Related papers: Stability of Lamb dipoles
This article is devoted to stationary solutions of Euler's equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the…
The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital…
We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point…
For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and its vorticity $\omega$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if…
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
In this investigation we revisit the question of the linear stability analysis of 2D steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of…
Zeitlin's model is a discretisation of the 2-D Euler equations that preserves the underlying geometric structure. This feature makes it suitable for studying the qualitative behaviour of the dynamics. Here, we utilise Arnold's geometric…
A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. In the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating…
We prove the asymptotic stability of the equilibrium solution of a class of vector Li\'enard equations by means of LaSalle invariance principle. The key hypothesis consists in assuming that the intersections of the manifolds in $\{\dot V =…
In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main…
This paper is concerned with steady vortex rings in an ideal fluid of uniform density, which are special global axi-symmetric solutions of the three-dimensional incompressible Euler equation. We systematically establish the existence,…
We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $\Omega^*=\alpha\cos (\mathbf{p} \cdot \mathbf{x} )+\beta \sin (\mathbf{p} \cdot…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
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…
In this manuscript we consider the stability of periodic solutions to Lambda-Omega lattice dynamical systems. In particular, we show that an appropriate ansatz casts the lattice dynamical system as an infinite-dimensional fast-slow…
We consider variational principles related to V. I. Arnold's stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined…
In this paper, we consider the degenerate semi-linear Schr\"odinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which…
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which…
We consider the cubic nonlinear Schr\"odinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation…
In this work, we present a numerical study of the wave stability of steady solitary waves over a localised topographic obstacle through the full Euler equations. There are two branches of the solutions: one from the perturbed uniform flow…