Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation, incorporatin...
We introduce an intuitive and semi-analytical polaron master equation approach to model pulse-driven population inversion and emitted single photons from a quantum dot exciton. The master equation theory allows one to identify important phonon-induce...
In computer graphics, the rendering equation is an integral equation that expresses the amount of light leaving a point on a surface as the sum of emitted
We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $Γ$ to optimal regularity, one derivative smoother than the Riemann curvatu...
We study the rate of decay of the energy functional of solutions of the wave equation with localized damping and a external force. We prove that the decay rates of the energy functional is determined from a forced differential equation....
This paper deals with nonlinear parabolic equation for which a local solution in time exists and then blows up in a finite time. We consider the Chipot-Weissler equation. We study the numerical approximation, we show that the numerical solution conve...
A fundamental relation in celestial mechanics is Kepler's equation, linking an orbit's mean anomaly to its eccentric anomaly and eccentricity. Being transcendental, the equation cannot be directly solved for eccentric anomaly by conventional treatmen...
Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physic...
Based on the MRDP theorem concerning the Hilbert tenth problem, there is a corresponding Diophantine equation called proof equation for every formula of the First-order Peano Arithmetic (PA). A formula is provable in PA, if and only if the correspo...
Davydova-Lashkin-Fokas-Lenells equation (DLFLE) is a gauged equivalent form of Fokas-Lenells equation (FLE) that addresses both spatio-temporal dispersion (STD) and nonlinear dispersion (ND) effects. The balance between those effects results a solito...
Starting from expressions in Connor et al. (1988) [1], we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna and Callen (1984) [2] and by Nishimura et al (1998) [3], but for more realistic toroidal equilib...
I reconsider Einstein's 1912 "Prague-Theory" of static gravity based on a scalar field obeying a non-linear field equation. I point out that this equation follows from the self-consistent implementation of the principle that all energies source the g...
This paper aims to show that making use of Newton's view on equations of motion of a physical system and of the Maxwell stress tensor we come to a natural nonlinearization of Maxwell equations in vacuum making use only of nonrelativistic terms. The...
We propose a simple algebraic method for generating classes of traveling wave solutions for a variety of partial differential equations of current interest in nonlinear science. This procedure applies equally well to equations which may or may not...
We apply the homotopy perturbation method to construct series solutions for the fractional Rosenau-Hyman (fRH) equation and study their dynamics. Unlike the classical RH equation where compactons arise from truncated periodic solutions, we show that...
We present a new set of kinematic equations for front motion in bistable media. The equations extend earlier kinematic approaches by coupling the front curvature with the order parameter associated with a parity breaking front bifurcation. In addit...
above equation, the first two terms are the modified Young's equation, while the third term is due to the Laplace pressure. This nonlinear equation correctly
Young's equation may refer to: Young–Laplace equation, describes the capillary pressure difference sustained across the interface between two static fluids
The gyrokinetic Vlasov-Maxwell equations are cast as an infinite-dimensional Hamiltonian system. The gyrokinetic Poisson bracket is remarkably simple and similar to the Morrison-Marsden-Weinstein bracket for the Vlasov-Maxwell equations. By identifyi...
In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a $q$-difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete...