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Discrete & Continuous Dynamical Systems - B

2009 , Volume 12 , Issue 3

A Special Issue on Recent Advances in Hydrodynamics

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Adrian Constantin and  Joachim Escher
2009, 12(3): i-iii doi: 10.3934/dcdsb.2009.12.3i +[Abstract](34) +[PDF](74.9KB)
Over the period 12-23 October 2009, the program "Recent advances in integrable systems of hydrodynamic type'' was organized by us and held at the Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria).

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On the propagation of tsunami waves, with emphasis on the tsunami of 2004
Adrian Constantin
2009, 12(3): 525-537 doi: 10.3934/dcdsb.2009.12.525 +[Abstract](37) +[PDF](200.9KB)
In this review paper we discuss the range of validity of nonlinear dispersive integrable equations for the modelling of the propagation of tsunami waves. For the 2004 tsunami the available measurements and the geophyiscal scales involved rule out a connection between integrable nonlinear wave equations and tsunami dynamics.
Recent progress on particle trajectories in steady water waves
Mats Ehrnström and  Gabriele Villari
2009, 12(3): 539-559 doi: 10.3934/dcdsb.2009.12.539 +[Abstract](39) +[PDF](542.4KB)
We survey recent results on particle trajectories within steady two-dimensional water waves. Particular emphasis is placed on the linear and exact mathematical theory of periodic and symmetric waves, and the effects of a (possibly rotational) background current. The different results vindicate and detail the classical Stokes drift, and also show the transition of orbits when waves propagate into running water. The classical approximation, depicting the trajectories as closed ellipses, is shown to be a mathematical rarity.
Stability of multi antipeakon-peakons profile
Khaled El Dika and  Luc Molinet
2009, 12(3): 561-577 doi: 10.3934/dcdsb.2009.12.561 +[Abstract](52) +[PDF](245.1KB)
The Camassa-Holm equation possesses well-known peaked solitary waves that can travel to both directions. The positive ones travel to the right and are called peakon whereas the negative ones travel to the left and are called antipeakons. Their orbital stability has been established by Constantin and Strauss in [20]. In [28] we have proven the stability of trains of peakons. Here, we continue this study by extending the stability result to the case of ordered trains of anti-peakons and peakons.
Generalised Fourier transform and perturbations to soliton equations
Georgi Grahovski and  Rossen Ivanov
2009, 12(3): 579-595 doi: 10.3934/dcdsb.2009.12.579 +[Abstract](54) +[PDF](262.3KB)
A brief survey of the theory of soliton perturbations is presented. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of "squared solutions'' of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data.
   The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can 'modify' the soliton parameters such as to incorporate the changes caused by the perturbation.
   As illustrative examples the perturbed equations of the KdV hierarchy, in particular the Ostrovsky equation, followed by the perturbation theory for the Camassa-Holm hierarchy are presented.
Infinite propagation speed for a two component Camassa-Holm equation
David Henry
2009, 12(3): 597-606 doi: 10.3934/dcdsb.2009.12.597 +[Abstract](82) +[PDF](181.0KB)
This paper is concerned with the solutions of a two-component generalisation of the Camassa-Holm equation. We examine the propagation behaviour of compactly supported solutions, namely whether solutions which are initially compactly supported will retain this property throughout their time of evolution. In the negative case, where we show that solutions have an infinite speed of propagation, we present a description of how the solutions retain weaker properties throughout their existence time, namely they decay at an exponentially fast rate for the duration of their existence.
Nonlinear resonances of water waves
Elena Kartashova
2009, 12(3): 607-621 doi: 10.3934/dcdsb.2009.12.607 +[Abstract](39) +[PDF](263.0KB)
In the last fifteen years great progress has been made in the understanding of nonlinear resonance dynamics of water waves. Notions of scale- and angle-resonances have been introduced, new type of energy cascade due to nonlinear resonances in the gravity water waves has been discovered, conception of a resonance cluster has been much and successfully employed, a novel model of laminated wave turbulence has been developed, etc. etc. Two milestones in this area of research have to be mentioned: a) development of the $q$-class method which is effective for computing integer points on resonance manifolds, and b) construction of marked planar graphs, instead of classical resonance curves, representing simultaneously all resonance clusters in a finite spectral domain, together with their dynamical systems. Among them, new integrable dynamical systems have been found that can be used for explaining numerical and laboratory results. The aim of this paper is to give a brief overview of our current knowledge about nonlinear resonances among water waves, and finally to formulate the three most important open problems.
KdV theory and the Chilean tsunami of 1960
Raphael Stuhlmeier
2009, 12(3): 623-632 doi: 10.3934/dcdsb.2009.12.623 +[Abstract](42) +[PDF](156.3KB)
We investigate the Chilean tsunami of 1960 to determine the role of KdV dynamics. On the basis of the scales involved, and making use of recent advances, we put on a rigorous foundation the fact that KdV dynamics were not influential in this catastrophic event.
Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation
Shuyin Wu , Joachim Escher and  Zhaoyang Yin
2009, 12(3): 633-645 doi: 10.3934/dcdsb.2009.12.633 +[Abstract](42) +[PDF](178.7KB)
This paper is concerned with the long time behaviour of a weakly dissipative Degasperis-Procesi equation. Our analysis discloses the co-existence of global in time solutions and finite time break down of strong solutions. Our blow-up criterion for the initial profile generalizes considerably results obtained earlier in [32].
On the Hunter--Saxton system
Marcus Wunsch
2009, 12(3): 647-656 doi: 10.3934/dcdsb.2009.12.647 +[Abstract](36) +[PDF](157.1KB)
We show local existence of solutions to a two-component Hunter--Saxton system. Moreover, we prove that the slopes of solutions can become unbounded. Finally, if initial data satisfy appropriate smallness conditions, the associated flow is global.
Blow up and propagation speed of solutions to the DGH equation
Yong Zhou and  Zhengguang Guo
2009, 12(3): 657-670 doi: 10.3934/dcdsb.2009.12.657 +[Abstract](66) +[PDF](212.1KB)
A wave-breaking mechanism for solutions with certain initial profiles and propagation speed are discussed in this paper. Firstly, we apply the best constant to give sufficient condition via an appropriate integral form of the initial data, which guarantees finite time singularity formation for the corresponding solution, then we establish blow up criteria via the conserved quantities. Finally, persistence properties of the strong solutions are presented and infinite propagation speed is also investigated in the sense that the corresponding solution $u(x,t)$ does not have compact spatial support for $t>0$ though $u_0 \in C_0^{\infty}(\mathbb{R})$.

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