| In this dissertation we study a two-dimensional nonlinear dispersive hyperelastic wave equation arising from an elastic plate. We prove that under certain assumptions the Cauchy problem is globally well-posed and the equation admits solitary wave solutions which are nonlinearly stable.;In the first part, we study finite deformations in a pre-stressed, hyperelastic compressible plate. For small-amplitude nonlinear waves, we obtain equations that involve an amplitude parameter epsilon. Using an asymptotic perturbation technique, we derive a new family of two-dimensional nonlinear dispersive equations. This family includes the KdV, Kadomtsev-Petviashvili and Camassa-Holm equations.;In the second part, we prove that when the plate is sufficiently stiff, then the Cauchy problem for the corresponding hyperelastic dispersive equation is globally well-posed in the natural energy space.;The third part deals with the solitary wave solution to the equations. We show that when the stiffness is positive then there exist solitary wave solutions u(x, y, t) = &phis; c(x-ct, y) that come from the associated variational problem. Such solitary waves are nonlinearly stable in the sense that if a solution is initially close to the set of such solitary waves, it remains close to this set for all time. In the case of zero stiffness, the equation becomes quasilinear and can change type. We prove that under certain assumptions this equation still permits solitary wave solutions.;In the last part we provide a criterion for nonexistence of solitary wave solutions. |
