In the present work, we derive exact solutions for $p=1,2,3,4$, starting from the travelling wave ODE system satisfied by $U$ and $\Psi$. The method is new: (i) obtain first integrals by use of multi-reduction symmetry theory; (ii) apply a hodograph transformation which leads to triangular (decoupled) system; (iii) introduce an ansatz for polynomial solutions of the base ODE; (iv) characterize conditions under which solutions yield solitary waves; (v) solve an algebraic system for the coefficients in the ansatz under those conditions. The resulting solitary waves exhibit a wide range of features: bright and dark peaks; single peaked and multi-peaked; zero and non-zero backgrounds.

For small values of the applied electric potential, a steady axisymmetric flow is observed. As this parameter is increased, a transition to rotating waves, followed by a secondary transition to amplitude-modulated rotating waves, occurs. Subsequently, period-doubling and symmetry breaking bifurcations lead to more complicated flows. Further increase in the parameter leads to flows that resemble a rotating wave with one or two isolated vortices. \

We investigate the bifurcation structure of solutions that connects these various types of flow using numerical continuation based on time-integration. In particular, a Newton-Krylov method, which exploits the rotating nature of the flows, is implemented for the continuation of rotating waves and modulated rotating waves, and linear stability analysis of a flow map is used to identify the flow transitions that result due to changes in the model parameters. \

This is joint work with Mary Pugh and Stephen Morris (University of Toronto).

Abstract: A common inverse problem is to recover the source of diffusing molecules from noisy arrivals to small reactive sites. In this talk I will present a perspective on this important problem via extreme statistics. The central premise is that when a single stochastic process exhibits large variability (unreliable), the extrema of multiple processes has a remarkably tight distribution (reliable). In this talk I will present some background on extreme statistics followed by specific applications to directional sensing - the process of acquiring the direction of diffusive sources. We find that extreme statistics provide new insights and corroborate real world observations.

The talk is based on joint work with Damir Kinzebulatov.

We examine these conditions by solving numerically a set of variational optimization problems. These problems aim to determine initial conditions $\mathbf{u}_0$ such tat the corresponding flow maximizes $\int_0^T \|\mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$ for different values of $T$ while satisfying specific constraints. We address these problems computationally, employing a large-scale adjoint-based gradient approach in Sobolev and Lebesgue spaces.

We extend earlier work by considering various values of $q,$ and different types of gradients to discretize gradient flows. We also studied the limiting case $q=3$ where the regularity condition is slightly different.

A recursion operator on invariants is presented, producing two hierarchies of higher-order invariants. One of them consist of Hamiltonian Casimirs. The other one holding non-Casimirs holds only for an entropic equation of state (EOS).

The Hamiltonian structure of the radial fluid flow equations in combination with these non-Casimir invariants provides a corresponding hierarchy of generalized symmetries. The Lie algebra of the first-order symmetries is non-abelian.

For the special cases of barotropic EOS and entropic EOS two new kinematic conserved integrals yield additional first-order generalized symmetries. These provide an explicit transformation group acting on solutions of the fluid equations.