Related papers: Numerical bifurcation and stability for the capill…
Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical…
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the authors established the well-posedness and…
In this paper, two-dimensional periodic capillary-gravity waves travelling under the effect of a vertical electric field are considered. The full system is a nonlinear, two-layered and free boundary problem. The interface dynamics arises…
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional…
We study travelling-wave spatially periodic solutions of a forced Cahn-Hilliard equation. This is a model for phase separation of a binary mixture, subject to external forcing. We look at arbitrary values of the mean mixture concentration,…
In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but…
Recent experimental observations have demonstrated interesting instability phenomenon during thermal drawing of microstructured glass/polymer fibers, and these observations motivate us to examine surface-tension-driven instabilities in…
We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin-Bona-Mahony and Camassa-Holm equations. To prove stability, we use the abstract results of Grillakis-Shatah-Strauss and the Floquet theory…
In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and…
Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A…
We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of the shallow water equations, provided that the slope…
A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation…
We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.
We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling…
Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincar\'e--Hopf bifurcation of traveling-wave solutions of viscous conservation…
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and…
We present a stability and convergence analysis of the space-time continuous finite element method for the Hamiltonian formulation of the wave equation. More precisely, we prove a continuous dependence of the discrete solution on the data…
Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic…
A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi…
We derive the Whitham modulation equations for the Zakharov-Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham…