Pub Date : 2024-07-03DOI: 10.1088/1361-6544/ad56ec
Fan Yang
We prove Anderson localization for all Diophantine frequencies and all non-resonant phases for a model that arises from a 2D quantum walk model subject to an external magnetic field, also known as the unitary almost Mathieu operator. Our work provides the first localization results for all Diophantine frequencies in quasi-periodic quantum walk and CMV matrix setting. We also obtain sharp asymptotics of the localized eigenfunctions.
{"title":"Anderson localization for the unitary almost Mathieu operator","authors":"Fan Yang","doi":"10.1088/1361-6544/ad56ec","DOIUrl":"https://doi.org/10.1088/1361-6544/ad56ec","url":null,"abstract":"We prove Anderson localization for all Diophantine frequencies and all non-resonant phases for a model that arises from a 2D quantum walk model subject to an external magnetic field, also known as the unitary almost Mathieu operator. Our work provides the first localization results for all Diophantine frequencies in quasi-periodic quantum walk and CMV matrix setting. We also obtain sharp asymptotics of the localized eigenfunctions.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"24 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-26DOI: 10.1088/1361-6544/ad5948
Jonathan P Keating and Fei Wei
In a companion paper (Keating and Wei 2023 Int. Math. Res. Not.2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.
{"title":"Joint moments of higher order derivatives of CUE characteristic polynomials II: structures, recursive relations, and applications","authors":"Jonathan P Keating and Fei Wei","doi":"10.1088/1361-6544/ad5948","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5948","url":null,"abstract":"In a companion paper (Keating and Wei 2023 Int. Math. Res. Not.2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-25DOI: 10.1088/1361-6544/ad4b8e
M Bertola, T Grava and G Orsatti
We develop the theory of integrable operators acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a -problem in the complex plane. When such a -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert–Carleman determinant of the operator . With suitable choices of the setup we show that the Hilbert–Carleman determinant is a τ-function of the Kadomtsev–Petviashvili (KP) or nonlinear Schrödinger hierarchies.
我们通过类比作用于复数平面轮廓的可积分算子理论，发展了作用于具有光滑边界的复数平面域的可积分算子理论。我们展示了如何从复数平面上的-问题的解中得到解析算子。当这样一个-问题取决于辅助参数时，我们类比等单旋转问题理论，定义了它的 Malgrange one 形式。我们证明了马尔格朗日一形式是封闭的，并且与算子希尔伯特-卡勒曼行列式的外部对数微分重合。通过适当的设置选择，我们证明了希尔伯特-卡勒曼行列式是卡多姆采夫-彼得维亚什维利（KP）或非线性薛定谔层次结构的τ函数。
{"title":"Integrable operators, ∂― -problems, KP and NLS hierarchy","authors":"M Bertola, T Grava and G Orsatti","doi":"10.1088/1361-6544/ad4b8e","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4b8e","url":null,"abstract":"We develop the theory of integrable operators acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a -problem in the complex plane. When such a -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert–Carleman determinant of the operator . With suitable choices of the setup we show that the Hilbert–Carleman determinant is a τ-function of the Kadomtsev–Petviashvili (KP) or nonlinear Schrödinger hierarchies.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"27 15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-24DOI: 10.1088/1361-6544/ad5781
Piotr B Mucha and Jan Peszek
The paper examines the issue of well-posedness of the Cucker-Smale model with communication restricted to the q-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy q-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.
{"title":"A fuzzy q-closest alignment model","authors":"Piotr B Mucha and Jan Peszek","doi":"10.1088/1361-6544/ad5781","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5781","url":null,"abstract":"The paper examines the issue of well-posedness of the Cucker-Smale model with communication restricted to the q-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy q-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"38 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-23DOI: 10.1088/1361-6544/ad5636
Jared Wunsch, Mengxuan Yang and Yuzhou Zou
We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.
{"title":"The Morse index theorem for mechanical systems with reflections","authors":"Jared Wunsch, Mengxuan Yang and Yuzhou Zou","doi":"10.1088/1361-6544/ad5636","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5636","url":null,"abstract":"We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"17 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-19DOI: 10.1088/1361-6544/ad506d
Shih-Wei Chou and Bo-Chih Huang
In this paper, we make modifications to the original hot-Jupiter model, which addresses the problem of hydrodynamic escape for the planetary atmosphere. The model involves the Euler equation with gravity, tidal effect, and heat. We employ the generalised Glimm technique to prove the presence of transonic solutions to the problem. By adjusting the dilation of the characteristic fields, we enhance the accuracy of the Glimm–Goodman wave interaction estimates. This allows us to establish a more general admissible condition for stabilizing the generalised Glimm scheme. Additionally, we derive the exact relationship between the lower bound of the gas velocity in the subsonic state and the adiabatic constant of the gas. Under certain constraints on the transonic initial and boundary data, the limit of the approximation solutions represents an entropy transonic solution with bounded variations. Furthermore, we are able to determine the feasible hydrodynamical region directly from the equation itself, without the need for any additional state equation.
