Related papers: Wave breaking in the Ostrovsky--Hunter equation
This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography…
Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak…
We study the modulational instability of small-amplitude periodic traveling wave solutions in a dispersion generalized Ostrovsky equation. Specifically, we investigate the invertibility of the associated linearized operator in the vicinity…
This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -\gamma u=0, \end{eqnarray*} with positive $\beta$ and…
Stability of the peaked periodic waves in the reduced Ostrovsky equation has remained an open problem for a long time. In order to solve this problem we obtain sharp bounds on the exponential growth of the $L^2$ norm of co-periodic…
We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitz\'eica equation and prove global existence of small-norm solutions in Sobolev space $H^3(R)$. This scenario is an alternative to finite-time wave…
Many supervised machine learning methods have revolutionised the empirical modelling of complex systems. These empirical models, however, are usually "black boxes" and provide only limited physical explanations about the underlying systems.…
The problem of interest in this article are waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. Perturbation theory is used to identify canonical…
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…
The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the KdV equation. We show…
In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter-Saxton equatio, and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to…
The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute…
We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we revisit the classical periodic traveling waves and for the short pulse model, we…
This paper considers a class of non-local equations that are weakly dispersive perturbations of the inviscid Burgers equation, which includes the Fornberg-Whitham equation as a special case. We precise the known results on finite time…
The Ostrovsky equation is a model for gravity waves propagating down a channel under the influence of Coriolis force. This equation is a modification of the famous Korteweg-de Vries equation and is also Hamiltonian. However the Ostrovsky…
In this paper, we consider the Kakutani-Matsuuchi model which describes the surface elevation of the water-waves under the effect of viscosity. We show wave breaking for the Kakutani-Matsuuchi model, namely, the solution remains bounded but…
This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg-Whitham equation) can exhibit wave breaking for initial data with…
In the context of fluid flows, the coupled Ostrovsky equations arise when two distinct linear long wave modes have nearly coincident phase speeds in the presence of background rotation. In this paper, nonlinear waves in a stratified fluid…
In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water)…
For models describing water waves, Constantin and Escher's works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be…