Partial differential systems with nonlocal nonlinearities: Generation and solutions

We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction-diffusion systems with nonlocal quadratic nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic nonlinearity. In each case we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended.Comment: 4 figure

Developing citizen science ecosystem:critical factors for quadruple helix stakeholders engagement

Purpose: The purpose of this paper is to provide answers regarding the factors that motivate or discourage the quadruple helix (QH) stakeholders and the wider public in citizen science (CS) activities. The research reveals a current overview of the perceptions, attitudes, concerns and motivation with regard to development of CS ecosystem in four countries: Greece, Lithuania, the Netherlands and Spain. Design/methodology/approach: The researchers deploy a mixed methodology, entailing an in-depth literature review and a large-scale quantitative survey (approximately 2,000 citizens) targeting QH stakeholders and general public from the local national ecosystems. The results contain both descriptive statistics and statistical analysis per country. After the comprehensive overview of drivers and barriers regarding the participation in CS activities in general, the focus is narrowed down on the engagement motivation of different QH stakeholders and the differences in enabling/hindering factors at the local ecosystems. Findings: Depending on the country and the pre-existing level of CS maturity, the results provide a complicated network of factors that unlock or block participation in CS activities. These factors include, to name a few, political maturity, civic engagement, technological infrastructures, economic growth, culture of stakeholder collaboration, psychological stimulus and surplus of resources. The implications of the findings necessitate the alignment of the envisioned CS ecosystem with the local dynamics in each country. Research limitations/implications: The quantitative nature of the survey method, limited sample size and only four countries context are noted as limitations of the study and offer future research potential for longitudinal settings and mixed-methods studies. Originality/value: The results contribute to the wider literature on CS that focuses on perspectives, possibilities and differences in local contexts with respect to the public engagement by developing CS ecosystem. At the same time, its added value lies in the overall practical proposition, and how the latter can effectively and efficiently attract and retain different stakeholder groups and citizens, under a collaborative approach.</p

Grassmannian flows and applications to non-commutative non-local and local integrable systems

We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by P\"oppe based on solving the corresponding underlying linear partial differential equation to generate an evolutionary Hankel operator for the `scattering data', and then solving a linear Fredholm equation akin to the Marchenko equation to generate the evolutionary solution to the nonlinear partial differential system. Our generalisation involves inflating the underlying linear partial differential system for the scattering data to incorporate corresponding adjoint, reverse time or reverse space-time data, and it also allows for Hankel operators with matrix-valued kernels. With this approach we show how to linearise the matrix nonlinear Schr\"odinger and modified Korteweg de Vries equations as well as nonlocal reverse time and/or reverse space-time versions of these systems. Further, we formulate a unified linearisation procedure that incorporates all these systems as special cases. Further still, we demonstrate all such systems are example Fredholm Grassmannian flows.Comment: 31 page

Applications of Grassmannian and graph flows to coagulation systems

We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as Grassmannian or nonlinear graph flows and are therefore linearisable, and/or integrable in this sense. We first prove that a general Smoluchowski-type equation with a constant frequency kernel, that encompasses a large class of such models, is realisable as a multiplicative Grassmannian flow, and then go on to establish that several other related constant kernel models can be realised as such. These include: the Gallay--Mielke coarsening model; the Derrida--Retaux depinning transition model and a general multiple merger coagulation model. We then introduce and explore the notion of nonlinear graph flows, which are related to the notion of characteristics for partial differential equations. These generalise flows on a Grassmann manifold from sets of graphs of linear maps to sets of graphs of nonlinear maps. We demonstrate that Smoluchowski's coagulation equation in the additive and multiplicative frequency kernel cases, are realisable as nonlinear graph flows, and are thus integrable provided we can uniquely retrace the initial data map along characteristics. The additive and multiplicative frequency kernel cases correspond to inviscid Burgers flow. We explore further applications of such nonlinear graph flows, for example, to the stochastic viscous Burgers equation. Lastly we consider an example stochastic partial differential equation with a nonlocal nonlinearity that generalises the convolution form associated with nonlinear coagulation interaction, and demonstrate it can be realised as an infinite dimensional Grassmannian flow. In our companion paper, Doikou et al. [DMSW-integrable], we consider the application of such infinite dimensional Grassmannian flows to classical non-commutative integrable systems such as the Korteweg--de Vries and nonlinear Schrodinger equations.Comment: 44 pages, 1 figure. This paper is a companion paper to our other title "Grassmannian flows and applications to integrable systems". arXiv admin note: substantial text overlap with arXiv:1905.0503

Le Petit Marocain

14 décembre 19451945/12/14 (A33,N9155)-1945/12/14

Matlab file for Riccati RDE solution from Partial differential systems with non-local nonlinearities: generation and solutions

We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction–diffusion systems with non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local cubic nonlinearity. In each case, we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’

Matlab file for Riccati generalized NLS solution from Partial differential systems with non-local nonlinearities: generation and solutions

We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction–diffusion systems with non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local cubic nonlinearity. In each case, we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’