Related papers: Capillary gravity water waves linearized at monoto…
We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of…
The Lamb-Chaplygin dipole (Lamb1895,Lamb1906,Chaplygin1903) is one of the few closed-form relative equilibrium solutions of the 2D Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear…
By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime…
This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water in a flow of constant vorticity over an impermeable flat bed. The motion of these waves is assumed to be governed both by…
The study of gravity-capillary water waves in two space dimensions has been an important question in mathematical fluid dynamics. By implementing the cubic modified energy method of Ifrim-Tataru in the context of gravity-capillary waves, we…
Processes of propagation and interaction of nonlinear gravity-capillary waves on the free surface of a deep non-conducting liquid with high dielectric constant under the action of a tangential electric field are numerically simulated. The…
We investigate theoretically and experimentally the capillary-gravity waves created by a small object moving steadily at the water-air interface along a circular trajectory. It is well established that, for straight uniform motion, no…
We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show…
In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 \div 0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy…
We review the Analogue Gravity description of a unidirectional water wave system, assuming no prior knowledge of General Relativity or differential geometry. In so doing, we generalize established results concerning an effective curved…
We investigate the Benjamin-Feir instability of small-amplitude gravity-capillary Stokes waves in deep water for the full water wave equations. While modulational instability has been classically predicted by formal asymptotic approaches,…
Although internal gravity waves are generally recognized as an important mechanism to distribute energy through the atmosphere, their dynamics near the instability is only partially understood to date. Many types of instabilities, notably…
We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey…
We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…
In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is…
We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in…
Steady two-dimensional surface capillary-gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions…
In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral…
Frequently encountered in nature, internal solitary waves in stratified fluids are well-observed and well-studied from the experimental, the theoretical, and the numerical perspective. From the mathematical point of view, these waves are…
In this paper, we consider the advective unstable Cahn-Hilliard equation in 2D with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon \Delta^2 u= \Delta(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb…