[Paper Review] New Hamiltonian formalism and Lagrangian representations for integrable hydrodynamic type systems

M. V. Pavlov|ArXiv.org|Aug 14, 2006

Nonlinear Waves and Solitons42 references3 citations

TL;DR

This paper introduces a new Hamiltonian formalism for integrable hydrodynamic type systems based on conjugate curvilinear coordinate nets and anti-flatness conditions, establishing a 'mirrored' structure to Dubrovin-Novikov and Ferapontov's formalisms. It derives local Lagrangian representations in the 'mirrored-flat' case and proves the existence of an infinite hierarchy of local Hamiltonian structures of all odd orders, generalizing Nutku–Olver and Sheftel–Teshukov systems.

ABSTRACT

New Hamiltonian formalism based on the theory of conjugate curvilinear coordinate nets is established. All formulas are ``mirrored'' to corresponding formulas in the Hamiltonian formalism constructed by B.A. Dubrovin and S.P. Novikov (in a flat case) and E.V. Ferapontov (in a non-flat case). In the ``mirrored-flat'' case the Lagrangian formulation is found. Multi-Hamiltonian examples are presented. In particular Egorov's case, generalizations of local Nutku--Olver's Hamiltonian structure and corresponding Sheftel--Teshukov's recursion operator are presented. An number of Hamiltonian structures of all odd orders is found.

Motivation & Objective

• To develop a new Hamiltonian formalism for integrable hydrodynamic type systems using the theory of conjugate curvilinear coordinate nets.
• To establish a 'mirrored' structure analogous to Dubrovin–Novikov and Ferapontov formalisms, particularly in the anti-flat case.
• To derive local Lagrangian representations for systems satisfying the 'mirrored-flat' condition.
• To construct infinite hierarchies of local Hamiltonian structures of odd orders, generalizing known examples like Nutku–Olver and Sheftel–Teshukov systems.
• To extend nonlocal Hamiltonian structures to arbitrary co-dimension via the anti-flatness condition.

Proposed method

• Formalism is built on the anti-flatness condition, which generalizes the flatness condition in Dubrovin–Novikov theory.
• The key equation is the anti-flatness condition (28), which defines a dual structure to flat systems.
• A 'mirrored' Hamiltonian formalism is constructed, with formulas analogous to Dubrovin–Novikov and Ferapontov but adapted to conjugate nets.
• Local Lagrangian representation is derived via Riemann invariants and the condition (23), inspired by ideal gas dynamics.
• Higher-order Hamiltonian structures are generated recursively using the recursion operator (38), leading to infinite odd-order structures.
• The method leverages differential-geometric Poisson brackets of the form (38), with quasi-rational dependence on higher derivatives.

Experimental results

Research questions

• RQ1 Can a new Hamiltonian formalism be constructed for integrable hydrodynamic type systems using conjugate curvilinear coordinate nets and anti-flatness?
• RQ2 Does the 'mirrored-flat' case admit a local Lagrangian formulation, and if so, how is it related to Riemann invariants?
• RQ3 Can an infinite hierarchy of local Hamiltonian structures of all odd orders be systematically derived from a single bi-Hamiltonian pair?
• RQ4 How do the generalized Nutku–Olver and Sheftel–Teshukov structures fit into this new formalism?
• RQ5 Can nonlocal Hamiltonian structures based on anti-flatness be extended to arbitrary co-dimension?

Key findings

• An infinite number of local Hamiltonian structures of all odd orders is constructed, generalizing the Nutku–Olver and Sheftel–Teshukov systems.
• The Egorov flat hydrodynamic type systems are shown to possess infinitely many local Hamiltonian structures of odd higher degrees.
• A local Lagrangian representation is explicitly found in the 'mirrored-flat' case, linking the system to Riemann invariants and ideal gas dynamics.
• The mixed bi-Hamiltonian structure (38) generates an infinite sequence of commuting flows with rational dependence on first derivatives.
• The nonlocal Hamiltonian formalism based on anti-flatness is extended to arbitrary co-dimension, generalizing previous results.
• The new formalism provides a unified framework for multi-Hamiltonian structures, showing equivalence to the Dubrovin–Novikov type in the local case.