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John Gibbons

Publications and source records attributed to John Gibbons.

9 recordsLinked to original sources
arXiv2024

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

nlin.SI↗
arXiv2012

Prime form and sigma function

In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r =x^s + λ_{1} x^{s-1} +...+ λ_{s-1} x + λ_s$. We give an expression for the prime form $\cE(P,Q)$, where $(P, Q \in X)$, in terms of the sigma function for some such curves, specifically any hyperelliptic curve $(r,s) = (2, 2g+1)$ as well as the cyclic trigonal curve $(r,s) = (3,4)$, $$ \cE(P,Q) =\frac{σ_{\natural_{r}}(u - v)}{\sqrt{du_1}\sqrt{d v_1}}, $$ where $\natural_r$ is a certain index of differentials. Here $u_1$ and $v_1$ are respectively the first components of $u = w(P)$ and $v = w(Q)$ which are given by the Abel map $w: X \to \CC^g$, where $g$ is the genus of $X$.

math.AG↗
arXiv2009

A genus six cyclic tetragonal reduction of the Benney equations

A reduction of Benney's equations is constructed corresponding to Schwartz-Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian sigma-function of the curve.

math-ph↗
arXiv2008

Geometry of Vlasov kinetic moments: a bosonic Fock space for the symmetric Schouten bracket

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie-Poisson bracket is extended to anisotropic interactions.

nlin.CD↗
arXiv2007

Vlasov moments, integrable systems and singular solutions

The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known Lie-Poisson Hamiltonian system. Remarkably, the operation of taking the moments of the Vlasov PDF preserves the Lie-Poisson structure. The individual particle motions correspond to singular solutions of the Vlasov equation. The paper focuses on singular solutions of the problem of geodesic motion of the Vlasov moments. These singular solutions recover geodesic motion of the individual particles.

nlin.SI↗
arXiv2007

Differential Geometry of Hydrodynamic Vlasov Equations

We consider hydrodynamic chains in $(1+1)$ dimensions which are Hamiltonian with respect to the Kupershmidt-Manin Poisson bracket. These systems can be derived from single $(2+1)$ equations, here called hydrodynamic Vlasov equations, under the map $A^n =\int_{-\infty}^\infty p^n f dp.$ For these equations an analogue of the Dubrovin-Novikov Hamiltonian structure is constructed. The Vlasov formalism allows us to describe objects like the Haantjes tensor for such a chain in a much more compact and computable way. We prove that the necessary conditions found by Ferapontov and Marshall in (arXiv:nlin.SI/0505013) for the integrability of these hydrodynamic chains are also sufficient.

nlin.SI↗
arXiv2006

Singular solutions for geodesic flows of Vlasov moments

The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known example of coadjoint motion. Remarkably, the property of coadjoint motion survives the process of taking moments. That is, the evolution of the moments of the Vlasov PDF is also coadjoint motion. We find that {\it geodesic} coadjoint motion of the Vlasov moments with respect to powers of the single-particle momentum admits singular (weak) solutions concentrated on embedded subspaces of physical space. The motion and interactions of these embedded subspaces are governed by canonical Hamiltonian equations for their geodesic evolution.

nlin.CD↗