非线性科学
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显示 2025年07月21日， 星期一 新的列表
- [1] arXiv:2507.13570 [中文pdf, pdf, html, 其他]
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标题： 耦合慢-快神经元模型的时间序列分析：从赫斯特指数到格兰杰因果关系标题： Time series analysis of coupled slow-fast neuron models: From Hurst exponent to Granger causality主题： 混沌动力学 (nlin.CD)
我们对小网络进行时间序列分析，其中每个节点都是Schaeffer和Cain提出的变性Morris--Lecar神经元的慢速-快速版本。 我们从文献中选择流行的耦合策略，并详细说明改变其强度如何驱动小网络的动力学。 时间序列分析的算法包括测量它们的持续性（记住过去值的能力）、不规则性、混沌和准周期性，以及网络中每个节点之间的时间序列同步性。 当耦合具有热敏感性且温度高于参考温度时，观察到抑制性耦合强度下的混沌。 当耦合非常弱时，观察到准周期性，而当耦合强度高度兴奋时，观察到同步爆发。 在某些情况下，我们也观察到衰减振荡。 最后，进行因果性测试以检测耦合系统中一个神经元的动力学是否影响另一个神经元的动力学。
We perform time series analysis of small networks where every node is the slow-fast version of the denatured Morris--Lecar neuron proposed by Schaeffer and Cain. We choose popular coupling strategies from the literature and provide a detailed account of how varying their strength drives the dynamics of the small networks. Algorithms for time series analysis range from measuring their persistence (ability to remember past values), irregularity, chaos and quasiperiodicity, to synchronization between time series from every node within a network. Chaos is observed for inhibitory coupling strengths and for temperature higher than a reference temperature when the coupling is thermally sensitive. We observe quasi-periodicity when the coupling is very weak and synchronized bursting for highly excitatory coupling strength. In certain cases we also observe decay oscillations. Finally, a causality test is performed to detect whether the dynamics of one neuron is influencing the dynamics of the other in the coupled system.
- [2] arXiv:2507.13632 [中文pdf, pdf, html, 其他]
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标题： 量子液滴和气泡的指数渐近分析标题： Exponential asymptotics of quantum droplets and bubbles评论： 21页，7图。已接受发表于《Physica D: 非线性现象》主题： 模式形成与孤子 (nlin.PS)
这项研究调查了局部状态的形成和稳定性，这些状态被称为量子液滴和气泡，在二次三次离散非线性薛定谔方程中出现。 在麦克斯韦点附近，这些状态从连接双稳平衡的两个前缘产生。 通过调整控制参数，我们确定了一个“钉扎区域”，其中多个稳定状态共存，并通过同宿蛇形相互连接。 我们分析了系统的行为，以揭示强耦合条件下的潜在机制。 使用指数渐近方法，我们确定了钉扎区域的宽度及其与耦合强度的关系，揭示了它们之间存在指数级小的关系。 此外，我们采用特征值计数来确定这些状态的稳定性，通过计算其相应线性化算子的临界特征值，证明了站点前缘不稳定而位点间前缘稳定。 这些理论结果通过数值模拟得到验证，结果与我们的解析预测高度一致。
This research investigates the formation and stability of localized states, known as quantum droplets and bubbles, in the quadratic-cubic discrete nonlinear Schr\"odinger equation. Near a Maxwell point, these states emerge from two fronts connecting the bistable equilibria. By adjusting a control parameter, we identify a "pinning region" where multiple stable states coexist and are interconnected through homoclinic snaking. We analyze the system's behavior to uncover the underlying mechanisms under strong coupling conditions. Using exponential asymptotics, we determine the pinning region's width and its dependence on coupling strength, revealing an exponentially small relationship between them. Additionally, we employ eigenvalue counting to establish the stability of these states by computing the critical eigenvalue of their corresponding linearization operator, proving onsite fronts unstable and intersite fronts stable. These theoretical results are validated through numerical simulations, which show excellent agreement with our analytical predictions.
