New finite-dimensional integrable systems and explicit solutions of Hirota–Satsuma coupled Kortweg–de Vries equation

Abstract

The nonlinearization approach is extended to the Hirota–Satsuma coupled Kortweg–de Vries (KdV) equation associated with a 4×4 matrix spectral problem, from which two new finite-dimensional integrable Hamiltonian systems are obtained in the Liouville sense. It is shown that the solutions of this coupled KdV equation are reduced to solving a compatible system of ordinary differential equations. The reductions of the two Hamiltonian systems and their separability are discussed. An interesting connection between the two reduced Hamiltonian systems in the case of one-parameter and known two-dimensional integrable systems is revealed. As application, some explicit solutions of the Hirota–Satsuma coupled KdV equation are derived.