Related papers: Shallow water soliton dynamics beyond KdV
We derived consistently, according to the second order perturbation approach, the extended KdV equation for an uneven bottom for the case of $\alpha=O(\beta)$ and $\delta=O(\beta^2)$. This equation can be obtained only when the bottom is…
Experiments on 2D random water wave propagation in a large wave tank are analyzed when the effect of dispersion changes. A stereoscopic profilometry technique is used to measure the water surface displacement resolved in both time and space…
The authors of the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" [1] claim that they have derived the full third order perturbed KdV equation for the case of uneven bottom. We…
The extended KdV equation is a nonlinear dispersive wave model that is asymptotically or variationally derived from the full dispersive Euler shallow water waves equations when gravity-capillary and higher order nonlinear effects are taken…
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,…
The magnetohydrodynamic equations system for heavy fluid over an arbitrary surface in shallow water approximation is studied in the present paper. It is shown that simple wave solutions exist only for underlying surfaces that are slopes of…
The results of recent experiments [1] on observing soliton lattices and their dislocations in vertical cylindrical channels filled with immiscible fluids with strongly different viscosities and but slightly different densities are…
Third-order approximate solutions for surface gravity waves in the finite water depth are studied in the context of potential flow theory. This solution provides explicit expressions for the surface elevation, free-surface velocity…
We study large-amplitude, very oblique Alfv\'en waves at low $\beta$, with small gradient length scales, comparable to the ion inertial scale $d_i$. Such waves have large density fluctuations, and slight dispersion from finite-frequency and…
Two-dimensional free-surface potential flows of an ideal fluid over a strongly inhomogeneous bottom are investigated with the help of conformal mappings. Weakly-nonlinear and exact nonlinear equations of motion are derived by the…
Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact…
The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and…
The mild-slope equation and its various modifications aim to model, with varying degrees of success, linear water wave propagation over sloping or undulating seabed topography. However, despite multiple modifications and attempted…
In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The…
New exact solutions to the KdV2 equation (known also as the extended KdV equation) are constructed. The KdV2 equation is a second order approximation of the set of Boussinesq's equations for shallow water waves which in first order…
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)…
The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is…
We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding…
In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough…