Related papers: Wave breaking in a shallow water model
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…
In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions…
Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile,…
In this paper we study wave breaking in the unidirectional non-local wave model describing the motion of a collision-free plasma in a magnetic field. By analyzing the monotonicity and continuity properties of a system of the Riccati-type…
The generation of an undular bore in the vicinity of a wave-breaking point is considered for the integrable Kaup-Boussinesq shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the…
We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive…
The Ostrovsky--Hunter equation governs evolution of shallow water waves on a rotating fluid in the limit of small high-frequency dispersion. Sufficient conditions for the wave breaking in the Ostrovsky--Hunter equation are found both on an…
The propagation of water waves of finite depth and flat bottom is studied in the case when the depth is not small in comparison to the wavelength. This propagation regime is complementary to the long-wave regime described by the famous KdV…
This article addresses linear hyperbolic partial differential equations with non-smooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper…
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…
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…
The modeling of wave breaking dissipation in coastal areas is investigated with a fully nonlinear and dispersive wave model. The wave propagation model is based on potential flow theory, which initially assumes non-overturning waves.…
In this short note, we derive a system of two nonlocal equations for the water-wave problem following the work of [AFM06]. Specifically, we consider a fluid with a one-dimensional free surface for an irrotational fluid both with, and…
Wave-breaking is studied analytically first and the results are compared with accurate numerical simulations of 3D wave-breaking. We focus on the time dependence of various quantities becoming singular at the onset of breaking. The power…
A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on…
We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local…
The paper deals with blow--up for the solutions of wave equation with nonlinear source and nonlinear boudary damping terms, posed in a bounded and regular domain. The initial data are posed in the energy space. The aim of the paper is to…
We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain…
We propose a shallow water model which combines the dispersion relation of water waves and the Boussinesq equations, and which extends the Whitham equation to permit bidirectional propagation. We establish that its sufficiently small,…
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.…