- [3] arXiv:2507.13643 [中文pdf, pdf, html, 其他]
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标题： 离散非线性薛定谔方程中暗孤子和亮孤子的指数渐近分析标题： Exponential asymptotics of dark and bright solitons in the discrete nonlinear Schrödinger equation评论： 21页，9图。已接受发表于《Physica D: 非线性现象》主题： 模式形成与孤子 (nlin.PS)
我们研究在强耦合 regime 中非线性 Schrödinger 晶格中孤子的存在性和线性稳定性。考虑聚焦和非聚焦非线性，产生亮孤子和暗孤子。在此 regime 中，晶格离散性的影响变得指数级小，需要超越所有阶次的分析。为此，我们采用指数渐近方法推导孤子解并系统地检查其稳定性。我们表明只有两种对称相关的孤子配置是允许的：位于晶格点上的孤子和位于相邻点之间的孤子。尽管由于实特征值对导致的孤子不稳定性的数值结果已知，但特别是对于暗孤子，缺乏严格的解析描述。我们的工作填补了这一空白，得出的解析预测与数值计算高度一致。我们还建立了位于晶格点上的亮孤子的线性稳定性。虽然由于连续谱覆盖整个虚轴，该方法无法直接解析位于晶格点上的暗孤子的四重特征值引起的不稳定性，但我们提出了一个特征值计数论点，支持它们的不稳定性。总体而言，我们对指数渐近方法的应用展示了该方法在解决离散非线性系统中的多尺度问题方面的多功能性。
We investigate the existence and linear stability of solitons in the nonlinear Schr\"odinger lattices in the strong coupling regime. Focusing and defocusing nonlinearities are considered, giving rise to bright and dark solitons. In this regime, the effects of lattice discreteness become exponentially small, requiring a beyond-all-orders analysis. To this end, we employ exponential asymptotics to derive soliton solutions and examine their stability systematically. We show that only two symmetry-related soliton configurations are permissible: onsite solitons centered at lattice sites and intersite solitons positioned between adjacent sites. Although the instability of intersite solitons due to real eigenvalue pairs is known numerically, a rigorous analytical account, particularly for dark solitons, has been lacking. Our work fills this gap, yielding analytical predictions that match numerical computations with high accuracy. We also establish the linear stability of onsite bright solitons. While the method cannot directly resolve the quartet eigenvalue-induced instability of onsite dark solitons due to the continuous spectrum covering the entire imaginary axis, we conjecture an eigenvalue-counting argument that supports their instability. Overall, our application of the exponential asymptotics method shows the versatility of this approach for addressing multiscale problems in discrete nonlinear systems.
新提交 (展示 3 之 3 条目 )
- [4] arXiv:2507.13418 (交叉列表自 cond-mat.mes-hall) [中文pdf, pdf, html, 其他]
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标题： 霍夫施塔特蝴蝶：连接凝聚态物理、拓扑学和数论标题： The Hofstadter Butterfly: Bridging Condensed Matter, Topology, and Number Theory主题： 中尺度与纳米尺度物理 (cond-mat.mes-hall) ; 混沌动力学 (nlin.CD)
庆祝其金禧年，霍夫斯塔特蝴蝶分形成为艺术与科学的卓越融合。 这个标志性的X形状的分形吸引着物理学家、数学家和爱好者，优雅地展示了在磁场影响下二维晶体晶格中电子的能量谱。 这个量子分形及其变体已成为拓扑绝缘体的范式模型，是21世纪物理学中的新物质状态，其中包含了源自拓扑的整数，这些整数作为霍尔电导率的量子。 本文通过几何学和数论的视角，探讨蝴蝶分形性的理论框架。 在这个诗意的数学中，我们见证了一种罕见的量子魔法：自然使用抽象分形来构建蝴蝶图本身。 在最简单形式中，蝴蝶图用梯形和三角形铺满二维平面，其中霍尔电导率的量子嵌入在梯形的整数斜率对角线上。 理论框架通过具有整数系数的单模矩阵简洁地表达出来，使抽象构造如法雷树、阿波罗尼亚环和毕达哥拉斯三元组树得以呈现。
Celebrating its golden jubilee, the Hofstadter butterfly fractal emerges as a remarkable fusion of art and science. This iconic X shaped fractal captivates physicists, mathematicians, and enthusiasts alike by elegantly illustrating the energy spectrum of electrons within a two dimensional crystal lattice influenced by a magnetic field. Enriched with integers of topological origin that serve as quanta of Hall conductivity, this quantum fractal and its variations have become paradigm models for topological insulators, novel states of matter in 21st century physics. This paper delves into the theoretical framework underlying butterfly fractality through the lenses of geometry and number theory. Within this poetic mathematics, we witness a rare form of quantum magic: Natures use of abstract fractals in crafting the butterfly graph itself. In its simplest form, the butterfly graph tessellates a two dimensional plane with trapezoids and triangles, where the quanta of Hall conductivity are embedded in the integer sloped diagonals of the trapezoids. The theoretical framework is succinctly expressed through unimodular matrices with integer coefficients, bringing to life abstract constructs such as the Farey tree, the Apollonian gaskets, and the Pythagorean triplet tree.
