In this work we establish the existence of at least one weak periodic solution in the spatial directions of a nonlinear system of two coupled differential equations associated with a 2D Boussinesq model which describes the evolution of long water waves with small amplitude under the effect of surface tension. For wave speed 0 < |c| < 1, the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space H 1 k (R) (k-periodic functions f ∈ L

more »

... 2 k (R) such that f ∈ L 2 k (R)). For wave speed |c| > 1, the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a 4 × 4 system with a special Hamiltonian structure. In the case |c| > 1, we also compute numerical approximations of these travelling waves by using a Fourier spectral discretization of the corresponding 1D travelling wave equations and a Newton-type iteration.