A Plethora of Integrable Bi-Hamiltonian Equations

This paper discusses several algorithmic ways of constructing integrable evolution equations based on the use of multi-Hamiltonian structures. The recognition that integrable soliton equations, such as the Korteweg-deVries (KdV) and nonlinear Schrodinger (NLS) equations, can be constructed using a biHamiltonian method dates back to the late 1970’s. An extension of the method was proposed by the first author and Fuchssteiner in the early 1980’s and was used to derive integrable generalizations of the KdV and of the modified KdV. However it was not until these models reappeared in physical problems, and their novel solutions such as compactons and peakons were discovered, that the method achieved recognition. In this paper, we describe the basic approach to constructing a wide variety of integrable bi-Hamiltonian equations. In addition to usual soliton equations, these new hierarchies include equations with nonlinear dispersion which support novel types of solitonic solutions.