<p style='text-indent:20px;'>Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits <i>emergent singular solutions</i> for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions.</p><p style='text-indent:20px;'>In this paper, we investigate:</p><p style='text-indent:20px;'>(1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and</p><p style='text-indent:20px;'>(2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.</p>

Citation: Ronald Brown. Kirill Mackenzie, Bangor and holonomy. Journal of Geometric Mechanics, 2022, 14 (2) : 359-362. doi: 10.3934/jgm.2022012

<p style='text-indent:20px;'>In this paper we look at the question of integrability, or not, of the two natural almost complex structures <inline-formula><tex-math id="M1">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> defined on the twistor space <inline-formula><tex-math id="M2">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> of an even-dimensional manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> with additional structures <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula> a <inline-formula><tex-math id="M6">\begin{document}$ g $\end{document}</tex-math></inline-formula>-connection. We measure their non-integrability by the dimension of the span of the values of <inline-formula><tex-math id="M7">\begin{document}$ N^{J^\pm_\nabla} $\end{document}</tex-math></inline-formula>. We also look at the question of the compatibility of <inline-formula><tex-math id="M8">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> with a natural closed <inline-formula><tex-math id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-form <inline-formula><tex-math id="M10">\begin{document}$ \omega^{J(M, g, \nabla)} $\end{document}</tex-math></inline-formula> defined on <inline-formula><tex-math id="M11">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M12">\begin{document}$ (M, g) $\end{document}</tex-math></inline-formula> we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection <inline-formula><tex-math id="M13">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula>. In all cases <inline-formula><tex-math id="M14">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> is a bundle of complex structures on the tangent spaces of <inline-formula><tex-math id="M15">\begin{document}$ M $\end{document}</tex-math></inline-formula> compatible with <inline-formula><tex-math id="M16">\begin{document}$ g $\end{document}</tex-math></inline-formula>. In the case <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.</p>

<p style='text-indent:20px;'>The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula> of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over <inline-formula><tex-math id="M2">\begin{document}$ TQ $\end{document}</tex-math></inline-formula>, where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over <inline-formula><tex-math id="M3">\begin{document}$ Q\times Q $\end{document}</tex-math></inline-formula>.</p>

<p style='text-indent:20px;'>This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmüller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.</p>

<p style='text-indent:20px;'>The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory.

<p style='text-indent:20px;'>We consider a family of birational maps <inline-formula><tex-math id="M1">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family <inline-formula><tex-math id="M2">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> using Poisson geometry tools, namely the properties of the restrictions of the maps <inline-formula><tex-math id="M3">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> and their fourth iterate <inline-formula><tex-math id="M4">\begin{document}$ \varphi^{(4)}_k $\end{document}</tex-math></inline-formula> to the symplectic leaves of an appropriate Poisson manifold <inline-formula><tex-math id="M5">\begin{document}$ (\mathbb{R}^4_+, P) $\end{document}</tex-math></inline-formula>. These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product <inline-formula><tex-math id="M6">\begin{document}$ SL(2, \mathbb{Z})\ltimes\mathbb{R}^2 $\end{document}</tex-math></inline-formula>. The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for <inline-formula><tex-math id="M7">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> characterized by the parameter values <inline-formula><tex-math id="M8">\begin{document}$ k = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ k = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ k\geq 3 $\end{document}</tex-math></inline-formula>.

We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$ T $ map where $ T\in \mathbb{R}^n $ is the period of the acting $ n $-torus.

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation.