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The $abcd$-Boussinesq system is a model of two equations that can describethe propagation of small-amplitude long waves in both directions in the waterof finite depth. Considering the Hamiltonian regimes, where the parameters $b$and $d$ in the system satisfy $b=d>0$, small solutions in the energy space areglobally defined. Then, a variational approach is applied to establish theexistence and nonlinear stability of the set of solitary-wave solutions for thegeneralized $abcb$-Boussinesq system. The main point of the analysis is to showthat the traveling-wave solutions of the generalized $abcb$-Boussinesq systemconverge to nontrivial solitary-wave solutions of the generalized Korteweg-deVries equation.

Stability of solitary waves for generalized $abcd$-Boussinesq system: The Hamiltonian case