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1078-0947
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1553-5231
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Discrete & Continuous Dynamical Systems - A
May 2008 , Volume 20 , Issue 2
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We consider the generalized Korteweg-de Vries equation
$\partial_t u + \partial_x (\partial_x^2 u + f(u))=0, (t,x)\in \mathbb{R}\times \mathbb{R}$(1)
with $C^3$ nonlinearity $f$.
Under an explicit condition on $f$ and $c>0$,
there exists a solution of (1) in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton.
In [11],
[13], it was proved that for $f(u)=u^p$, $p=2,3,4$,
the family of solitons
is asymptotically stable in some local sense in $H^1$, i.e.
if $u(t)$ is close to $Q_{c}$,
then $u(t,.+\rho(t))$ locally converges in the energy space
to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+$~$c$ and some function $\rho(t)$ such that $\rho'(t)$~$c^+$.
Then,
in [9] and [14], these results were extended with shorter proofs
under general assumptions on $f$.
The first objective of this paper is to give more information about the function
$\rho(t)$.
In the case $f(u)=u^p$, $p=2,3,4$ and under the additional assumption $x_+ u\in L^2(\mathbb{R})$, we prove
that the function $\rho(t)-c^+ t$ has a finite limit as $t\to +\infty$.
Second, we prove stability and asymptotic stability
results for two solitons for a general nonlinearity $f(u)$, in the case
where the ratio of the speeds of the two solitons is small.
In this paper we correct the proofs of some statements that Colliander, Kenig and Staffilani made for the KP-I initial-value problem in [2]. These corrections actually give stronger well-posedness results than the one claimed in the above mentioned paper. The new proofs are inspired by those used by Ionescu-Kenig ([3, 4, 5]) in works on the Benjamin-Ono equation and on the Schrödinger map problems.
In this paper, we study how stability properties of solutions to general reaction-diffusion systems are related to the diffusion coefficients, response rate, and spatial inhomogeneity. In particular, all eigenvalues of steady states to general shadow systems are completely determined, and some consequences are discussed.
Entropy sets are defined both topologically and for a measure. The set of topological entropy sets is the union of the sets of entropy sets for all invariant measures. For a topological system $(X,T)$ and an invariant measure $\mu$ on $(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of the set of all entropy sets (resp. $\mu$- entropy sets) in the hyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp. $(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has an invariant measure with full support and infinite topological entropy.
Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropy sets) is a dense $G_\delta$ in $H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed.
Concerning entropy capacities, it is shown that when $\mu$ is ergodic with $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ such that the Bowen entropy $h(E)\ge h_\mu(T)$ is residual in $H^\mu(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.
Let $X$ be a separable metric space not necessarily compact, and let $f: X\rightarrow X$ be a continuous transformation. From the viewpoint of Hausdorff dimension, the authors improve Bowen's method to introduce a dynamical quantity distance entropy, written as $ent_{H}(f;Y)$, for $f$ restricted on any given subset $Y$ of $X$; but it is essentially different from Bowen's entropy(1973). This quantity has some basic properties similar to Hausdorff dimension and is beneficial to estimating Hausdorff dimension of the dynamical system. The authors show that if $f$ is a local lipschitzian map with a lipschitzian constant $l$ then $ent_{H}(f;Y)\le\max\{0, $HD$(Y)\log l}$ for all $Y\subset X$; if $f$ is locally expanding with skewness $\lambda$ then $ent_{H}(f;Y)\ge $HD$(Y)\log\lambda$ for any $Y\subset X$. Here HD$(-)$ denotes the Hausdorff dimension. The countable stability of the distance entropy $ent_{H}$ proved in this paper, which generalizes the finite stability of Bowen's $h$-entropy (1971), implies that a continuous pointwise periodic map has the distance entropy zero. In addition, the authors show examples which demonstrate that this entropy describes the real complexity for dynamical systems over noncompact-phase space better than that of various other entropies.
Given a topologically hyperbolic attracting set of a smooth three dimensional Kupka-Smale diffeomorphism, it is proved under some dissipation hypothesis, that either the set is hyperbolic or the diffeomorphism is $C^1-$approximated by another one exhibiting either a heterodimensional cycle or a homoclinic tangency.
We consider equation $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x))$(*) , when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=\kappa>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(\kappa)) =$ [ c*, c * ] such that (*) has at least one (possibly, non-monotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here c*$ >0$ is finite and c * $ \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound c*is sharp so that there are not wavefronts moving with speed $c < $ c*. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations (*) can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. (*) includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with non-monotone nonlinearity.
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