Search results for "Nonlinear equations"
showing 7 items of 7 documents
Asymptotic mean value formulas for parabolic nonlinear equations
2021
In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge–Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of dynamic programming principles for certain two-player, zero-sum games. peerReviewed
Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type
2014
Abstract The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
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…
Modulational instability and generation of self-induced transparency solitons in resonant optical fibers
2009
International audience; We consider continuous-wave propagation through a fiber doped with two-level resonant atoms, which is described by a system of nonlinear Schrodinger-Maxwell-Bloch (NLS-MB) equations. We identify the modulational instability (MI) conditions required for the generation of ultrashort pulses, in cases of both anomalous and normal GVD (group-velocity dispersion). It is shown that the self-induced transparency (SIT) induces non-conventional MI sidebands. The main result is a prediction of the existence of both bright and dark SIT solitons in the anomalous and normal GVD regimes.
Local behaviour of singular solutions for nonlinear elliptic equations in divergence form
2012
We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and $${\mathcal{A}}$$ is a positive C 1(0,1] function which is regularly varying at zero with index $${\vartheta}$$ in (2−N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if $${\Phi\not\in L^q(B_1(0))}$$ , where $${\Phi}$$ denotes the fundamental solution of $${-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}$$ in $${\mathcal D'(B_1(0))}$$ and δ0 is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of al…
A class of quasi-Newton generalized Steffensen methods on Banach spaces
2002
AbstractWe consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich–Ostrowski's conditions. The majorizing sequence will be a Newton's type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.
On Discovering Low Order Models in Biochemical Reaction Kinetics
2007
We develop a method by which a large number of differential equations representing biochemical reaction kinetics may be represented by a smaller number of differential equations. The basis of our technique is a conjecture that the high dimension equations of biochemical kinetics, which involve reaction terms of specific forms, are actually implementing a low dimension system whose behavior requires right hand sides that can not be biochemically implemented. For systems that satisfy this conjecture, we develop a simple approximation scheme based on multilinear algebra that extracts the low dimensional system from simulations of the high dimension system. We demonstrate this technique on a st…