Related papers: Numerical bifurcation and stability for the capill…
We study fully three-dimensional droplets that slide down an incline by employing a thin-film equation that accounts for capillarity, wettability, and a lateral driving force in small-gradient (or long-wave) approximation. In particular, we…
We consider propagation of instability fronts in conservative nonlinear wave systems by the Whitham method. Whitham modulation equations for periodic solutions of the generalized Klein-Gordon equation are solved in the limit of…
We consider a layer of an inviscid fluid with free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency)…
We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time…
We analyze the weak solution concept for the Fornberg-Whitham equation in case of traveling waves with a piecewise smooth profile function. The existence of discontinuous weak traveling wave solutions is shown by means of analysis of a…
We study stationary, periodic solutions to the thermocapillary thin-film model \begin{equation*} \partial_t h + \partial_x \Bigl(h^3(\partial_x^3 h - g\partial_x h) + M\frac{h^2}{(1+h)^2}\partial_xh\Bigr) = 0,\quad t>0,\ x\in \mathbb{R},…
The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of…
We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and…
We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass…
This paper establishes suficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas.…
In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial…
We derive the Whitham modulation equations for the nonlinear Schr\"odinger equation in the plane (2d NLS) with small dispersion. The modulation equations are derived in terms of both physical and Riemann variables; the latter yields…
We study the onset of the wave-resistance due to the generation of capillary-gravity waves by a partially immersed moving object in the case where the object is hold at a fixed immersion depth. We show that, in this case, the wave…
Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid models is considered. These models have linear dynamics with discrete spectra, when…
The inclusion of stochastic terms in equations of motion for fluid problems enables a statistical representation of processes which are left unresolved by numerical computation. Here, we derive stochastic equations for the behaviour of…
In this paper, we study the three-dimensional gravity-capillary water wave problem involving an irrotational, perfect fluid with gravity and surface tension. We focus on steady waves propagating uniformly in one direction. Assuming constant…
In this work, we study the nonlinear traveling waves in density stratified fluids with depth varying shear currents. Beginning the formulation of the water-wave problem due to [1], we extend the work of [4] and [18] to examine the interface…
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schr\"odinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it…
The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in…
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…