A three-parameter family of Boussinesq type systems in two spacedimensions is considered. These systems approximate thethree-dimensional Euler equations, and consist of three nonlineardispersive wave equations that describe two-way propagation oflong surface waves of small amplitude in ideal fluids over ahorizontal bottom. For a subset of these systems it is proved thattheir Cauchy problem is locally well-posed in suitable Sobolevclasses. Further, a class of these systems is discretized byGalerkin-finite element methods, and error estimates are provedfor the resulting continuous time semidiscretizations. Results ofnumerical experiments are also presented with the aim of studyingproperties of line solitary waves and expanding wave solutions of these systems.
