40 citations · 154 across the 10 of their papers we have counts for
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Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes
John Gibbons, Alexander Stokes, Alexander P. Veselov
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 thi…
Theory of Heat Equations for Sigma Functions
J. C. Eilbeck, J. Gibbons, Y. Ônishi +1
We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat…
Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type
John Gibbons, Paolo Lorenzoni, Andrea Raimondo
We study purely nonlocal Hamiltonian structures for systems of hydrodynamic type. In the case of a semi-Hamiltonian system, we show that such structures are related to quadratic ex…
Hamiltonian structure of reductions of the Benney system
John Gibbons, Paolo Lorenzoni, Andrea Raimondo
We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dKP) starting from the family of conformal maps associated to it.
Vlasov moments, integrable systems and singular solutions
John Gibbons, Darryl D Holm, Cesare Tronci
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 o…
Differential Geometry of Hydrodynamic Vlasov Equations
John Gibbons, Andrea Raimondo
We consider hydrodynamic chains in dimensions which are Hamiltonian with respect to the Kupershmidt-Manin Poisson bracket. These systems can be derived from single …