We present a method to turn a pair of compatible Hamiltonian operators into a pair of symplectic operators. This can then be used to write two different Lagrangians for the equation of interest, which in turn leads to two different Lagrangian multiforms. This indicates that the analogue in multiform theory of a bi-Hamiltonian system is a system with two distinct Lagrangian multiforms.
(Joint work with Pierandrea Vergallo)
The harmonic locus consists of the monodromy-free Schroedinger operators with rational potential quadratically growing at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via Wronskian map for Hermite polynomials.
We show that the harmonic locus can also be identified with the subset of the CalogeroMoser spaces, introduced by Wilson, which is invariant under a natural symplectic action of C*. As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing partition in terms of the spectrum of the corresponding Moser's matrix. We also compute the characters of the C*-action at the fixed points, proving a conjecture of Conti and Masoero.
A heteroclinic network is a type of solution to a dynamical system consisting of a set of equilibrium solutions and connecting orbits between them. Heteroclinic networks can be thought of as an embedding of a directed graph into the phase space of the dynamical system, where vertices correspond to equilibria and directed edges to heteroclinic trajectories. The dynamics near a heteroclinic network is characterized by intermittent behaviour: solutions spend a long period of time close to one equilibrium before rapidly switching to another. The manner in which the transitions between equilibria occur can be incredibly rich: trajectories may visit all the equilibria in the network, or only a subset of them; the order in which equilibria are visited may be regular, or apparently chaotic. In spatially extended systems (modelled by partial differential equations), solutions near heteroclinic networks can arise as travelling or spiral waves. In this talk I will present some results demonstrating this exotic behaviour near heteroclinic networks, and discuss some of the ways in which we are able to analyse this behaviour.
In this talk, I will review some of the recent advances in developing mathematical and computational methods for 1D localised patterns (patterns that are embedded in a quiescent state) to 2D localised patterns. These patterns occur in a wide range of applications from buckling of cylinders, vegetation patches near deserts, to fluid mechanics. While the mathematical theory of these patterns in 1D is well-established in higher-dimensions, new tools are required. This work has appeared on the front cover of JFM 2015 and the January 2024 Nonlinearity journal, nominated for the IMA Lighthill-Thwaites prize 2021, and subject of the SIAM 2024 T. Brooke Benjamin prize.
