Abstract
In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical r-matrix structure was unravelled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: (i) the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical r-matrix structure; (ii) the zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; (iii) the Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion {I, H} = 0 for a first integral I.
Highlights
We prove the following results: (i) the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical r-matrix structure; (ii) the zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; (iii) the Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion {I, H} = 0 for a first integral I
As we showed in [21], some of the ideas in that area provide a solution to our conceptual problem: can we construct a covariant Poisson bracket such that the Lax form possesses a classical r-matrix structure and such that the zero curvature equation can be written as a covariant Hamilton equation? This success in giving the r-matrix a covariant interpretation was illustrated explicitly on three models: the nonlinear Schrödinger (NLS) equation, the modified Korteweg–de Vries equation and the sineGordon model in laboratory coordinates
This work constitutes progress towards the understanding of the role played by multi-time consistency and the application of covariant Hamiltonian field theory to integrable systems, which is a line of research that started with [21] and continued with [22, 50]
Summary
The seminal work of Gardner, Greene, Kruskal and Miura [1] quickly followed by that of Zakharov and Shabat [2] launched the modern era of integrable systems by introducing the. This ‘dual’ description was still unsatisfactory in the sense that one had to choose one of the independent variables or the other as a starting point, but it did not seem possible to construct a classical r-matrix formalism capable of including both independent variables simultaneously This flaw of the (traditional) Hamiltonian formulation of a field theory has been known for a long time. We call them multi-time Poisson bracket (for the obvious reason that it can be used to generate the flows with respect to all the ‘times’ xk of the hierarchy) and Hamiltonian multiform respectively The latter terminology comes from the fact that this object is derived from what is called a Lagrangian multiform, a notion introduced by Lobb and Nijhoff in [23] and which encodes integrability in a variational way. Some of those long, and not necessarily illuminating, proofs are gathered in appendix B
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