# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 2009 , Volume 23 , Issue 4

A special issue on
Asymptotic Description of Natural Phenomena

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2009, 23(4): i-ii doi: 10.3934/dcds.2009.23.4i +[Abstract](1204) +[PDF](35.1KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems grows out of a focused session at the AIMS 6 meeting held in Poitiers, France, in 2006. The session struck its organizers, who were perhaps not unbiased, as very successful. As a consequence, when Shouchuan Hu approached us about editing a special issue based on the papers delivered in the session, we were enthusiastic at the prospect.
The focus of the session was asymptotic models of physical phenomena. This is a large subject, and one special session cannot hope to do it justice. Consequently, some focal point was required, and in the event, most of the lectures in the session were centered upon nonlinear wave equations arising in plasma physics and fluid mechanics. This is also the subject around which most of this special issue turns.

2009, 23(4): 1099-1139 doi: 10.3934/dcds.2009.23.1099 +[Abstract](1479) +[PDF](1519.6KB)
Abstract:
Laser sources nowadays deliver optical pulses reaching few cycles in duration and peak powers exceeding several terawatt (TW). When such pulses propagate in transparent media, they first self-focus in space, until they generate a tenuous plasma by photo-ionization. These pulses evolve as self-guided objects, resulting from successive equilibria between the Kerr focusing process, the defocusing action of the electron plasma and the chromatic dispersion of the medium. Discovered ten years ago, this self-channeling mechanism reveals a new physics, having direct applications in the long-distance propagation of TW beams in air, supercontinuum emission as well as pulse self-compression. This review presents the major progress in this field. Particular emphasis is laid to the derivation of the propagation equations, for single as well as coupled wave components. Physics is discussed from numerical simulations and explained by analytical arguments. Attention is also paid to the multifilamentation instability, which breaks up broad beams into small-scale cells. Several experimental data validate theoretical descriptions.
2009, 23(4): 1141-1168 doi: 10.3934/dcds.2009.23.1141 +[Abstract](1480) +[PDF](296.7KB)
Abstract:
Studied here is the large-time behavior and eventual periodicity of solutions of initial-boundary-value problems for the BBM equation and the KdV equation, with and without a Burgers-type dissipation appended. It is shown that the total energy of a solution of these problems grows at an algebraic rate which is in fact sharp for solutions of the associated linear equations. We also establish that solutions of the linear problems are eventually periodic if the boundary data are periodic.
2009, 23(4): 1169-1190 doi: 10.3934/dcds.2009.23.1169 +[Abstract](1389) +[PDF](1591.3KB)
Abstract:
We present here a highly efficient and accurate numerical scheme for initial and boundary value problems of a two-dimensional Boussinesq system which describes three-dimensional water waves over a moving and uneven bottom with surface pressure variation. The scheme is then used to study in details the waves generated from rectangular sources and the two-dimensional wave patterns.
2009, 23(4): 1191-1204 doi: 10.3934/dcds.2009.23.1191 +[Abstract](1894) +[PDF](1239.0KB)
Abstract:
We consider a Boussinesq system of BBM-BBM type in two space dimensions. This system approximates the three-dimensional Euler equations and consists of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. We show that the initial-boundary value problem for this system, posed on a bounded smooth plane domain with homogeneous Dirichlet or Neumann or reflective (mixed) boundary conditions, is locally well-posed in $H^1$. After making some remarks on the temporal interval of validity of these models, we discretize the system by a standard Galerkin-finite element method and present the results of some numerical experiments aimed at simulating two-dimensional surface wave flows in complex plane domains with a variety of initial and boundary conditions.
2009, 23(4): 1205-1240 doi: 10.3934/dcds.2009.23.1205 +[Abstract](1513) +[PDF](318.0KB)
Abstract:
The Benjamin-Ono equation is known as a model of internal long waves in stratified fluids or two-fluid systems. In this paper, we consider the validity of this type of modeling of a two-phase problem for capillary-gravity waves, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. We show that the solutions of the free boundary problem split up into two waves and the shape of each wave is governed by the Benjamin-Ono equation in a slow time scale.
2009, 23(4): 1241-1252 doi: 10.3934/dcds.2009.23.1241 +[Abstract](2022) +[PDF](201.7KB)
Abstract:
The regularized long-wave or BBM equation

$u_{t}+u_{x}+u u_{x}-u_{x x t} = 0$

was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg-de Vries equation. Considered here is the initial-value problem wherein $u$ is specified everywhere at a given time $t = 0$, say, and inquiry is then made into its further development for $t>0$. It is proven that this initial-value problem is globally well posed in the $L^2$-based Sobolev class $H^s$ if $s \geq 0$. Moreover, the map that associates the relevant solution to given initial data is shown to be smooth. On the other hand, if $s < 0$, it is demonstrated that the correspondence between initial data and putative solutions cannot be even of class $C^2$. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in $H^s$-based spaces for $s < 0$. One is thus led to surmise that the initial-value problem for the BBM equation is not even locally well posed in $H^s$ for negative values of $s$.

2009, 23(4): 1253-1275 doi: 10.3934/dcds.2009.23.1253 +[Abstract](1688) +[PDF](257.0KB)
Abstract:
In this essay, we study the initial-value problem

$u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,$
$u(x,0)=u_0(x), \qquad x\in\mathbb R, (0.1)$

where $u=u(x,t)$ is a real-valued function, $L$ is a Fourier multiplier operator with real symbol $\alpha(\xi),$ say, and $g$ is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for $L_p$, $H^s$ and bore-like or kink-like initial data. In the special case where $\alpha(\xi)=|\xi|^{r}$ wherein $r>1$ and $g(u)=1/2u^2,$ (0.1) is globally well-posed in time if $s$ and $r$ satisfy a simple algebraic relation.

2009, 23(4): 1277-1294 doi: 10.3934/dcds.2009.23.1277 +[Abstract](1582) +[PDF](256.0KB)
Abstract:
We study the global well-posedness (GWP) and small data scattering of radial solutions of the semirelativistic Hartree type equations with nonlocal nonlinearity $F(u) = \lambda (|\cdot|^{-\gamma}$ * $|u|^2)u$, $\lambda \in \mathbb{R} \setminus \{0\}$, $0 < \gamma < n$, $n \ge 3$. We establish a weighted $L^2$ Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.
2009, 23(4): 1295-1311 doi: 10.3934/dcds.2009.23.1295 +[Abstract](1548) +[PDF](220.8KB)
Abstract:
We prove the local-well posedness of the generalized Benjamin-Ono equations in $H^1(\T)$.
2009, 23(4): 1313-1325 doi: 10.3934/dcds.2009.23.1313 +[Abstract](1715) +[PDF](216.1KB)
Abstract:
We consider the Cauchy problem for the damped nonlinear Schrödinger equations, and prove some blowup and global existence results which depend on the size of the damping coefficient. We also discuss the $L^2$ concentration phenomenon of blowup solutions in the critical case.

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