In this paper, two different techniques will be employed to study the cnoidal wave solutions of the Boussinesq systems. First, the existence of periodic travelling-wave solutions for a large family of systems is established by using a topological method. Although this result guarantees the existence of cnoidal wave solutions in a parameter region in the period and phase speed plane, it does not provide the uniqueness nor the non-existence of such solutions in other parameter regions. The explicit solutions are then found by using the Jacobi elliptic function series. Some of these explicit solutions fall in the parameter region where the cnoidal wave solutions are proved to exist, and others do not; so the method with Jacobi elliptic functions provides additional cnoidal wave solutions. In addition, the explicit solutions can be used in many ways, such as in testing numerical code and in testing the stability of these waves.

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In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (i.e both components are nonzero) of the nonlinear Schrödinger system iut + uxx + (a|u| 2 + b|v| 2)u = 0, ivt + vxx + (b|u| 2 + c|v| 2)v = 0, where u, v are complex-valued functions of (x, t) ∈ R 2 , and a, b, c ∈ R are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters a, b and c, further information about the structures of this set are given.