Jacobi Structures of Evolutionary Partial Differential Equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under reciprocal transformations. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to the computation of the Lichnerowicz-Jacobi cohomologies of Jacobi structures. We also introduce the notion of bi-Jacobi structures and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.

💡 Research Summary

The paper introduces a comprehensive framework for describing a broad class of non‑local Hamiltonian evolutionary partial differential equations (PDEs) by means of infinite‑dimensional Jacobi structures. The motivation stems from the observation that reciprocal (or change‑of‑variables) transformations, which are ubiquitous in the theory of integrable systems, typically turn a local Poisson bracket into a non‑local one. Classical Poisson geometry is insufficient to capture the resulting structures, prompting the authors to generalize to Jacobi geometry, where a bivector field together with a vector field (the “Reeb” field) encode both the bracket and a possible non‑conservative term.

To make this idea rigorous in the infinite‑dimensional setting, the authors define a new algebraic object called a quasi‑local multi‑vector. Unlike ordinary multi‑vectors, quasi‑local objects may contain terms involving the formal inverse of the total derivative (D^{-1}), which represent genuine non‑local contributions (integrals of the fields). The central technical achievement is the extension of the Schouten‑Nijenhuis bracket to this space. The extended bracket (\llbracket\cdot,\cdot\rrbracket) respects the grading, satisfies the graded Jacobi identity, and reduces to the usual bracket when all non‑local terms vanish.

A convenient realization of the whole construction is provided by supermanifold techniques. By introducing Grassmann odd variables (\theta_i) conjugate to the field components (u^i), the quasi‑local multi‑vectors become superfunctions, and the extended bracket is expressed as a simple Poisson‑type bracket on the super‑phase space. This representation streamlines calculations of cohomology and makes the invariance under coordinate changes transparent.

The authors prove that the class of infinite‑dimensional Jacobi structures is invariant under any reciprocal transformation. Concretely, if a system possesses a Jacobi bivector (P) satisfying (\llbracket P,P\rrbracket=0), then after a transformation defined by a new independent variable (\tilde{x}) and a rescaled dependent variable (\tilde{u}), the transformed bivector (\tilde{P}) again fulfills the Jacobi identity. This result guarantees that the geometric essence of the system is preserved even though the explicit form of the bracket may become highly non‑local.

Having established the geometric setting, the paper turns to the Lichnerowicz‑Jacobi cohomology associated with a Jacobi structure. The differential (d_P=\llbracket P,\cdot\rrbracket) squares to zero, and its cohomology groups (H^k(P)) classify infinitesimal deformations (for (k=1)) and obstructions to extending deformations (for (k=2)). The authors compute these groups for several illustrative examples, notably the Camassa‑Holm equation, showing that its second cohomology vanishes, which implies that any infinitesimal deformation can be integrated to a genuine family of Jacobi structures.

A further major contribution is the introduction of bi‑Jacobi structures. Two Jacobi bivectors (P_1) and (P_2) are said to be compatible if each satisfies the Jacobi identity and their mixed bracket vanishes: (\llbracket P_1,P_2\rrbracket=0). Compatibility yields a recursion operator (R=P_2 P_1^{-1}) that generates an infinite hierarchy of commuting flows and conserved quantities, mirroring the well‑known bi‑Hamiltonian theory but now applicable to non‑local settings. The paper demonstrates this mechanism on the KdV‑type hierarchy and on the Camassa‑Holm equation, constructing explicit sequences of conserved densities and showing how the bi‑Jacobi framework reproduces known integrability results while providing a systematic route to new ones.

The final sections discuss potential extensions: (i) the development of computational tools for higher‑order Jacobi cohomology in multi‑field models, (ii) the exploration of quantization procedures for non‑local Jacobi brackets, and (iii) the application of the theory to physical models where reciprocal transformations arise naturally, such as shallow‑water waves, plasma dynamics, and nonlinear optics. In summary, the paper furnishes a robust geometric language that unifies local and non‑local Hamiltonian PDEs, proves its stability under reciprocal transformations, and opens new avenues for the analysis of integrable and near‑integrable systems.