91 research outputs found
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon
This paper deals with the dead-water phenomenon, which occurs when a ship
sails in a stratified fluid, and experiences an important drag due to waves
below the surface. More generally, we study the generation of internal waves by
a disturbance moving at constant speed on top of two layers of fluids of
different densities. Starting from the full Euler equations, we present several
nonlinear asymptotic models, in the long wave regime. These models are
rigorously justified by consistency or convergence results. A careful
theoretical and numerical analysis is then provided, in order to predict the
behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit
On the Galilean invariance of some dispersive wave equations
Surface water waves in ideal fluids have been typically modeled by asymptotic
approximations of the full Euler equations. Some of these simplified models
lose relevant properties of the full water wave problem. One of them is the
Galilean symmetry, which is not present in important models such as the BBM
equation and the Peregrine (Classical Boussinesq) system. In this paper we
propose a mechanism to modify the above mentioned classical models and derive
new, Galilean invariant models. We present some properties of the new
equations, with special emphasis on the computation and interaction of their
solitary-wave solutions. The comparison with full Euler solutions shows the
relevance of the preservation of Galilean invariance for the description of
water waves.Comment: 29 pages, 13 figures, 2 tables, 71 references. Other author papers
can be downloaded at http://www.denys-dutykh.com
Variational derivation of two-component Camassa-Holm shallow water system
By a variational approach in the Lagrangian formalism, we derive the
nonlinear integrable two-component Camassa-Holm system (1). We show that the
two-component Camassa-Holm system (1) with the plus sign arises as an
approximation to the Euler equations of hydrodynamics for propagation of
irrotational shallow water waves over a flat bed. The Lagrangian used in the
variational derivation is not a metric.Comment: to appear in Appl. Ana
Practical use of variational principles for modeling water waves
This paper describes a method for deriving approximate equations for
irrotational water waves. The method is based on a 'relaxed' variational
principle, i.e., on a Lagrangian involving as many variables as possible. This
formulation is particularly suitable for the construction of approximate water
wave models, since it allows more freedom while preserving the variational
structure. The advantages of this relaxed formulation are illustrated with
various examples in shallow and deep waters, as well as arbitrary depths. Using
subordinate constraints (e.g., irrotationality or free surface impermeability)
in various combinations, several model equations are derived, some being
well-known, other being new. The models obtained are studied analytically and
exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Computer Algebra Applied to a Solitary Waves Study
International audienceWe apply computer algebra techniques, such as algebraic computations of resultants and discriminants, certified drawing (with a guaranteed topology) of plane curves, to a problem in fluid dynamics: We investigate " capillary-gravity " solitary waves in shallow water, relying on the framework of the Serre-Green-Naghdi equations. So, we deal with two-dimensional surface waves, propagating in a shallow water of constant depth. By a differential elimination process, the study reduces to describing the solutions of an ordinary non linear first order differential equation, depending on two parameters. The paper is illustrated with examples and pictures computed with the computer algebra system Maple
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