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Discrete & Continuous Dynamical Systems - A

Open Access Articles

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system
Masaru Hamano and Satoshi Masaki
2020 doi: 10.3934/dcds.2020323 +[Abstract](239) +[HTML](116) +[PDF](492.71KB)

In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation
Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa and Shoji Yotsutani
2020, 40(8): 4907-4925 doi: 10.3934/dcds.2020205 +[Abstract](527) +[HTML](196) +[PDF](815.51KB)

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [4] and [5], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

On substitution tilings and Delone sets without finite local complexity
Jeong-Yup Lee and Boris Solomyak
2019, 39(6): 3149-3177 doi: 10.3934/dcds.2019130 +[Abstract](2368) +[HTML](636) +[PDF](606.02KB)

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
Tadahiro Oh, Mamoru Okamoto and Oana Pocovnicu
2019, 39(6): 3479-3520 doi: 10.3934/dcds.2019144 +[Abstract](2817) +[HTML](674) +[PDF](764.62KB)

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document}, \begin{document}$ d = 5,6 $\end{document}, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on \begin{document}$ \mathbb{R}^d $\end{document}, \begin{document}$ d = 5, 6 $\end{document}, in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions
Etienne Emmrich and Robert Lasarzik
2018, 38(9): 4617-4635 doi: 10.3934/dcds.2018202 +[Abstract](3391) +[HTML](983) +[PDF](430.79KB)

We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.

On the universality of the incompressible Euler equation on compact manifolds
Terence Tao
2018, 38(3): 1553-1565 doi: 10.3934/dcds.2018064 +[Abstract](4955) +[HTML](1054) +[PDF](422.44KB)

The incompressible Euler equations on a compact Riemannian manifold \begin{document}$(M,g)$\end{document} take the form

\begin{document}$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$ \end{document}

We show that any quadratic ODE \begin{document}$\partial_t y =B(y,y)$\end{document}, where \begin{document}$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$\end{document} is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold \begin{document}$M$\end{document} if and only if \begin{document}$B$\end{document} obeys the cancellation condition \begin{document}$\langle B(y,y), y \rangle =0$\end{document} for some positive definite inner product \begin{document}$\langle,\rangle$\end{document} on \begin{document}$\mathbb{R}^n$\end{document}. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.

Traveling wave solutions of a highly nonlinear shallow water equation
Anna Geyer and Ronald Quirchmayr
2018, 38(3): 1567-1604 doi: 10.3934/dcds.2018065 +[Abstract](6438) +[HTML](2496) +[PDF](12373.99KB)

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

Degenerate with respect to parameters fold-Hopf bifurcations
Gheorghe Tigan
2017, 37(4): 2115-2140 doi: 10.3934/dcds.2017091 +[Abstract](3039) +[HTML](1701) +[PDF](813.0KB)

In this work we study degenerate with respect to parameters fold-Hopfbifurcations in three-dimensional differential systems. Such degeneraciesarise when the transformations between parameters leading to a normal formare not regular at some points in the parametric space. We obtain newgeneric results for the dynamics of the systems in such a degenerateframework. The bifurcation diagrams we obtained show that in a degeneratecontext the dynamics may be completely different than in a non-degenerateframework.

Peter Bates, Danielle Hilhorst, Hiroshi Matano and Yoshihisa Morita
2017, 37(2): i-iii doi: 10.3934/dcds.201702i +[Abstract](1704) +[HTML](903) +[PDF](84.4KB)
Professor Paul Chase Fife was born in Cedar City, Utah, on February 14, 1930. After undergraduate studies at the University of Chicago, he obtained a Master's degree in physics from the University of California Berkeley where he also received a Phi Beta Kappa Award. He then completed a PhD in Applied Mathematics at New York University, Courant Institute, in June 1959. While at NYU he met Jayne Winters, and they married on December 22, 1959. They then moved to Palo Alto, California, where Paul joined the Department of Mathematics at Stanford University.

