Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support

A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons (solitary-wave solutions having compact support), or cusped and/or peaked solitons. A general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified Korteweg–de Vries equation, the nonlinear Schrödinger equation, the integrable Boussinesq system used to model the two-way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These hierarchies include a remarkable variety of interesting integrable nonlinear differential equations.