Search for: All records

Abstract In Jayanti and Trivisa (2022 J. Math. Fluid Mech. 24 46), the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in Pitaevskii (1959 Sov. Phys. JETP 8 282–287) and couples the nonlinear Schrödinger equation and the Navier–Stokes equations. In this article, we prove a weak–strong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results. 

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

ABSTRACT Quorum sensing (QS) enables coordinated, population-wide behavior. QS-active bacteria “communicate” their number density using autoinducers which they synthesize, collect, and interpret. Tangentially, chemotactic bacteria migrate, seeking out nutrients and other molecules. It has long been hypothesized that bacterial behaviors, such as chemotaxis, were the primordial progenitors of complex behaviors of higher-order organisms. Recently, QS was linked to chemotaxis, yet the notion that these behaviors can together contribute to higher-order behaviors has not been shown. Here, we mathematically link flocking behavior, commonly observed in fish and birds, to bacterial chemotaxis and QS by constructing a phenomenological model of population-scale QS-mediated phenomena. Specifically, we recast a previously developed mathematical model of flocking and found that simulated bacterial behaviors aligned well with well-known QS behaviors. This relatively simple system of ordinary differential equations affords analytical analysis of asymptotic behavior and describes cell position and velocity, QS-mediated protein expression, and the surrounding concentrations of an autoinducer. Further, heuristic explorations of the model revealed that the emergence of “migratory” subpopulations occurs only when chemotaxis is directly linked to QS. That is, behaviors were simulated when chemotaxis was coupled to QS and when not. When coupled, the bacterial flocking model predicts the formation of two distinct groups of cells migrating at different speeds in their journey toward an attractant. This is qualitatively similar to phenomena spotted in our Escherichia coli chemotaxis experiments as well as in analogous work observed over 50 years ago. IMPORTANCE Our modeling efforts show how cell density can affect chemotaxis; they help to explain the roots of subgroup formation in bacterial populations. Our work also reinforces the notion that bacterial mechanisms are at times exhibited in higher-order organisms.