New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map

Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations

In this work, we recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov-Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.

Perspectives on the formation of peakons in the stochastic Camassa–Holm equation

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Berry phases in the reconstructed KdV equation

We consider the KdV equation on a circle and its Lie-Poisson reconstruction, which is reminiscent of an equation of motion for fluid particles. For periodic waves, the stroboscopic reconstructed motion is governed by an iterated map whose Poincaré rotation number yields the drift velocity. We show that this number has a geometric origin: it is the sum of a dynamical phase, a Berry phase, and an "anomalous phase." The last two quantities are universal: they are solely due to the underlying Virasoro group structure. The Berry phase, in particular, was previously described by Oblak [J. High Energy Phys. 10, 114 (2017)] for two-dimensional conformal field theories and follows from adiabatic deformations produced by the propagating wave. We illustrate these general results with cnoidal waves, for which all phases can be evaluated in closed form thanks to a uniformizing map that we derive. Along the way, we encounter "orbital bifurcations" occurring when a wave becomes non-uniformizable: there exists a resonance wedge, in the cnoidal parameter space, where particle motion is locked to the wave, while no such locking occurs outside of the wedge.

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge-Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.