Multi-Lagrangians, hereditary operators and Lax pairs for the Korteweg–de Vries positive and negative hierarchies

Abstract

We present an approach to the construction of action principles (the inverse problem of the calculus of variations), for first order (in time derivatives) differential equations, and generalize it to field theory in order to construct systematically, for integrable equations which are based on the existence of a Nijenhuis (or hereditary) operator, a (multi-Lagrangian) ladder of action principles which is complementary to the well-known multi-Hamiltonian formulation. We work out results for the Korteweg–de Vries (KdV) equation, which is a member of the positive hierarchy related to a hereditary operator. Three negative hierarchies of (negative) evolution equations are defined naturally from the hereditary operator as well, in a concise way, suitable for field theory. The Euler–Lagrange equations arising from the action principles are equivalent to deformations of the original evolution equation, and the deformations are obtained explicitly in terms of the positive and negative evolution vectors. We recognize, after appropriate coordinate transformations, the Liouville, Sinh–Gordon, Hunter–Zheng, and Camassa–Holm equations as negative evolution equations. The multi-Lagrangian ladder for KdV is directly mappable to a ladder for any of these negative equations and other positive evolution equations (e.g., the Harry–Dym and a special case of the Krichever–Novikov equations). For example, several nonequivalent, nonlocal time-reparametrization invariant action principles for KdV are constructed, and a new nonlocal action principle for the deformed system Sinh–Gordon+spatial translation vector is presented. Local and nonlocal Hamiltonian operators are obtained in factorized form as the inverses of all the nonequivalent symplectic two-forms in the ladder. Alternative Lax pairs for all negative evolution vectors are constructed, using the negative vectors and the hereditary operator as only input. This result leads us to conclude that, basically, all positive and negative evolution equations in the hierarchies share the same infinite-dimensional sets of local and nonlocal constants of the motion for KdV, which are explicitly obtained using symmetries and the local and nonlocal action principles for KdV.