- [5] arXiv:2507.13683 (交叉列表自 cond-mat.quant-gas) [中文pdf, pdf, html, 其他]
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标题： 二元玻色-爱因斯坦凝聚体混溶-不可混溶转变的非线性调控标题： Nonlinear management of the miscibility-immiscibility transition in binary Bose-Einstein condensates评论： 13页，9图。将发表于《物理评论E》主题： 量子气体 (cond-mat.quant-gas) ; 模式形成与孤子 (nlin.PS)
我们研究了非线性管理（NM，即组分间排斥强度的周期性变化）在两组分BEC临界点处混溶-不混溶（MIM）转变中的应用，包括组分之间存在线性混合（Rabi耦合，RC）和不存在线性混合的情况。 为此，我们首先通过变分近似和数值解确定了在没有管理的情况下系统所支持的多种静态域墙（DW）结构。 发现DW的近似解析解与数值解高度一致。 还给出了在受限系统中由于囚禁势的压力导致MIM转变上移的解析估计。 对于包含Pöschl-Teller势的系统，得到了精确的DW解，当势是排斥的（吸引的）时，该解是稳定的（不稳定的）。 此外，我们研究了空间均匀混合态中的线性激发谱，从而确定了系统对分离稳定/不稳定的参数区域。 特别是，RC会提高MIM转变开始的组分间排斥强度临界值。 通过监测扰动状态的演化，从数值模拟中识别了DW态上的激发特征频率。 在DW特征频率处施加弱NM揭示了非线性共振的特征。 更强的NM使系统周期性地穿过MIM转变点，从而限制了混溶性。
We investigate application of the nonlinearity management (NM, i.e., periodic variation of the strength of the inter-component repulsion) to the miscibility-immiscibility (MIM) transition across the critical point of a two-component BEC, both with and without the linear mixing (Rabi coupling, RC) between the components. To this end, we first identify, by means of a variational approximation and numerical solution, diverse stationary domain-wall (DW) structures supported by the system in the absence of the management. The approximate analytical solutions for the DWs are found to be in excellent agreement with their numerical counterparts. An analytical estimate is also produced for the upshift of the MIM transition caused by the pressure of the trapping potential in the case of a confined system. An exact DW solution is produced for the system including the P\"{o}schl-Teller potential, which is stable (unstable) if the potential is repulsive (attractive). Further, we find the spectrum of linear excitations in the spatially uniform mixed state, and thus establish parameter regions where the system is stable/unstable against demixing. In particular, RC upshifts the critical strength of the inter-component repulsion for the onset of the MIM transition. Eigenfrequencies of excitations on top of DW states are identified from numerical simulations through monitoring the evolution of perturbed states. Weak NM applied at the DW eigenfrequency reveals features of the nonlinear resonance. Stronger NM, under which the system periodically crosses the MIM-transition point, restricts the miscibility.
