# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2019 , Volume 39 , Issue 8

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2019, 39(8): 4331-4344 doi: 10.3934/dcds.2019175 +[Abstract](636) +[HTML](101) +[PDF](430.22KB)
Abstract:

Given a one-dimensional shift \begin{document}$X$\end{document} and a word \begin{document}$v$\end{document} in the language of \begin{document}$X$\end{document}, the follower set of \begin{document}$v$\end{document} is the set of all finite words which can legally follow \begin{document}$v$\end{document} in some point of \begin{document}$X$\end{document}. The predecessor set of \begin{document}$v$\end{document} is the set of all finite words which can legally precede \begin{document}$v$\end{document} in some point of \begin{document}$X$\end{document}. We construct the follower set sequence of \begin{document}$X$\end{document} by recording, for each \begin{document}$n$\end{document}, the number of distinct follower sets of words of length \begin{document}$n$\end{document} in \begin{document}$X$\end{document}. We construct the predecessor set sequence of \begin{document}$X$\end{document} by recording, for each \begin{document}$n$\end{document}, the number of distinct predecessor sets of words of length \begin{document}$n$\end{document} in \begin{document}$X$\end{document}. Extender sets are a generalization of follower sets (see [6]), and we define the extender set sequence similarly. In this paper, we examine achievable differences in limiting behavior of follower, predecessor, and extender set sequences. This is done through the classical \begin{document}$\beta$\end{document}-shifts, first introduced in [10]. We show that the follower set sequences of \begin{document}$\beta$\end{document}-shifts must grow at most linearly in \begin{document}$n$\end{document}, while the predecessor and extender set sequences may demonstrate exponential growth rate in \begin{document}$n$\end{document}, depending on choice of \begin{document}$\beta$\end{document}.

2019, 39(8): 4345-4358 doi: 10.3934/dcds.2019176 +[Abstract](767) +[HTML](127) +[PDF](395.67KB)
Abstract:

We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.

2019, 39(8): 4359-4398 doi: 10.3934/dcds.2019177 +[Abstract](591) +[HTML](110) +[PDF](608.67KB)
Abstract:

In this paper we prove local smoothing estimates for the Dirac equation on some non-flat manifolds; in particular, we will consider asymptotically flat and warped products metrics. The strategy of the proofs relies on the multiplier method.

2019, 39(8): 4399-4414 doi: 10.3934/dcds.2019178 +[Abstract](644) +[HTML](154) +[PDF](481.38KB)
Abstract:

We present an exact solution to the nonlinear governing equations in the \begin{document}$\beta$\end{document}-plane approximation for geophysical waves propagating at arbitrary latitude on a zonal current. Such an exact solution is explicit in the Lagrangian framework and represents three-dimensional, nonlinear oceanic wave-current interactions. Based on the short-wavelength instability approach, we prove criteria for the hydrodynamical instability of such waves.

2019, 39(8): 4415-4427 doi: 10.3934/dcds.2019179 +[Abstract](588) +[HTML](119) +[PDF](355.49KB)
Abstract:

We consider the ocean flow of the Antarctic Circumpolar Current. Using a recently-derived model for gyres in rotating spherical coordinates, and mapping the problem on the sphere onto the plane using the Mercator projection, we obtain a boundary-value problem for a semi-linear elliptic partial differential equation. For constant and linear oceanic vorticities, we investigate existence, regularity and uniqueness of solutions to this elliptic problem. We also provide some explicit solutions. Moreover, we examine the physical relevance of these results.

2019, 39(8): 4429-4441 doi: 10.3934/dcds.2019180 +[Abstract](609) +[HTML](108) +[PDF](413.37KB)
Abstract:

We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results.

