Lax Representations of Multi-Hamiltonian Systems

Abstract

As mentioned in the Introduction, the existence of nontrivial, i.e. not coming from a local conservation law [44], Lax representation or zero-curvature representation is a principal condition necessary for the solution of the Cauchy problem for the underlying dynamical system. It suggests that a Lax representation also carries important information about algebraic properties of related evolution equations. The recursion relations for differential equations in Lax form first appeared in the context of the inverse scattering method. The infinite family of equations, which leave the eigenvalues of the KdV spectral problem and the AKNS ones invariant in time, were found by Gardner et al. [90] and Ablowitz at al. [1] in 1974. As mentioned in the previous Chapter, the real importance of the recursion scheme was recognized by Olver in 1977. Then, Adler [4] in 1979 and Kupershmidt and Wilson [111] in 1981 derived the Hamiltonian structure of some dynamical systems directly from their Lax representations. Finally, in 1982, Fokas and Anderson [72] proved that the recursion operators constructed from Lax equations have the hereditary property.