- [6] arXiv:2507.13849 (交叉列表自 physics.soc-ph) [中文pdf, pdf, html, 其他]
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标题： 分形社会动力学作为共识与不平等的驱动因素标题： Fractal Social Dynamics as a Driver of Consensus and Inequality评论： 11页，1图主题： 物理与社会 (physics.soc-ph) ; 适应性与自组织系统 (nlin.AO)
人类社会行为组织在分层的、等级化的网络中，有一个大约5人的支持群体，每一层按比例扩展，最多达到每个个体约150次频繁互动。 这被称为社会大脑假说，其研究结果得到了心理学和神经学证据的支持。 分形网络框架为诸如虚假新闻传播和技术发展等社会现象提供了有价值的见解。 本研究使用分形网络对经济社会互动进行建模，其中群体规模按固定因子扩展，以分析共识是如何形成的。 使用$q$-微积分，该模型揭示了层次结构如何影响信息传播，突出了由幂律支配的普遍特征。 结果遵循$q$-高斯分布，显示出与全球社会中观察到的不平等相一致的重尾特性。 结果表明，不平等来自于经济社会网络的分形结构。
Human social behavior is organized in stratified, hierarchical networks, with a support group with about 5 members, expanding proportionally at each layer up to a maximum of approximately 150 frequent interactions per individual. This is known as Social Brain Hypothesis, and its findings are supported by psychological and neurological evidence. The fractal network framework provides valuable insights into social phenomena such as the spread of fake news and the development of technology. This study models socioeconomic interactions using fractal networks, where group sizes scale by a fixed factor, to analyze how consensus is formed. Using $q$-calculus, the model reveals how hierarchical structures influence information spread, highlighting universal features governed by power laws.. The results follow $q$-Gaussian distributions, showing heavy-tails that align with observed inequalities in societies worldwide. The results show that inequalities arise from the fractal structure of the socioeconomic network.
- [7] arXiv:2507.13879 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]
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标题： 具有内部自由度的磁单极子标题： Magnetic monopoles with an internal degree of freedom评论： 10页，5图主题： 高能物理 - 理论 (hep-th) ; 其他凝聚态物理 (cond-mat.other) ; 高能物理 - 现象学 (hep-ph) ; 模式形成与孤子 (nlin.PS)
我们考虑一类自发电磁$SU(2)$规范理论，具有伴随标量，并在Bogomol'nyi--Prasad--Sommerfield（BPS）极限下寻找精确的磁单极解。 我们发现其中一些解表现出一种新的内部自由度（模空间参数），该参数控制单极子的能量密度分布，同时保持总能量（质量）不变。
We consider a class of spontaneously broken $SU(2)$ gauge theories with adjoint scalar and look for exact magnetic monopole solutions in the Bogomol'nyi--Prasad--Sommerfield (BPS) limit. We find that some of the resulting solutions exhibit a new internal degree of freedom (a moduli space parameter) that controls the energy density profile of the monopole while keeping the total energy (mass) constant.
- [8] arXiv:2507.13911 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]
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标题： 约旦自旋链用于$AdS_5\times S^5$中的扭力弦标题： Jordanian spin chains for twisted strings in $AdS_5\times S^5$评论： 52页，1图，2附录主题： 高能物理 - 理论 (hep-th) ; 统计力学 (cond-mat.stat-mech) ; 数学物理 (math-ph) ; 精确可解与可积系统 (nlin.SI)
我们研究了Jordanian变形的$AdS_5\times S^5$超弦的可积自旋链公式，该变形通过Drinfel'd扭曲实现。 在这些模型中，我们首先识别出一个局限于$SL(2,\mathbb{R})$扇区且具有常数标量场的唯一超引力变形。 然后，我们建立了一个闭合Drinfel'd扭曲自旋链的一般框架，并构建了一个到具有扭曲边界条件的未变形模型的显式映射。 应用于非紧致的$\mathrm{XXX}_{-1/2}$自旋链，Jordanian扭曲破坏了标记磁子激发的Cartan生成元，阻碍了标准的Bethe方法。 相反，使用扭曲边界形式，我们在热力学极限和短链情况下基于剩余根生成元启动了谱问题。 我们发现基态被非平凡地变形，并与经典弦结果一致，而我们的分析未能捕捉到高自旋激发态。 我们还研究了相关$Q$-系统的行为，其表现良好并与边界扭曲相容。 虽然对谱的完全理解仍然开放，但我们的工作为非阿贝尔扭曲可积模型的谱描述提供了具体的步骤。
We study the integrable spin chain formulation of Jordanian deformations of the $AdS_5\times S^5$ superstring, realised via Drinfel'd twists. Among these models, we first identify a unique supergravity deformation confined to an $SL(2,\mathbb{R})$ sector and with constant dilaton. We then develop a general framework for closed Drinfel'd twisted spin chains and construct an explicit map to undeformed models with twisted-boundary conditions. Applied to the non compact $\mathrm{XXX}_{-1/2}$ spin chain, the Jordanian twist breaks the Cartan generator labelling magnon excitations, obstructing the standard Bethe methods. Instead, using the twisted-boundary formulation, we initiate the spectral problem based on a residual root generator both in the thermodynamic limit and for short chains. We find that the ground state is non-trivially deformed, and is in agreement with the classical string result, while our analysis does not capture higher-spin excited states. We also study the asymptotics of the associated $Q$-system, which is well-behaved and compatible with the boundary-twist. While a full understanding of the spectrum remains open, our work provides concrete steps toward a spectral description of non-abelian twisted integrable models.
