Related papers: Computation and Stability of Traveling Waves in Se…
We consider the damped hyperbolic equation in one space dimension $\epsilon u_{tt} + u_t = u_{xx} + F(u)$, where $\epsilon$ is a positive, not necessarily small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the interval…
In previous works [4, 5], existence and uniqueness of travelling waves for the nonlinear Schr\"odinger equations have been shown for speeds close to the speed of sound. Furthermore, it has been proved that a chain of dark solitons of…
The Kawahara equation is a weakly nonlinear long-wave model of dispersive waves that emerges when leading order dispersive effects are in balance with the next order correction. Traveling wave solutions of the Kawahara equation satisfy a…
The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the…
In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show…
The main goal of this paper is to present orbital stability results of periodic standing waves for the one-dimensional logarithmic Klein-Gordon equation. To do so, we first use compactness arguments and a non-standard analysis to obtain the…
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…
We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schr\"odinger equations in one space dimension.
We study the stability and dynamics of traveling-front solutions of a modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion…
This note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in \cite{BNPR} but their asymptotic behavior was still unclear. We use here a…
In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow…
In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of…
We investigate numerically a model consisting in a kinetic equation for the biased motion of bacteria following a run-and-tumble process, coupled with two reaction-diffusion equations for chemical signals. This model exhibits asymptotic…
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling-wave solutions of viscous systems of conservation laws for systems of…
In this paper, we present the first result concerning the orbital stability of periodic traveling waves for the modified Kawahara equation. Our method is based on the Fourier expansion of the periodic wave in order to know the behaviour of…
We investigate the existence and spectral stability of traveling wave solutions for a class of fourth-order semilinear wave equations, commonly referred to as beam equations. Using variational methods based on a constrained maximization…
We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with…
The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics,…
We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where data is…
A nonlinear Schr\"odinger equation with repulsive (defocusing) nonlinearity is considered. As an example, a system with a spatially varying coefficient of the nonlinear term is studied. The nonlinearity is chosen to be repelling except on a…