A conservation law with spatially localized sublinear damping

Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function

We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to...

One-dimensional Conservation Law with Boundary Conditions: General Results and Spatially Inhomogeneous Case

The note presents the results of the recent work [5] of K. Sbihi and the author on existence and uniqueness of entropy solutions for boundary-value problem for conservation law ut + φ(u)x = 0 (here, we focus on the simplified one-dimensional setting). Then, using nonlinear semigroup theory, we extend these well-posedness results to the case of spatially dependent flux φ(x, u).

Stabilization of a Boussinesq system with localized damping

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damp...

Particle interactions with spatially localized wavepackets

Spatially Localized Unstable Periodic Orbits

Using an innovative damped-Newton method, we report the first calculation of many distinct unstable periodic orbits (UPOs) of a large high-dimensional extensively chaotic system. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov fractal dimension D = 8.8 can be es...