- [9] arXiv:2507.13997 (交叉列表自 math.DS) [中文pdf, pdf, html, 其他]
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标题： 基于等稳坐标系的慢流形识别与计算标题： Identification and Computation of Slow Manifolds Using the Isostable Coordinate System主题： 动力系统 (math.DS) ; 混沌动力学 (nlin.CD)
Koopman分析可用于以线性但通常为无限维算子的方式理解非线性动力系统的行为。 等相位坐标系关注最慢衰减的主要Koopman特征模式。 这项工作利用等相位坐标框架来识别具有固定点吸引子的动力系统的慢流形，这些流形定义为最快衰减的等相位坐标为零的表面。 与快慢时间尺度之间分离相关的数值挑战需要开发新的计算方法来识别这些慢流形。 开发了两种策略，这些策略近似从靠近固定点开始并在超出线性区域的慢流形上求解逆时间解。 应用于各种示例展示了这些方法的实用性及其在模型降阶方面的潜在用途。
Koopman analysis can be used to understand the dynamics of a nonlinear dynamical system in terms a linear, but generally infinite dimensional operator. The isostable coordinate system focuses on the slowest decaying principal Koopman eigenmodes. This work leverages the isostable coordinate framework in the identification of slow manifolds for dynamical systems with fixed point attractors, defined as surfaces for which the fastest decaying isostable coordinates are zero. Numerical challenges associated with separation between fast and slow timescales necessitate the development of new computational approaches to identify these slow manifolds. Two such strategies are developed which approximate backward-time solutions on the slow manifold starting near the fixed point and extending far beyond the linear regime. Application to a variety of examples illustrates the utility of these methods and their potential use for model order reduction purposes.
- [10] arXiv:2507.14040 (交叉列表自 math.DS) [中文pdf, pdf, html, 其他]
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标题： 马尔可夫矩阵扰动以优化动态和熵泛函标题： Markov matrix perturbations to optimize dynamical and entropy functionals主题： 动力系统 (math.DS) ; 混沌动力学 (nlin.CD)
在应用动力系统中，一个重要的问题是计算引发所需可观测量最大响应的外部扰动。 为此，我们研究了马尔可夫矩阵的扰动理论，并将其与统计物理中的线性响应理论联系起来。 我们使用微扰展开来推导出优化物理相关量的线性算法，例如：马尔可夫矩阵及其相关概率向量的熵、Kullback-Liebler散度和熵产生。 这些优化算法被应用于离散和连续流动的马尔可夫链表示，包括平衡内外的情况。 我们考虑源自转移算子的Ulam型近似和基于不稳定周期轨道理论的湍流流动的降阶模型的马尔可夫矩阵表示。 我们还提出了一种数值协议，将矩阵扰动重新表述为矢量场扰动。 结果允许以数据驱动的方式物理地解释获得的优化扰动，而无需了解底层方程。
An important problem in applied dynamical systems is to compute the external forcing that provokes the largest response of a desired observable quantity. For this, we investigate the perturbation theory of Markov matrices in connection with linear response theory in statistical physics. We use perturbative expansions to derive linear algorithms to optimize physically relevant quantities such as: entropy, Kullback-Liebler-divergence and entropy production of Markov matrices and their related probability vectors. These optimization algorithms are applied to Markov chain representations of discrete and continuous flows in and out of equilibrium. We consider Markov matrix representations originating from Ulam-type approximations of transfer operators and a reduced order model of a turbulent flow based on unstable periodic orbits theory. We also propose a numerical protocol to recast matrix perturbations into vector field perturbations. The results allow to physically interpret the obtained optimizing perturbations without knowledge of the underlying equations, in a data-driven way.
