Related papers: One-dimensional weakly nonlinear model equations f…
Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For…
We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modeling the propagation of large amplitude waves in shallow water. The novelty here is that we allow for a general vorticity, hereby…
We solve the Vlasov equation for the longitudinal distribution function and find stationary wave patterns when the distribution in the energy error is Maxwellian. In the long wavelength limit a stability criterion for linear waves has been…
The turbulent energy flow of the onedimensional Majda-McLaughlin-Tabak equation is studied numerically. The system exhibits weak turbulence for weak driving forces, while weak turbulence coexists with strongly nonlinear intermittent…
In this article we present a system of coupled non-linear PDEs modeling an anisotropic electrokinetic flow. We show the existence of suitable weak solutions in three spatial dimensions, that is weak solutions which fulfill an energy…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
This article deals with rotating magnetohydrodynamic flows of a thin stratified layer of astrophysical plasma in a gravitational field with a free-surface in a vertical external magnetic field. Magnetohydrodynamic equations are obtained in…
We prove the existence of weak solutions to steady, compressible non-Newtonian Navier-Stokes system on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power $r$…
We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on…
The Benjamin-Bona-Mahony (BBM) equation has proven to be a good approximation for the unidirectional propagation of small amplitude long waves in a channel where the crosswise variation can be safely ignored. The…
We present a brief discussion on the nonlinear Schr{\"o}dinger equation for modeling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions that can be connected to the sudden formation of…
A general equation for drift waves is derived incorporating both nonlinear electron density perturbation and ion vorticity effects. It is emphasized that the well-known Hasegawa-Mima (HM) equation for drift waves [A. Hasegawa and K. Mima,…
Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, an approximate solution sequence is constructed, and then a priori estimates are…
The focus of this study is to investigate primary and secondary bifurcations to weakly nonlinear flows (weak branch) in convective rotating spheres in a regime where only strongly nonlinear oscillatory sub- and super-critical flows (strong…
Based on the Vlasov-Maxwell equations describing the self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. In an earlier…
A weakly nonlinear model for two-dimensional Faraday waves over infinite depth is derived and studied. Sideband instability of monochromatic standing waves as well as non-monochromatic solutions are studied analytically. Persistent…
This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as…
We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac…
The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth…
We study the statistical properties of an ensemble of weak gravitational waves interacting nonlinearly in a flat space-time. We show that the resonant three-wave interactions are absent and develop a theory for four-wave interactions in a…