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The Boussinesq $abcd$ system arises in the modeling of long wave smallamplitude water waves in a channel, where the four parameters $(a,b,c,d)$satisfy one constraint. In this paper we focus on the solitary wave solutionsto such a system. In particular we work in two parameter regimes where thesystem does not admit a Hamiltonian structure (corresponding to $b \ne d$). Weprove via analytic global bifurcation techniques the existence of solitarywaves in such parameter regimes. Some qualitative properties of the solutionsare also derived, from which sharp results can be obtained for the globalsolution curves. Specifically, we first construct solutions bifurcating fromthe stationary waves, and obtain a global continuous curve of solutions thatexhibits a loss of ellipticity in the limit. The second family of solutionsbifurcate from the classical Boussinesq supercritical waves. We show that thecurve associated to the second class either undergoes a loss of ellipticity inthe limit or becomes arbitrarily close to having a stagnation point.

Global bifurcation of solitary waves to the Boussinesq $abcd$ system