Xiaoyutao Luo, Vicentiu D. Rădulescu, Maoding Zhen
{"title":"Standing waves with prescribed norm for the coupled Hartree-Fock system","authors":"Xiaoyutao Luo, Vicentiu D. Rădulescu, Maoding Zhen","doi":"10.3934/dcds.2023043","DOIUrl":"https://doi.org/10.3934/dcds.2023043","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90925256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other such results have been proved in this case. However, situations in which the defocusing modified KdV (mKdV) equation is expected to be the modulation equation have been much less studied. As seen in numerical experiments, the kink solution of the mKdV seems essential in understanding the $beta$-FPUT recurrence. In this paper, we derive explicit approximation results for solutions of the FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for solutions to be non-localized as to allow approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT.
{"title":"Long-time approximations of small-amplitude, long-wavelength FPUT solutions","authors":"Trevor Norton, C. Eugene Wayne","doi":"10.3934/dcds.2023131","DOIUrl":"https://doi.org/10.3934/dcds.2023131","url":null,"abstract":"It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other such results have been proved in this case. However, situations in which the defocusing modified KdV (mKdV) equation is expected to be the modulation equation have been much less studied. As seen in numerical experiments, the kink solution of the mKdV seems essential in understanding the $beta$-FPUT recurrence. In this paper, we derive explicit approximation results for solutions of the FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for solutions to be non-localized as to allow approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $is considered, where $ mathcal B = { x in mathbb{R}^N : |x|1 $ on $ [0, R] $, $ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $, and $ q>1 $. The existence of positive solutions is proved for every $ lambda>lambda_1(q) $, where $ lambda_1(q) $ is the first eigenvalue of $ q $-Laplacian.
考虑$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $问题，其中$ mathcal B = { x in mathbb{R}^N : |x|<R } $、$ N ge 1 $、$ Delta_{p(|x|)} u = mbox{div} (|nabla u|^{p(|x|)-2}nabla u) $、$ p(r) $是连续的，满足$ [0, R] $、$ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $、$ q>1 $上的$ p(r)>1 $。证明了每个$ lambda>lambda_1(q) $正解的存在性，其中$ lambda_1(q) $是$ q $ -拉普拉斯算子的第一个特征值。
{"title":"An eigenvalue problem for a variable exponent problem, via topological degree","authors":"Raúl Manásevich, Satoshi Tanaka","doi":"10.3934/dcds.2023134","DOIUrl":"https://doi.org/10.3934/dcds.2023134","url":null,"abstract":"The problem$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $is considered, where $ mathcal B = { x in mathbb{R}^N : |x|<R } $, $ N ge 1 $, $ Delta_{p(|x|)} u = mbox{div} (|nabla u|^{p(|x|)-2}nabla u) $, $ p(r) $ is continuous and satisfies $ p(r)>1 $ on $ [0, R] $, $ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $, and $ q>1 $. The existence of positive solutions is proved for every $ lambda>lambda_1(q) $, where $ lambda_1(q) $ is the first eigenvalue of $ q $-Laplacian.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135560507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the boundary layer problem on a chemotaxis system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen and no-flux boundary conditions for bacteria, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $ varepsilon $ goes to zero and the boundary-layer thickness is of order $ mathcal{O}(varepsilon^alpha) $ with $ 0
{"title":"Boundary layer problem on the chemotaxis model with Robin boundary conditions","authors":"Qianqian Hou","doi":"10.3934/dcds.2023108","DOIUrl":"https://doi.org/10.3934/dcds.2023108","url":null,"abstract":"This paper is concerned with the boundary layer problem on a chemotaxis system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen and no-flux boundary conditions for bacteria, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $ varepsilon $ goes to zero and the boundary-layer thickness is of order $ mathcal{O}(varepsilon^alpha) $ with $ 0<alpha<frac{1}{2} $.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135700134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we advance the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows, which is one of the oldest problems in mathematical physics. More precisely, in this work we study the nonlinear stability in the Lyapunov sense of one equilibrium solution in autonomous Hamiltonian systems with $ n $-degrees of freedom, assuming the existence of two vectors of resonance, both of order four, with interaction in one frequency. We provide conditions to obtain a type of formal stability, called Lie stability. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems with three and four degrees of freedom. In addition, we give sufficient conditions for the instability in the sense of Lyapunov. We apply some of our results in the spatial satellite problem at one of its equilibrium points, which is a novelty in this problem.
本文提出了哈密顿流平衡解的Lyapunov稳定性和不稳定性的研究，这是数学物理中最古老的问题之一。更准确地说，在本工作中，我们研究了具有$ n $自由度的自治哈密顿系统在Lyapunov意义下的一个平衡解的非线性稳定性，假设存在两个共振向量，它们都是四阶的，并且在一个频率上相互作用。我们提供条件来获得一种形式稳定性，称为李氏稳定性。随后，我们保证了三自由度和四自由度李稳定系统在Nekhoroshev意义上的指数稳定的一些充分条件。此外，我们给出了Lyapunov意义下的不稳定性的充分条件。我们将我们的一些结果应用到空间卫星问题的平衡点上，这是该问题中的一个新问题。
{"title":"Nonlinear stability of elliptic equilibria in Hamiltonian systems with resonances of order four with interactions","authors":"Claudio Sierpe, Claudio Vidal","doi":"10.3934/dcds.2023119","DOIUrl":"https://doi.org/10.3934/dcds.2023119","url":null,"abstract":"In this paper, we advance the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows, which is one of the oldest problems in mathematical physics. More precisely, in this work we study the nonlinear stability in the Lyapunov sense of one equilibrium solution in autonomous Hamiltonian systems with $ n $-degrees of freedom, assuming the existence of two vectors of resonance, both of order four, with interaction in one frequency. We provide conditions to obtain a type of formal stability, called Lie stability. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems with three and four degrees of freedom. In addition, we give sufficient conditions for the instability in the sense of Lyapunov. We apply some of our results in the spatial satellite problem at one of its equilibrium points, which is a novelty in this problem.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135057376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Changwu Zou, Guangfeng Dong, Changjian Liu, Jiazhong Yang
In this paper we shall study $ Sigma $-center problem on piecewise smooth systems with two zones divided by a straight line in the real plane. We first give some necessary and sufficient conditions for the crossing $ Sigma $-center by taking advantage of the forms of the first integrals. Then for applications, we completely describe the $ Sigma $-centers from both the algebra and geometry points of view when the both of two sub-systems are linear.
{"title":"The center problem on piecewise smooth differential systems with two zones","authors":"Changwu Zou, Guangfeng Dong, Changjian Liu, Jiazhong Yang","doi":"10.3934/dcds.2023113","DOIUrl":"https://doi.org/10.3934/dcds.2023113","url":null,"abstract":"In this paper we shall study $ Sigma $-center problem on piecewise smooth systems with two zones divided by a straight line in the real plane. We first give some necessary and sufficient conditions for the crossing $ Sigma $-center by taking advantage of the forms of the first integrals. Then for applications, we completely describe the $ Sigma $-centers from both the algebra and geometry points of view when the both of two sub-systems are linear.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136256740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global dynamics of evolution systems with asymptotic annihilation","authors":"Taishan Yi, Xiao-Qiang Zhao","doi":"10.3934/dcds.2023025","DOIUrl":"https://doi.org/10.3934/dcds.2023025","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77252085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Emergence of large densities in a chemotaxis system with signaling loops, nonlinear signal productions and competitions sources under nonradial symmetry case","authors":"Guangyu Xu","doi":"10.3934/dcds.2023090","DOIUrl":"https://doi.org/10.3934/dcds.2023090","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：