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Discrete & Continuous Dynamical Systems - A

July 2019 , Volume 39 , Issue 7

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A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation
John P. Albert
2019, 39(7): 3635-3670 doi: 10.3934/dcds.2019149 +[Abstract](934) +[HTML](152) +[PDF](580.18KB)
Abstract:

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particular we prove that this is indeed the case for \begin{document}$ 2 $\end{document}-soliton solutions of the fourth-order stationary equation.

Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations
Shengfu Deng
2019, 39(7): 3671-3716 doi: 10.3934/dcds.2019150 +[Abstract](841) +[HTML](148) +[PDF](633.37KB)
Abstract:

The KdV-KdV system of Boussinesq equations belongs to the class of Boussinesq equations modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid. It has been numerically shown that this system possesses solutions with two humps which tend to a periodic solution with much smaller amplitude at infinity (called generalized two-hump wave solutions). This paper presents the first rigorous proof. The traveling form of this system can be formulated into a dynamical system with dimension 4. The classical dynamical system approach provides the existence of a solution with an exponentially decaying part and an oscillatory part (small-amplitude periodic solution) at positive infinity, which has a single hump at the origin and is reversible near negative infinity if some free constants, such as the amplitude and the phase shit of the periodic solution, are activated. This eventually yields a generalized two-hump wave solution. The method here can be applied to obtain generalized $2^k$-hump wave solutions for any positive integer $k$.

Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains
Dingshi Li, Kening Lu, Bixiang Wang and Xiaohu Wang
2019, 39(7): 3717-3747 doi: 10.3934/dcds.2019151 +[Abstract](833) +[HTML](132) +[PDF](619.13KB)
Abstract:

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in ${\mathbb R}^{n+1}$ and an equation on a domain in ${\mathbb R}^{n}$ are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors.

Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces
Wen Tan, Bo-Qing Dong and Zhi-Min Chen
2019, 39(7): 3749-3765 doi: 10.3934/dcds.2019152 +[Abstract](976) +[HTML](127) +[PDF](531.3KB)
Abstract:

This paper is devoted to the study of the modified quasi-geostrophic equation

in \begin{document}$ \mathbb{R}^2 $\end{document}. By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case \begin{document}$ \beta = \alpha-1 $\end{document} and the existence of regular solutions for large time \begin{document}$ t>T $\end{document} with respect to the supercritical case \begin{document}$ \beta >\alpha -1 $\end{document} in Besov spaces. Earlier results for the equation in Hilbert spaces \begin{document}$ H^s $\end{document} spaces are improved.

Classification of linear skew-products of the complex plane and an affine route to fractalization
Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó and Joan Carles Tatjer
2019, 39(7): 3767-3787 doi: 10.3934/dcds.2019153 +[Abstract](694) +[HTML](117) +[PDF](1800.55KB)
Abstract:

Linear skew products of the complex plane,

where \begin{document}$ \theta\in {\mathbb T} $\end{document}, \begin{document}$ z\in {\mathbb C} $\end{document}, \begin{document}$ \frac{\omega}{2\pi} $\end{document} is irrational, and \begin{document}$ \theta\mapsto a(\theta) \in {\mathbb C}\setminus \{0\} $\end{document} is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of \begin{document}$ \theta\mapsto a(\theta) $\end{document}. We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present.

Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions
Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas and Kun Zhao
2019, 39(7): 3789-3838 doi: 10.3934/dcds.2019154 +[Abstract](783) +[HTML](127) +[PDF](6535.99KB)
Abstract:

We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in $H^3(\Omega)$, under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as the chemical diffusion rate tends to zero under the Neumann-Stress-free type boundary conditions. Numerical analysis is performed for a discretization of the model with the Dirichlet-Stress-free type boundary conditions, and a monotonic exponential decay to the equilibrium solution (analogous to the continuous case) is proven. Numerical simulations are supplemented to illustrate the exponential decay, test the assumptions of the exponential decay theorem, and to predict boundary layer formation under the Dirichlet-Stress-free type boundary conditions.

General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay
Ammar Khemmoudj and Yacine Mokhtari
2019, 39(7): 3839-3866 doi: 10.3934/dcds.2019155 +[Abstract](872) +[HTML](169) +[PDF](529.54KB)
Abstract:

In this paper we consider a viscoelastic modified nonlinear Von-Kármán system with a linear delay term. The well posedness of solutions is proved using the Faedo-Galerkin method. We use minimal and general conditions on the relaxation function and establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $
E. Compaan and N. Tzirakis
2019, 39(7): 3867-3895 doi: 10.3934/dcds.2019156 +[Abstract](707) +[HTML](133) +[PDF](545.05KB)
Abstract:

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrödinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the \begin{document}$L^2$\end{document} conservation law of the Schrödinger part with a careful iteration of the rough wave part in lower order Sobolev norms in the spirit of the work in [5].

