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We discuss the response of both moving and trapped solitary wave solutions of a nonlinear two-component nonlinear Schr\"odinger system in 1+1 dimensions to an anti-$\mathcal{PT}$ external periodic complex potential. The dynamical behavior…
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical {\alpha}-stable L\'evy processes via modulation or amplitude equations. We study SPDEs with a cubic…
This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting,…
Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in…
We describe a map-based model which reproduces many of the behaviors seen in partial differential equations (PDE's). Like PDE's, we show that this model can support an infinite number of stationary solutions, traveling solutions, breathing…
Understanding, predicting, and controlling physical processes often relies on the analysis of the dynamics of partial differential equations (PDEs). In this context, the present study offers an in-depth investigation into the nonlinear…
In this paper we focus on nonlinear SPDEs with singularities included in both drift and noise coefficients, for which the Gelfand-triple argument developed for (local) monotone SPDEs turns out to be invalid. We propose a general framework…
Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore researchers often seek simpler…
Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. We…
We present collective dynamics of the Cucker-Smale (C-S) ensemble under random communications. As an effective modeling of the C-S ensemble, we introduce a stochastic kinetic C-S equation with a multiplicative white noise. For the proposed…
Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which where a so-called shape functional is constrained by a partial differential equation (PDE)…
In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as…
Simple analytic considerations are applied to recently discovered patterns in a generalized Fisher equation for population dynamics. The generalization consists of the inclusion of non-local competition interactions among individuals. We…
We present a data-driven framework for learning hydrodynamic equations from particle-based simulations of active matter. Our method leverages coarse-graining in both space and time to bridge microscopic particle dynamics with macroscopic…
Systems involving Partial Differential Equations (PDEs) have recently become more popular among the machine learning community. However prior methods usually treat infinite dimensional problems in finite dimensions with Reduced Order…
Machine learning has emerged as a promising approach to path loss prediction, yet its effectiveness often degrades when measurement data are scarce. To address this limitation, we propose an ensemble-based machine learning framework that…
In this paper, we propose a method, that is based on equivariant moving frames, for development of high order accurate invariant compact finite difference schemes that preserve Lie symmetries of underlying partial differential equations. In…
In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the…
Colloidal mixtures represent a versatile model system to study transport in complex environments. They allow for a systematic variation of the control parameters, namely size ratio, total volume fraction and composition. We study the…