Search results for " PD"
showing 10 items of 651 documents
Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation
2007
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a…
Some evolution equations arising in physics
1983
In this paper we consider a new series of evolution equations generalizing the Korteweg-deVries (KdV) and Burgers equations, and we report recent advances on these equations together with the physical phenomena where they arise. In particular we consider a generalized Burgers' equation and we sketch a method for solution in series by using the theory of Sobolevskij and Tanabe. Then we study the KdV equation with nonuniformity terms and we describe various physical interpretation of this equation. We consider various particular cases in which varying solitonic solutions exist. Also we sketch a unicity theorem. Finally modified Burgers-KdV equations are considered.
Polar Sets in a Nonlinear Potential Theory
1988
In this lecture we discuss nonlinear potential theory based on “A-super-harmonic functions”; the theory can be viewed as a (nonlinear) extension of the classical study of superharmonic functions in ℝn.
The fixed angle scattering problem with a first order perturbation
2021
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measurements up to a natural gauge. We also show that one can recover the full first order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and M. Salo to Hamiltonians with first order perturbations, and it is based on wave equation methods and Carleman estimates.
High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex array
2008
Numerical solutions of Prandtl’s equation and Navier Stokes equations are considered for the two dimensional flow induced by an array of periodic rec- tilinear vortices interacting with an infinite plane. We show how this initial datum develops a separation singularity for Prandtl equation. We investigate the asymptotic validity of boundary layer theory considering numerical solu- tions for the full Navier Stokes equations at high Reynolds numbers.
A direct method to find solutions of some type of coupled Korteweg-de Vries equations using hyperelliptic functions of genus two
2008
Abstract We suggest how one can obtain exact solutions of some type of coupled Korteweg–de Vries equations by means of hyperelliptic functions of genus two.
On the regularity of critical and minimal sets of a free interface problem
2015
We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin [Calc Var. Partial Differential Equations, 1993].
Numerical study of the stability of the Peregrine solution
2017
International audience; The Peregrine solution to the nonlinear Schrödinger equations is widely discussed as a model for rogue waves in deep water. We present here a detailed fully nonlinear numerical study of high accuracy of perturbations of the Peregrine solution as a solution to the nonlinear Schrödinger (NLS) equations.We study localized and nonlocalized perturbations of the Peregrine solution in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.
Existence and orbital stability of standing waves to nonlinear Schr��dinger system with partial confinement
2018
We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -\Delta u_2 + (x_1^2+x_2^2)u_2&= \lambda_2 u_2 + \mu_2 |u_2|^{p_2 -2}u_2 +\beta r_2 |u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \end{equation*} under the constraint \begin{align*} \int_{\mathbb{R}^3}|u_1|^2 \, dx = a_1>0,\quad \int_{\mathbb{R}^3}|u_2|^2 \, dx = a_2>0, \end{align*} where $\mu_1, \mu_2, \beta >0, 2 1, r_1 + r_2 < \frac{10}{3}$. In the system, the parameters $\lambda_1, \lambda_2 \in \R$ are unknown …
INFLUENCE OF UNCERTAINTIES ON PD TUNING
2013
The aim of this work is to present a method for tuning the parameters of PD controller under the influences of the uncertainties, in order to stabilize the position of a rotor supported by active magnetic bearings (AMBs). The uncertainties are relative to mass, transverse and polar moment of inertia of the rotor. The introduction of the uncertainties is due to an incomplete modeled dynamic of the system or in the case the system being subjected to a parametric variation. The presence of the uncertainties produces a set of differences among the values of the output. Poles displacement method is used to reach the asymptotically stability condition characterized by a periodic oscillation durin…