The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations
Hayato Chiba and Georgi S. Medvedev
2019, 39(7): 3897-3921 doi: 10.3934/dcds.2019157 +[Abstract](714) +[HTML](114) +[PDF](659.33KB)
Abstract:

In our previous work [3], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model.

In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Rényi, small-world, as well as certain weighted graphs on a circle.

The twisted cohomological equation over the geodesic flow
Zhenqi Jenny Wang
2019, 39(7): 3923-3940 doi: 10.3934/dcds.2019158 +[Abstract](726) +[HTML](108) +[PDF](477.65KB)
Abstract:

We study the twisted cohomoligical equation over the geodesic flow on \begin{document}$ SL(2, \mathbb{R} )/\Gamma $\end{document}. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in products of rank-one groups in future works.

Weighted topological and measure-theoretic entropy
Tao Wang and Yu Huang
2019, 39(7): 3941-3967 doi: 10.3934/dcds.2019159 +[Abstract](781) +[HTML](129) +[PDF](527.8KB)
Abstract:

Feng and Huang in 2016 defined a new notion called weighted topological entropy (pressure) and obtained the corresponding variational principle for compact dynamical systems. In this paper, it was our hope to carry out a further study from the following three aspects:

(1) Inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships.

(2) The classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version.

(3) A formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.

Cohomological equation and cocycle rigidity of discrete parabolic actions
James Tanis and Zhenqi Jenny Wang
2019, 39(7): 3969-4000 doi: 10.3934/dcds.2019160 +[Abstract](599) +[HTML](109) +[PDF](543.67KB)
Abstract:

We study the cohomological equation for discrete horocycle maps on \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} and \begin{document}$ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $\end{document} via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document}. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of \begin{document}$ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $\end{document}, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} in which all cases of irreducible, unitary representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} can be studied simultaneously.

Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

Traveling waves for nonlocal models of traffic flow
Johanna Ridder and Wen Shen
2019, 39(7): 4001-4040 doi: 10.3934/dcds.2019161 +[Abstract](707) +[HTML](134) +[PDF](868.62KB)
Abstract:

We consider several nonlocal models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with nonlocal flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition.

Minimality and gluing orbit property
Peng Sun
2019, 39(7): 4041-4056 doi: 10.3934/dcds.2019162 +[Abstract](720) +[HTML](115) +[PDF](355.11KB)
Abstract:

We show that a dynamical system with gluing orbit property is either minimal or of positive topological entropy. Moreover, for equicontinuous systems, we show that topological transitivity, minimality and orbit gluing property are equivalent. These facts reflect the similarity and dissimilarity of gluing orbit property with specification like properties.

Asymptotically sectional-hyperbolic attractors
B. San Martín and Kendry J. Vivas
2019, 39(7): 4057-4071 doi: 10.3934/dcds.2019163 +[Abstract](714) +[HTML](114) +[PDF](1603.11KB)
Abstract:

The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.

On the oscillation behavior of solutions to the one-dimensional heat equation
Dong-Ho Tsai and Chia-Hsing Nien
2019, 39(7): 4073-4089 doi: 10.3934/dcds.2019164 +[Abstract](661) +[HTML](103) +[PDF](443.38KB)
Abstract:

We study the oscillation behavior of solutions to the one-dimensional heat equation and give some interesting examples. We also demonstrate a simple ODE method to find explicit solutions of the heat equation with certain particular initial conditions.

Random dynamics of fractional nonclassical diffusion equations driven by colored noise
Renhai Wang, Yangrong Li and Bixiang Wang
2019, 39(7): 4091-4126 doi: 10.3934/dcds.2019165 +[Abstract](687) +[HTML](152) +[PDF](683.22KB)
Abstract:

The random dynamics in \begin{document}$ H^s(\mathbb{R}^n) $\end{document} with \begin{document}$ s\in (0,1) $\end{document} is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

Stability and separation property of radial solutions to semilinear elliptic equations
Shoichi Hasegawa
2019, 39(7): 4127-4136 doi: 10.3934/dcds.2019166 +[Abstract](704) +[HTML](118) +[PDF](399.63KB)
Abstract:

We study stability and separation property of solutions to Hénontype equations. In particular, assuming separation property of radial solutions, we shall show the stability of solutions. Moreover, we shall also study those properties of solutions to generalized Eddington equations.

Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case
José Godoy, Nolbert Morales and Manuel Zamora
2019, 39(7): 4137-4156 doi: 10.3934/dcds.2019167 +[Abstract](764) +[HTML](152) +[PDF](378.37KB)
Abstract:

We analyze the existence of T−periodic solutions to the second-order indefinite singular equation

which depends on a positive parameter β > 0. Here, h is a sign-changing function with h = 0 and where the nonlinear term of the equation has two singularities. For the first time, the degenerate case is studied, displaying an unexpected feature which contrasts with the results known in the literature for indefinite singular equations.

Scaling properties of the Thue–Morse measure
M. Baake, P. Gohlke, M. Kesseböhmer and T. Schindler
2019, 39(7): 4157-4185 doi: 10.3934/dcds.2019168 +[Abstract](721) +[HTML](118) +[PDF](298.27KB)
Abstract:

The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $
Zhongjie Liu and Duanzhi Zhang
2019, 39(7): 4187-4206 doi: 10.3934/dcds.2019169 +[Abstract](648) +[HTML](121) +[PDF](490.5KB)
Abstract:

In this paper, we consider multiplicity of brake orbits on compact symmetric dynamically convex reversible hypersurfaces in $\mathbb{R}^{2n}$. We prove that there exist at least \([\frac{n+1}{2}]\) geometrically distinct closed characteristics on dynamically convex hypersurface $\Sigma$ in $R^{2n}$ with the symmetric and reversible conditions, i.e. $\Sigma = -\Sigma$ and $N\Sigma = \Sigma$, where $N = diag(-I_{n}, I_{n})$. For $n\geq2$, we prove that there are at least 2 symmetric brake orbits on $\Sigma$, which generalizes Kang's result in [8] of 2013.

Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts
Tim Gutjahr and Karsten Keller
2019, 39(7): 4207-4224 doi: 10.3934/dcds.2019170 +[Abstract](694) +[HTML](123) +[PDF](495.43KB)
Abstract:

In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.

On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability
Xuemei Li and Zaijiu Shang
2019, 39(7): 4225-4257 doi: 10.3934/dcds.2019171 +[Abstract](689) +[HTML](138) +[PDF](630.37KB)
Abstract:

In this paper, we establish a KAM-theorem about the existenceof invariant tori in non-conservative dynamical systems with finitely differentiable vector fields and multiple degeneracies under the assumption that theintegrable part is finitely differentiable with respect to parameters, instead ofthe usual assumption of analyticity. We prove these results by constructingapproximation and inverse approximation lemmas in which all functions arefinitely differentiable in parameters.

A polygonal scheme and the lower bound on density for the isentropic gas dynamics
Geng Chen, Ronghua Pan and Shengguo Zhu
2019, 39(7): 4259-4277 doi: 10.3934/dcds.2019172 +[Abstract](676) +[HTML](105) +[PDF](846.5KB)
Abstract:

Positive density lower bound is one of the major obstacles toward large data theory for one dimensional isentropic compressible Euler equations, also known as p-system in Lagrangian coordinates. The explicit example first studied by Riemann shows that the lower bound of density can decay to zero as time goes to infinity of the order \begin{document}$ O\left( {\frac{1}{{1 + t}}} \right)$\end{document}, even when initial density is uniformly positive. In this paper, we establish a proof of the lower bound on density in its optimal order \begin{document}$ O\left( {\frac{1}{{1 + t}}} \right)$\end{document} using a method of polygonal scheme.

The semirelativistic Choquard equation with a local nonlinear term
Bartosz Bieganowski and Simone Secchi
2019, 39(7): 4279-4302 doi: 10.3934/dcds.2019173 +[Abstract](692) +[HTML](113) +[PDF](476.45KB)
Abstract:

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in \begin{document}$ \mathbb{R}^N $\end{document}

where \begin{document}$ m > 0 $\end{document} and the potential \begin{document}$ V $\end{document} is decomposed as the sum of a \begin{document}$ \mathbb{Z}^N $\end{document}-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space \begin{document}$ \mathbb{R}_{+}^{N+1} $\end{document}.

The Boltzmann equation with frictional force for very soft potentials in the whole space
Yingzhe Fan and Yuanjie Lei
2019, 39(7): 4303-4329 doi: 10.3934/dcds.2019174 +[Abstract](649) +[HTML](106) +[PDF](501.24KB)
Abstract:

We develop a general energy method for proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, for both hard potentials and soft potentials. With the help of this method, we establish the global existence and temporal convergence rates of solution near a given global Maxwellian to the Cauchy problem on the Boltzmann equation with frictional force for very soft potentials i.e. \begin{document}$ -3<\gamma<-2 $\end{document}.

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