2019, 39(8): 4443-4453 doi: 10.3934/dcds.2019181 +[Abstract](511) +[HTML](107) +[PDF](349.36KB)
Abstract:

A class of non-stationary surface gravity waves propagating in the zonal direction in the equatorial region is described in the f-plane approximation. These waves are described by exact solutions of the equations of hydrodynamics in Lagrangian formulation and are generalizations of Gerstner waves. The wave shape and non-uniform pressure distribution on a free surface depend on two arbitrary functions. The trajectories of fluid particles are circumferences. The solutions admit a variable meridional current. The dynamics of a single breather on the background of a Gerstner wave is studied as an example.

2019, 39(8): 4455-4469 doi: 10.3934/dcds.2019182 +[Abstract](506) +[HTML](107) +[PDF](522.95KB)
Abstract:

Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution.

2019, 39(8): 4471-4486 doi: 10.3934/dcds.2019183 +[Abstract](544) +[HTML](106) +[PDF](2186.44KB)
Abstract:

The existence of internal geophysical waves of extreme form is confirmed and an explicit solution presented. The flow is confined to a layer lying above an eastward current while the mean horizontal flow of the solutions is westward, thus incorporating flow reversal in the fluid.

2019, 39(8): 4487-4518 doi: 10.3934/dcds.2019184 +[Abstract](656) +[HTML](112) +[PDF](738.95KB)
Abstract:

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

2019, 39(8): 4519-4532 doi: 10.3934/dcds.2019185 +[Abstract](570) +[HTML](102) +[PDF](351.29KB)
Abstract:

A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is infinitely deep, with a higher density than the upper layer which is bounded above by a flat surface. The fluids are incompressible and inviscid. A Hamiltonian formulation for the dynamics in the presence of a depth-varying current is presented and it is shown that an appropriate scaling leads to the integrable Benjamin-Ono equation.

2019, 39(8): 4533-4545 doi: 10.3934/dcds.2019186 +[Abstract](528) +[HTML](101) +[PDF](858.78KB)
Abstract:

Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a detailed analysis of the effect of a given density distribution on the depth-structure of the corresponding traveling waves.

2019, 39(8): 4547-4563 doi: 10.3934/dcds.2019187 +[Abstract](591) +[HTML](121) +[PDF](417.21KB)
Abstract:

There are not too many known explicit solutions to the \begin{document}$2$\end{document}-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the \begin{document}$19$\end{document}th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the \begin{document}$1980$\end{document}s- obtained new explicit solutions with a similar feature.

We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.

In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.

2019, 39(8): 4565-4612 doi: 10.3934/dcds.2019188 +[Abstract](722) +[HTML](110) +[PDF](768.09KB)
Abstract:

We undertake a comprehensive study for the fractional nonlinear Schrödinger equation

where \begin{document}$\frac{d}{2d-1} \leq s <1$\end{document}, \begin{document}$0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s}$\end{document}. Firstly, we establish the local and global well-posedness results for non-radial and radial \begin{document}$H^s$\end{document} initial data, radial \begin{document}$\dot{H}^{s_c}\cap \dot{H}^s$\end{document} initial data, where \begin{document}$s_c = \frac{d}{2}-\frac{2s}{\alpha_2}$\end{document}. Secondly, we study the asymptotic behavior of global radial \begin{document}$H^s$\end{document} solutions. Of particular interest is the \begin{document}$L^2$\end{document}-critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the \begin{document}$L^2$\end{document}-critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial \begin{document}$\dot{H}^{s_c} \cap \dot{H}^s$\end{document} solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with \begin{document}$L^2$\end{document}-critical and \begin{document}$L^2$\end{document}-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both \begin{document}$L^2$\end{document}-critical and \begin{document}$L^2$\end{document}-supercritical cases, including mass-concentration and limiting profile.