{"title":"Global transonic solutions of hot-Jupiter model for exoplanetary atmosphere","authors":"Shih-Wei Chou and Bo-Chih Huang","doi":"10.1088/1361-6544/ad506d","DOIUrl":"https://doi.org/10.1088/1361-6544/ad506d","url":null,"abstract":"In this paper, we make modifications to the original hot-Jupiter model, which addresses the problem of hydrodynamic escape for the planetary atmosphere. The model involves the Euler equation with gravity, tidal effect, and heat. We employ the generalised Glimm technique to prove the presence of transonic solutions to the problem. By adjusting the dilation of the characteristic fields, we enhance the accuracy of the Glimm–Goodman wave interaction estimates. This allows us to establish a more general admissible condition for stabilizing the generalised Glimm scheme. Additionally, we derive the exact relationship between the lower bound of the gas velocity in the subsonic state and the adiabatic constant of the gas. Under certain constraints on the transonic initial and boundary data, the limit of the approximation solutions represents an entropy transonic solution with bounded variations. Furthermore, we are able to determine the feasible hydrodynamical region directly from the equation itself, without the need for any additional state equation.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-19DOI: 10.1088/1361-6544/ad5637
Mike R Jeffrey
A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their ‘underlying catastrophes’. Here we apply the idea to the homogeneous steady states of a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these ‘underlying’ catastrophes? We then show they essentially define a restricted class of ‘solvable’ rather than ‘all classifiable’ singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom’s elementary catastrophes in much broader contexts.
最近有人提出了一种实用方法，通过寻找 n 维矢量场 F 中的 "底层灾难 "来找到局部分岔点。在此，我们将这一想法应用于偏微分方程的同质稳定状态，以此为例说明灾变在反应扩散中的作用。这些 "潜在 "灾变是什么？通过识别矢量场 F 的退化零点而不考虑其矢量特性，我们证明它们本质上定义了一类受限的 "可解 "奇点，而非 "所有可分类 "奇点。因此，它们是由一组最小的 r 分析条件定义的，这些条件提供了求解它们的实用方法，并大大减少了对奇点进行分类所需的计算量，我们也将在此列举这些计算量。这样，底层灾变似乎可以让我们在更广阔的背景下应用托姆的基本灾变。
{"title":"Elementary catastrophes underlying bifurcations of vector fields and PDEs","authors":"Mike R Jeffrey","doi":"10.1088/1361-6544/ad5637","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5637","url":null,"abstract":"A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their ‘underlying catastrophes’. Here we apply the idea to the homogeneous steady states of a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these ‘underlying’ catastrophes? We then show they essentially define a restricted class of ‘solvable’ rather than ‘all classifiable’ singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom’s elementary catastrophes in much broader contexts.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"192 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-18DOI: 10.1088/1361-6544/ad5639
Koichi Komada and Satoshi Masaki
In this paper, we consider the focusing, L2-supercritical and -subcritical fourth-order nonlinear Schrödinger equations. We show the scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less than a certain value.
{"title":"Scattering of solutions with group invariance for the fourth-order nonlinear Schrödinger equation","authors":"Koichi Komada and Satoshi Masaki","doi":"10.1088/1361-6544/ad5639","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5639","url":null,"abstract":"In this paper, we consider the focusing, L2-supercritical and -subcritical fourth-order nonlinear Schrödinger equations. We show the scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less than a certain value.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-18DOI: 10.1088/1361-6544/ad52e2
Niklas Jöckel
We study the dispersion-generalized Benjamin–Ono equation in the periodic setting. This equation interpolates between the Benjamin–Ono equation ( ) and the inviscid Burgers’ equation ( ). We obtain local well-posedness in for and by using the short-time Fourier restriction method.
我们研究了周期性背景下的弥散广义本杰明-奥诺方程。该方程介于本杰明-奥诺方程（ ）和不粘性布尔格斯方程（ ）之间。我们通过使用短时傅立叶限制方法，在 for 和 中获得了局部好求解性。
{"title":"Well-posedness of the periodic dispersion-generalized Benjamin–Ono equation in the weakly dispersive regime","authors":"Niklas Jöckel","doi":"10.1088/1361-6544/ad52e2","DOIUrl":"https://doi.org/10.1088/1361-6544/ad52e2","url":null,"abstract":"We study the dispersion-generalized Benjamin–Ono equation in the periodic setting. This equation interpolates between the Benjamin–Ono equation ( ) and the inviscid Burgers’ equation ( ). We obtain local well-posedness in for and by using the short-time Fourier restriction method.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"42 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-17DOI: 10.1088/1361-6544/ad55a0
Yukihiro Sawada, Yasuhiro Takeuchi and Yueping Dong
In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.
{"title":"Dynamics of a two-patch logistic model with diffusion and time delay","authors":"Yukihiro Sawada, Yasuhiro Takeuchi and Yueping Dong","doi":"10.1088/1361-6544/ad55a0","DOIUrl":"https://doi.org/10.1088/1361-6544/ad55a0","url":null,"abstract":"In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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：