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Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
Ellen Baake and Michael Baake
2016, 36(12): 6645-6656 doi: 10.3934/dcds.2016088 +[Abstract](2368) +[PDF](429.7KB)
The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.
Low-dimensional Galerkin approximations of nonlinear delay differential equations
Mickaël D. Chekroun, Michael Ghil, Honghu Liu and Shouhong Wang
2016, 36(8): 4133-4177 doi: 10.3934/dcds.2016.36.4133 +[Abstract](2927) +[PDF](2260.8KB)
This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.
Alexandre J. Chorin and Jeffrey Rauch
2016, 36(8): i-ii doi: 10.3934/dcds.2016.36.8i +[Abstract](1562) +[PDF](2472.6KB)
The papers in this special issue of Discrete and Continuous Dynamical Systems are dedicated to Professor Peter D. Lax, of the Courant Institute, on the occasion of his ninetieth birthday, by some of his friends, associates, and students.

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The unsteady transonic small disturbance equation: Data on oblique curves
Mary Chern and Barbara Lee Keyfitz
2016, 36(8): 4213-4225 doi: 10.3934/dcds.2016.36.4213 +[Abstract](1782) +[PDF](360.7KB)
We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.
On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface
Takayuki Kubo, Yoshihiro Shibata and Kohei Soga
2016, 36(7): 3741-3774 doi: 10.3934/dcds.2016.36.3741 +[Abstract](2288) +[PDF](553.2KB)
In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.
Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion
Anotida Madzvamuse, Hussaini Ndakwo and Raquel Barreira
2016, 36(4): 2133-2170 doi: 10.3934/dcds.2016.36.2133 +[Abstract](3466) +[PDF](5124.7KB)
This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.
Kung-Ching Chang, Minyou Chi, Wei-Ming Ni and Zhuoqun Wu
2016, 36(2): i-iii doi: 10.3934/dcds.2016.36.2i +[Abstract](1819) +[PDF](1914.3KB)
Professor Rou-Huai Wang (October 30, 1924 - November 5, 2001) was a mathematician who proved fundamental results for partial differential equations, helped to introduce modern PDE theory to the Chinese mathematics community since early 50's, and played a leading role in revitalizing PDE research in China after the disastrous "Cultural Revolution" (1966-1976). He was regarded by many Chinese mathematicians of younger generations as a visionary, generous and caring mentor.

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Polynomial and linearized normal forms for almost periodic differential systems
Weigu Li, Jaume Llibre and Hao Wu
2016, 36(1): 345-360 doi: 10.3934/dcds.2016.36.345 +[Abstract](2390) +[PDF](418.1KB)
For almost periodic differential systems $\dot x= \varepsilon f(x,t,\varepsilon)$ with $x\in \mathbb{C}^n$, $t\in \mathbb{R}$ and $\varepsilon>0$ small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system $\dot x= \varepsilon \lim_{T \to \infty} \frac {1} {T} \int_0^T f(x,t,0) dt$, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non--resonant.
The general recombination equation in continuous time and its solution
Ellen Baake, Michael Baake and Majid Salamat
2016, 36(1): 63-95 doi: 10.3934/dcds.2016.36.63 +[Abstract](2609) +[PDF](581.9KB)
The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.
Baojun Bian, Shanjian Tang and Qi Zhang
2015, 35(11): i-iv doi: 10.3934/dcds.2015.35.11i +[Abstract](1951) +[PDF](132.3KB)
The workshop on ``Analysis and Control of Stochastic Partial Differential Equations" was held in Fudan University on December 3--6, 2012, which was jointly organized and financially supported by Fudan University and Tongji University. Many of the contributions in the special issue were reported in the workshop, and there are also some few others which are solicited from renowned researchers in the fields of stochastic partial differential equations (SPDEs). The contents of the special issue are divided into the following three parts.

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Jean-Baptiste Caillau, Maria do Rosário de Pinho, Lars Grüne, Emmanuel Trélat and Hasnaa Zidani
2015, 35(9): i-iv doi: 10.3934/dcds.2015.35.9i +[Abstract](1779) +[PDF](131.6KB)
This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction of new ideas and techniques, and thanks to various new applications.

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