2019, 39(8): 4613-4646 doi: 10.3934/dcds.2019189 +[Abstract](529) +[HTML](121) +[PDF](650.93KB)
Abstract:

We propose the Chern-Simons gauged sigma model from $\mathbb{R}^{1+2}$ into the hyperbolic plane $\mathbb{H}^2$. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near $\infty$

2019, 39(8): 4647-4711 doi: 10.3934/dcds.2019190 +[Abstract](542) +[HTML](110) +[PDF](967.32KB)
Abstract:

Let \begin{document}$T$\end{document} be any topological semigroup and \begin{document}$(T, X)$\end{document} with phase mapping \begin{document}$(t, x)\mapsto tx$\end{document} a semiflow on a compact \begin{document}$\text{T}_2$\end{document} space \begin{document}$X$\end{document}. If \begin{document}$tX = X$\end{document} for all \begin{document}$t$\end{document} in \begin{document}$T$\end{document} then \begin{document}$(T, X)$\end{document} is called surjective; if \begin{document}$x\mapsto tx$\end{document}, for each \begin{document}$t$\end{document} in \begin{document}$T$\end{document}, is 1-1 onto, then \begin{document}$(T, X)$\end{document} is termed invertible and the latter induces a right-action semiflow \begin{document}$(X, T)$\end{document} with the phase mapping \begin{document}$(x, t)\mapsto xt: = t^{-1}x$\end{document}. We show that \begin{document}$(T, X)$\end{document} is equicontinuous surjective iff it is uniformly distal iff \begin{document}$(X, T)$\end{document} is equicontinuous surjective. We then consider minimality, distality, point-distality, and sensitivity of \begin{document}$(X, T)$\end{document} when \begin{document}$(T, X)$\end{document} possesses these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of flow on a zero-dimensional space with phase group \begin{document}$\mathbb{Z}$\end{document}.

2019, 39(8): 4713-4729 doi: 10.3934/dcds.2019191 +[Abstract](546) +[HTML](137) +[PDF](466.96KB)
Abstract:

Of concern are steady two-dimensional periodic geophysical water waves of small amplitude near the equator. The analysis presented here is based on the bifurcation theory due to Crandall-Rabinowitz. Dispersion relations for various choices of the vorticity distribution, including constant, affine, and some nonlinear vorticities are obtained.

2019, 39(8): 4731-4746 doi: 10.3934/dcds.2019192 +[Abstract](544) +[HTML](99) +[PDF](995.37KB)
Abstract:

We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.

2019, 39(8): 4747-4770 doi: 10.3934/dcds.2019193 +[Abstract](542) +[HTML](127) +[PDF](563.71KB)
Abstract:

We consider nonlinear traveling waves in a two-dimensional fluid subject to the effects of vorticity, stratification, and in-plane Coriolis forces. We first observe that the terms representing the Coriolis forces can be completely eliminated by a change of variables. This does not appear to be well-known, and helps to organize some of the existing literature.

Second we give a rigorous existence result for periodic waves in a two-layer system with a free surface and constant densities and vorticities in each layer, allowing for the presence of critical layers. We augment the problem with four physically-motivated constraints, and phrase our hypotheses directly in terms of the explicit dispersion relation for the problem. This approach smooths the way for further generalizations, some of which we briefly outline at the end of the paper.

2019, 39(8): 4771-4781 doi: 10.3934/dcds.2019194 +[Abstract](569) +[HTML](129) +[PDF](354.16KB)
Abstract:

We present some results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the Antarctic Circumpolar Current.

2019, 39(8): 4783-4796 doi: 10.3934/dcds.2019195 +[Abstract](544) +[HTML](111) +[PDF](375.76KB)
Abstract:

The aim of the paper is to develop an exact solution relating to a system of model equations representing ocean flows with Equatorial Undercurrent and thermocline in the presence of linear variation of density with depth. The system of equations is generated from the Euler equations represented in a suitable rotating frame by following a careful asymptotic approach.The study in this paper is motivated by the recently developed Constantin-Johnson model [13] for Pacific flows with undercurrent and the exact results provided therein. The model formulated is two-layered, three-dimensional and nonlinear with a symmetric structure about the equator. The equations contain Coriolis effect and is consistent with \begin{document}$\beta$\end{document} - plane approximation. Exact results of the asymptotic system of equations have been derived in a region close to the equator.

