Related papers: Conservation Laws With Random and Deterministic Da…
It is shown that, in the non-relativistic limit, causal fermion systems give rise to an effective collapse theory. The nonlinear and stochastic correction terms to the Schr\"odinger equation are derived from the causal action principle. The…
The conservation laws of nonrelativistic and relativistic systems are reviewed and some simple illustrations are provided for the restrictive nature of the relativistic conservation law involving the center of energy compared to the…
In this paper we study the long time dynamics of the solutions to the initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its…
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path…
This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for…
Scientists have long aimed to discover meaningful formulae which accurately describe experimental data. A common approach is to manually create mathematical models of natural phenomena using domain knowledge, and then fit these models to…
We develop deterministic particle schemes to solve non-local scalar conservation laws with congestion. We show that the discrete approximations converge to the unique entropy solution with an explicit rate of convergence under more general…
We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time…
Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in…
We study the nonlinear wave equation for arbitrary function with fourth order dissipation. A special case that is analysed exclusively is the model of nerve membranes; we consider this model, both, in the presence and absence of the fourth…
The regularized theories are non-local at the scale of the cutoff, leading so to the usual difficulties of non-local theories. In this work the conservation laws and causality are investigated for classical field theories with multi-cluster…
We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the…
While fields like Artificial Life have made huge strides in quantifying the mechanisms that distinguish living systems from non-living ones, particular mechanisms remain difficult to reproduce in silico. Known as open-endedness, we've been…
Two-dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n*3 dimensions. Even driven-dissipative deterministic dynamical systems…
Well known biological approximations are universal, i.e. invariant to transformations from one species to another. With no other experimental data, such invariance yields exact conservation (with respect to biological diversity and…
Noether's theorem connects symmetries to invariants in continuous systems, however its extension to discrete systems has remained elusive. Recognizing the lowest-order finite difference as the foundation of local continuity, a viable method…
We look for traveling waves of the semi-discrete conservation law $4\dot u_j +u_{j+1}^2-u_{j-1}^2 = 0$, using variational principles related to concepts of ``hidden convexity'' appearing in recent studies of various PDE (partial…
The purpose of this article is to provide a perspective -- admittedly, a rather subjective one -- of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of…