We construct the large B-type and D-type universal character hierarchies (LBUC and LDUC) by enlarging the fermionic Fock space. We introduce the corresponding D-type and B-type Lie algebras gD and gB, together with their spin groups GD and GB. Their fermionic representations lead to the bilinear relations of the LBUC and LDUC hierarchies. We also prove that these bilinear relations characterize the corresponding group orbits under the product Cartan maps. In particular, the four parity sectors of the fermionic Fock space yield the decomposition LBUC=LDUC+mLDUC, where mLDUC consists of three types of mixed relations. Using the boson-fermion correspondence, we derive the Hirota bilinear equations of the LBUC hierarchy. The same decomposition is also proved at the Hirota level. Furthermore, we construct tau-functions of the LBUC hierarchy and obtain their expansions in universal characters with Pfaffian coefficients. Finally, we prove that these LBUC tau-functions can also be represented as shifted compositions of two large BKP (LBKP) tau-functions.
This study explores the nonlinear dynamics and the bifurcation structure of an active spherical Janus particle (JP) subjected to combined gravitational and AC electrophoretic forces within an unbounded electrolyte. We extend the existing ’weak-field’ dipolophoretic (DIP) models under a thin Debye layer, to include the effect of gravity and also introduce the concept of a frequency-dependent correction factor to adjust for the contribution of the induced-charge electrophoretic (ICEP) torque. Using a genetic algorithm, we develop an optimized model for this adjustment factor and validate it against experimental benchmarks. The model facilitates examination of the critical role of a JP gravitational tilt angle on the bifurcation structure and transient dynamics of a metallodielectric nano/micro sphere. The analysis reveals the existence of three distinct bifurcation regions that are governed by the density ratio between the two JP hemispheres, including two regions of unique stable equilibria which are separated by two supercritical pitchfork bifurcations that confine a region with bistable equilibria. These new insights advance our understanding of active Janus particle nonlinear fluid-structure interaction under realistic gravity conditions, providing implications for applications in microfluidics, targeted drug delivery, and active matter systems.
We extend the “Brownian bees” model of Berestycki et al. (2021, 2022) to cooperative reproduction, kA→(k+1)A, of a population of N symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit N → ∞, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all k and prove their linear stability for k ≤ 2 and instability for k ≥ 4. In the marginal case k=3, there is a whole continuous family of steady states at a single critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For k ≥ 4, the unstable steady state separates regimes of a finite-time collapse and diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.
We explore the advection-diffusion of a passive vector described by ∂tu+U·∇u=−∇p+νΔu, where both U and u are divergence-free velocity fields. We approach this equation from an input/output perspective, with U as the input and u as the output. This input/output viewpoint has been widely applied in recent studies on passive scalar equation, in the context of anomalous dissipation, mixing, optimal scalar transport, and nonuniqueness problems. What makes the passive vector equation considerably more challenging compared to the passive scalar equation is the lack of a Lagrangian perspective due to the presence of pressure.In this paper, rather than requiring U and u to be identical (as in the Navier-Stokes equation), we require U and u to be identical only in certain characteristics. We focus on the case where the characteristics in question are anomalous and enhanced dissipation. We study the advection-diffusion of a passive vector in a Couette flow configuration. The main result of this paper is a construction of a family of velocity fields {Uν} for which the energy dissipation scales as (logν−1)−2 such that the energy dissipation in velocity fields {uν} scales at least as (logν−1)−2. This means that both U and u exhibit near-anomalous dissipation, where the rate of energy dissipation decreases more slowly than any power-law να for any α > 0. The result in this paper is not just a mathematical construct; it closely resembles the behavior of turbulent flow in a channel. The (logν−1)−2 decrease of energy dissipation is predicted by phenomenological theories of wall-bounded turbulence, a prediction that has been extensively validated through experiments and numerical simulations. Finally, our study motivates the potential to close the gap between U and u-at least with the help of computational tools-and to investigate whether the (logν−1)−2 decrease in energy dissipation can be maintained.