2019, 39(8): 4797-4840 doi: 10.3934/dcds.2019196 +[Abstract](532) +[HTML](107) +[PDF](641.61KB)
Abstract:

In this paper, we study the convergence and pathwise dynamics of random differential equations driven by colored noise. We first show that the solutions of the random differential equations driven by colored noise with a nonlinear diffusion term uniformly converge in mean square to the solutions of the corresponding Stratonovich stochastic differential equation as the correlation time of colored noise approaches zero. Then, we construct random center manifolds for such random differential equations and prove that these manifolds converge to the random center manifolds of the corresponding Stratonovich equation when the noise is linear and multiplicative as the correlation time approaches zero.

2019, 39(8): 4841-4861 doi: 10.3934/dcds.2019197 +[Abstract](587) +[HTML](98) +[PDF](454.35KB)
Abstract:

We provide an example of a Schrödinger cocycle over a mixing Markov shift for which the integrated density of states has a very weak modulus of continuity, close to the log-Hölder lower bound established by W. Craig and B. Simon in [6]. This model is based upon a classical example due to Y. Kifer [15] of a random Bernoulli cocycle with zero Lyapunov exponents which is not strongly irreducible. It follows that the Lyapunov exponent of a Bernoulli cocycle near this Kifer example cannot be Hölder or weak-Hölder continuous, thus providing a limitation on the modulus of continuity of the Lyapunov exponent of random cocycles.

2019, 39(8): 4863-4873 doi: 10.3934/dcds.2019198 +[Abstract](568) +[HTML](116) +[PDF](429.8KB)
Abstract:

We consider the Euler equation of functionals involving a term of the form

with \begin{document}$1<p<q<p+1$\end{document} and \begin{document}$a(x)\geq 0$\end{document}. We prove weak comparison principle and summability results for the second derivatives of solutions.

2019, 39(8): 4875-4893 doi: 10.3934/dcds.2019199 +[Abstract](518) +[HTML](108) +[PDF](461.69KB)
Abstract:

The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces \begin{document}$\hat{H}^{s, r}$\end{document}. Our aim is to minimize \begin{document}$s = s(r)$\end{document} in the range \begin{document}$1<r \le 2$\end{document}. If \begin{document}$r \to 1$\end{document} we show that we almost reach the critical regularity dictated by scaling. In the classical case \begin{document}$r = 2$\end{document} the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulfill a null condition.

2019, 39(8): 4895-4928 doi: 10.3934/dcds.2019200 +[Abstract](542) +[HTML](99) +[PDF](1627.83KB)
Abstract:

We are concerned with conservative systems \begin{document}$\ddot q = \nabla V(q)$\end{document}, \begin{document}$q\in{\mathbb R}^{N}$\end{document} for a general class of potentials \begin{document}$V\in C^1({\mathbb R}^N)$\end{document}. Assuming that a given sublevel set \begin{document}$\{V\leq c\}$\end{document} splits in the disjoint union of two closed subsets \begin{document}$\mathcal{V}^{c}_{-}$\end{document} and \begin{document}$\mathcal{V}^{c}_{+}$\end{document}, for some \begin{document}$c\in{\mathbb R}$\end{document}, we establish the existence of bounded solutions \begin{document}$q_{c}$\end{document} to the above system with energy equal to \begin{document}$-c$\end{document} whose trajectories connect \begin{document}$\mathcal{V}^{c}_{-}$\end{document} and \begin{document}$\mathcal{V}^{c}_{+}$\end{document}. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of \begin{document}$\nabla V$\end{document} on \begin{document}$\partial \mathcal{V}^{c}_{\pm}$\end{document}. Next, we illustrate applications of the existence result to double-well potentials \begin{document}$V$\end{document}, and for potentials associated to systems of duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions \begin{document}$(q_{c})$\end{document}.

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