory solutions \(u\) can be decomposed into a sum of a longitudinal wave \( u _{1} \) and of a transversal wave \( u _{2}\) satisfying respectively \(\text{rot}\, u _{1}=0\) and \(\mathrm{div} u _{2}=0\), which have two different propagation speeds. The author establishes a related decomposition in the Kähler case, in which a solution can be decomposed into a sum of four terms which all have different propagation speeds.