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Discrete & Continuous Dynamical Systems - A

2007 , Volume 19 , Issue 3

Special Issue on
Geometric Mechanics

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Preface
Adrian Constantin and Boris Kolev
2007, 19(3): i-ii doi: 10.3934/dcds.2007.19.3i +[Abstract](379) +[PDF](38.7KB)
Abstract:
The present volume is related to the conference "Geometric Mechanics'', to be held from November 19 to November 23, 2007, at the CIRM (Centre International de Rencontres Mathématiques) in Marseille, France.

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Solitons from the Lagrangian perspective
Adrian Constantin
2007, 19(3): 469-481 doi: 10.3934/dcds.2007.19.469 +[Abstract](571) +[PDF](178.2KB)
Abstract:
The soliton solution of the Korteweg-de Vries equation provides a good approximation to the shape of a solitary wave solution to the governing equations for water waves. However, the corresponding velocity field below the soliton is not an accurate approximation. We propose an approach that provides us with a better approximation. By describing the particle paths below the free surface, we show that the qualitative features of the entire flow in a solitary water wave is captured by our approximation of the velocity field.
Deep-water waves with vorticity: symmetry and rotational behaviour
Mats Ehrnström
2007, 19(3): 483-491 doi: 10.3934/dcds.2007.19.483 +[Abstract](397) +[PDF](153.8KB)
Abstract:
We show that for steady, periodic, and rotational gravity deep-water waves, a monotone surface profile between troughs and crests implies symmetry. It is observed that if the vorticity function has a bounded derivative, then it vanishes as one approaches great depths.
Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation
Joachim Escher, Olaf Lechtenfeld and Zhaoyang Yin
2007, 19(3): 493-513 doi: 10.3934/dcds.2007.19.493 +[Abstract](1590) +[PDF](233.8KB)
Abstract:
After some remarks on a possible zero-curvature formulation we first establish local well-posedness for the 2-component Camassa-Holm equation. Then precise blow-up scenarios for strong solutions to the system are derived. Finally we present two blow-up results for strong solutions to the system.
On unique continuation for the modified Euler-Poisson equations
A. Alexandrou Himonas, Gerard Misiołek and Feride Tiǧlay
2007, 19(3): 515-529 doi: 10.3934/dcds.2007.19.515 +[Abstract](560) +[PDF](203.4KB)
Abstract:
It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.
Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity
Delia Ionescu-Kruse
2007, 19(3): 531-543 doi: 10.3934/dcds.2007.19.531 +[Abstract](534) +[PDF](186.0KB)
Abstract:
We describe the physical hypotheses underlying the derivation of an approximate model of water waves. For unidirectional surface shallow water waves moving over an irrotational flow as well as over a non-zero vorticity flow, we derive the Camassa-Holm equation by an interplay of variational methods and small-parameter expansions.
Conformal and Geometric Properties of the Camassa-Holm Hierarchy
Rossen I. Ivanov
2007, 19(3): 545-554 doi: 10.3934/dcds.2007.19.545 +[Abstract](571) +[PDF](165.2KB)
Abstract:
Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this contribution.
    The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions of some 2+1-dimensional generalizations of CH can be constructed via the IST for the CH hierarchy.
Poisson brackets in Hydrodynamics
Boris Kolev
2007, 19(3): 555-574 doi: 10.3934/dcds.2007.19.555 +[Abstract](454) +[PDF](251.4KB)
Abstract:
This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a valid and usable bracket to study rotational fluid flows with a free boundary. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open.
Global conservative solutions of the Dullin-Gottwald-Holm equation
Octavian G. Mustafa
2007, 19(3): 575-594 doi: 10.3934/dcds.2007.19.575 +[Abstract](485) +[PDF](222.2KB)
Abstract:
A new approach to the analysis of wave-breaking solutions to the Dullin-Gottwald-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of solutions is also built. The solutions have constant $H^{1}$-energy for almost every time with respect to the Lebesgue measure.
On Geometric Mechanics
Jean-Marie Souriau
2007, 19(3): 595-607 doi: 10.3934/dcds.2007.19.595 +[Abstract](526) +[PDF](183.4KB)
Abstract:
This survey paper introduces the reader to the origins of the Geometric Mechanics theory and traces its subsequent history.

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