Search results for "PDE"

MR3112896 Saichev, Alexander I.; Woyczyński, Wojbor A. Distributions in the physical and engineering sciences. Vol. 2. Linear and nonlinear dynamics …

2014

On a regularized approach for the method of fundamental solution

2018

The method of fundamental solution is a boundary meshless method recently adopted in the framework of non-invasive neu- roimaging techniques. The method approximates the solution of a BVP by a linear combination of fundamental solutions of the governing PDE. A crucial feature of the method is the placement of the fictitious boundary to avoid the singularities of fundamental solutions. In this paper we report on our experiences with a regularized MFS method in the neuroimaging context.

Shock formation in the dispersionless Kadomtsev-Petviashvili equation

2016

The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the $(x,y)$ plane, where the solution of the dKP equation exists in a weak sense only, and a…

Numerical study of the Kadomtsev–Petviashvili equation and dispersive shock waves

2018

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit.

Spectral long-range interaction of temporal incoherent solitons.

2014

We study the interaction of temporal incoherent solitons sustained by a highly noninstantaneous (Raman-like) nonlinear response. The incoherent solitons exhibit a nonmutual interaction, which can be either attractive or repulsive depending on their relative initial distance. The analysis reveals that incoherent solitons exhibit a long-range interaction in frequency space, which is in contrast with the expected spectral short-range interaction described by the usual approach based on the Raman-like spectral gain curve. Both phenomena of anomalous interaction and spectral long-range behavior of incoherent solitons are described in detail by a long-range Vlasov equation.

Impact of self-steepening on incoherent dispersive spectral shocks and collapse-like spectral singularities

2014

International audience; Incoherent dispersive shock waves and collapselike singularities have been recently predicted to occur in the spectral evolution of an incoherent optical wave that propagates in a noninstantaneous nonlinear medium. Here we extend this work by considering the generalized nonlinear Schrödinger equation. We show that self-steepening significantly affects these incoherent spectral singularities: (i) It leads to a delay in the development of incoherent dispersive shocks, and (ii) it arrests the incoherent collapse singularity. Furthermore, we show that the spectral collapselike behavior can be exploited to achieve a significant enhancement (by two orders of magnitudes) of…

Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

2011

Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\mu$, where $Q>1$. Suppose that $(X,d,\mu)$ supports a (local) $(1,2)$-Poincar\'e inequality and a suitable curvature lower bound. For the Poisson equation $\Delta u=f$ on $(X,d,\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\"older continuity with optimal exponent of solutions is obtained.

Maximal potentials, maximal singular integrals, and the spherical maximal function

2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

Regularity of the Inverse of a Sobolev Homeomorphism

2011

We give necessary and sufficient conditions for the inverse ofa Sobolev homeomorphism to be a Sobolev homeomorphism and conditions under which the inverse is of bounded variation.

Generalized dimension estimates for images of porous sets under monotone Sobolev mappings

2014

We give an essentially sharp estimate in terms of generalized Hausdorff measures for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions.