Integrable quadratic structures in peakon models

J. Avan; L. Frappat; E. Ragoucy

Home/Archive/arXiv

arXiv · 2203.13593

Integrable quadratic structures in peakon models

J. Avan·L. Frappat·E. Ragoucy

View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose realizations of the Poisson structures for the Lax representations of three integrable $n$-body peakon equations, Camassa--Holm, Degasperis--Procesi and Novikov. The Poisson structures derived from the integrability structures of the continuous equations yield quadratic forms for the $r$-matrix representation, with the Toda molecule classical $r$-matrix playing a prominent role. We look for a linear form for the $r$-matrix representation. Aside from the Camassa--Holm case, where the structure is already known, the two other cases do not allow such a presentation, with the noticeable exception of the Novikov model at $n=2$. Generalized Hamiltonians obtained from the canonical Sklyanin trace formula for quadratic structures are derived in the three cases.

Explore related subjects

nlin.SI math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

J. Avan, L. Frappat, E. Ragoucy. 2022-03-25. Integrable quadratic structures in peakon models. https://arxiv.org/abs/2203.13593

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Geometric, algebraic and analytic properties of $\mathrm{al}_{ab}$ function for hyperelliptic curves of genus $g$

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ functions. We then demonstrate the differential identities of the $\mathrm{al}_{ab}$ function. These identities are novel integrable partial nonlinear differential equations as an extension of the differential identities in terms of the $\mathrm{al}_a$ function known as the hyperelliptic solutions of the modified Korteweg-de Vries equation. Thus, we also show that by the identities, the $\mathrm{al}_{ab}$ function is useful for expressing hyperelliptic solutions to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations in an explicit form as an extension of the elliptic $\mathrm{sn}$ function solutions.

nlin.SI↗

arXiv2026

Equations of state of hydrodynamic type and particle statistics of a Dyson gas in an analytic confining potential

We investigate the equilibrium thermodynamics of a Dyson gas in connection with a set of integrable statistical mechanical observables satisfying the Toda Lattice hierarchy. We prove that in the thermodynamic limit, the integrable observables are state functions satisfying a set algebraic equations of state in closed form, obtained from direct integration of the Toda Lattice hierarchy in the continuum limit. We then explore the connection between regularity and critical behaviour of the state functions and the Dyson gas particle statistics via Monte Carlo simulations. We show that the properties of the integrable observables, such as regularity, multivaluedness, cusp singularities, carry information on the macroscopic particle statistics and its qualitative changes but with some limitations.

nlin.SI↗

arXiv2026

The Nakamura Conjecture Revisited: Toda Molecules and Stationary Axisymmetric Gravity

We revisit the Nakamura conjecture, which relates the Tomimatsu-Sato solutions of stationary axisymmetric gravity to finite Toda molecules. While the conjecture has been established partially, its general rotating sector remains an open problem. We show that the Toda determinants underlying the conjecture possess a natural weight grading. In particular, the two functions entering the Ernst potential have weights n^2 and n^2-1, and this grading extends systematically to shifted determinants labelled by partitions. In coordinates adapted to the Toda generators, each differentiation corresponds to adding one box to the associated Young diagram and increases the weight by one. The same integer n^2 also appears in the zero-order term of the Nakamura bilinear operator, revealing a compatibility between the differential equation and the determinant grading. The partition structure further explains the previously unresolved behavior of second derivatives. Repeated differentiation in one direction produces an internal sector and an external sector requiring only a one-step extension of the Wronskian hierarchy; the latter is reduced by a local three-term Pluecker relation. Thus weight grading, Young-diagram growth, Wronskian enlargement, and Pluecker reduction emerge as parts of a single determinant structure. The unit weight relation n^2 = (n^2-1) + 1 also singles out the elementary Toda seed as a natural third object, suggesting a possible route toward a genuine trilinear formulation. Although no trilinear closure is assumed here, the present construction reduces the remaining general-n Nakamura problem to definite determinant-minor identities and provides a structural framework in which such a formulation can be investigated.

nlin.SI↗