- [11] arXiv:2507.14062 (交叉列表自 q-bio.PE) [中文pdf, pdf, html, 其他]
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标题： 进化生态网络中的方向性度量：来自复杂自然模型的见解标题： Directionality measures in evolutionary ecological networks: Insights from the Tangled Nature model评论： 两个补充信息文件已提供主题： 种群与进化 (q-bio.PE) ; 适应性与自组织系统 (nlin.AO)
个体生物之间无数的微观相互作用在宏观尺度上共同导致了进化趋势。 能够检测这些趋势的方向性对于理解和管理自然系统的动态至关重要。 然而，确定能够捕捉这种方向性行为的关键可观测量是一个重大挑战。 在本研究中，我们提出将生态数据转换为网络框架是一种衡量系统稳定性和演化的有效策略。 我们以纠缠自然模型为例进行研究，评估网络熵、物种多样性以及聚类系数作为网络稳定性和方向性的度量指标。
The myriad microscopic interactions among the individual organisms that constitute an ecological system collectively give rise, at the macroscopic scale, to evolutionary trends. The ability to detect the directionality of such trends is crucial for understanding and managing the dynamics of natural systems. Nevertheless, identifying the key observable quantities that capture such directional behaviour poses a major challenge. In this study, we propose that translating ecological data into a network framework is a valuable strategy to measure system stability and evolution. We examine the Tangled Nature model as a test case, evaluating network entropy, species diversity, and the clustering coefficient as metrics of network stability and directionality.
交叉提交 (展示 8 之 8 条目 )
- [12] arXiv:2403.16845 (替换) [中文pdf, pdf, html, 其他]
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标题： 离散拉格朗日多形式用于ABS方程 II：四面体和八面体方程标题： Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations评论： v2被修订为两部分中的第二部分，第一部分arXiv:2501.13012提供了v1中缺失的基础讨论。v3被缩短以减少与arXiv:2501.13012的重叠。v4是发表版本期刊参考： SIGMA 21 (2025), 059, 27页主题： 精确可解与可积系统 (nlin.SI) ; 数学物理 (math-ph)
我们介绍了与ABS列表中的可积四边形方程相关的四种离散拉格朗日2-形式。 这些包括传统上用于ABS方程的拉格朗日多形式描述的三角形拉格朗日量，以及在本文第一部分中起关键作用的三叉戟拉格朗日量，还有两种在多形式设置中尚未研究的拉格朗日量。 其中两个拉格朗日2-形式的欧拉-拉格朗日方程为四边形方程，或与四边形方程等价的系统，而另一个则产生四面体方程。 这与三角形拉格朗日2-形式形成对比，后者产生的方程比四边形方程更弱（它们等价于两个八面体方程）。 我们利用拉格朗日2-形式之间的关系证明四边形方程组等价于四面体方程和八面体方程的组合系统。 此外，对于每个拉格朗日2-形式，我们研究外微分的双重零点性质。 特别是，这为八面体方程提供了一种可能的变分解释。
We present four types of discrete Lagrangian 2-form associated to the integrable quad equations of the ABS list. These include the triangle Lagrangian that has traditionally been used in the Lagrangian multiform description of ABS equations, the trident Lagrangian that was central to Part I of this paper, and two Lagrangians that have not been studied in the multiform setting. Two of the Lagrangian 2-forms have the quad equations, or a system equivalent to the quad equations, as their Euler-Lagrange equations, and one produces the tetrahedron equations. This is in contrast to the triangle Lagrangian 2-form, which produces equations that are weaker than the quad equations (they are equivalent to two octahedron equations). We use relations between the Lagrangian 2-forms to prove that the system of quad equations is equivalent to the combined system of tetrahedron and octahedron equations. Furthermore, for each of the Lagrangian 2-forms, we study the double zero property of the exterior derivative. In particular, this gives a possible variational interpretation to the octahedron equations.
- [13] arXiv:2501.13012 (替换) [中文pdf, pdf, html, 其他]
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标题： 离散拉格朗日多形式用于ABS方程 I：四边形方程标题： Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations评论： 第二部分是arXiv:2403.16845，v2：小的更正和澄清，v3：发表版本期刊参考： SIGMA 21 (2025), 058, 30页主题： 精确可解与可积系统 (nlin.SI) ; 数学物理 (math-ph)
离散拉格朗日多重形式理论是从变分角度研究在多维一致性意义上可积的格子方程的一种方法。 ABS分类方程的拉格朗日多重形式是该理论的起点，但通常在此背景下考虑的拉格朗日多重形式所产生的方程比ABS方程稍弱。 在本工作中，我们提出了替代的拉格朗日多重形式，其欧拉-拉格朗日方程与ABS方程等价。 此外，现有文献中对ABS拉格朗日多重形式的处理未能承认其定义中的复函数具有分支切割。 分支的选择会影响ABS方程的加法三腿形式的存在性以及拉格朗日多重形式的闭合性质。 我们给出了这两个性质的反例，但通过在作用量求和中包含与分支选择相关的整数值场，我们恢复了这些性质。
Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.
- [14] arXiv:2505.07363 (替换) [中文pdf, pdf, html, 其他]
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标题： 拉格朗日动力系统中的平衡传播学习标题： Equilibrium Propagation for Learning in Lagrangian Dynamical Systems评论： 9页，1图主题： 混沌动力学 (nlin.CD) ; 机器学习 (cs.LG) ; 数据分析、统计与概率 (physics.data-an)
我们提出了一种使用平衡传播训练由拉格朗日力学控制的动力系统的方法。 我们的方法通过利用作用极值原理，将最初为基于能量的模型开发的平衡传播扩展到动力轨迹。 通过轻轻推动轨迹向期望的目标靠近，并测量与待训练参数共轭的变量如何响应来实现训练。 这种方法特别适用于具有周期性边界条件或固定初始和最终状态的系统，能够在不需要通过时间的显式反向传播的情况下实现高效的参数更新。 在周期性边界条件的情况下，这种方法给出了量子平衡传播的半经典极限。 还讨论了在耗散系统中的应用。
We propose a method for training dynamical systems governed by Lagrangian mechanics using Equilibrium Propagation. Our approach extends Equilibrium Propagation - initially developed for energy-based models - to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring how the variables conjugate to the parameters to be trained respond. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of Quantum Equilibrium Propagation. Applications to systems with dissipation are also discussed.
- [15] arXiv:2210.11586 (替换) [中文pdf, pdf, html, 其他]
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标题： 球面和平面滚珠轴承——可积情况的研究标题： Spherical and Planar Ball Bearings -- a Study of Integrable Cases评论： 15页，3张图，最终版本 这篇论文是arXiv:2208.03009 [math-ph]的延续期刊参考： 《常规和混沌动力学》，28（2023）62-77主题： 数学物理 (math-ph) ; 动力系统 (math.DS) ; 精确可解与可积系统 (nlin.SI)
我们考虑非完整系统中的$n$个均匀球$\mathbf B_1,\dots,\mathbf B_n$，它们具有相同的半径$r$，在固定球$\mathbf S_0$上无滑动滚动，该固定球的中心为$O$，半径为$R$。 此外，假设一个动态非对称球 $\mathbf S$ 的中心与固定球 $\mathbf S_0$ 的中心 $O$ 相重合，在与运动球 $\mathbf B_1,\dots,\mathbf B_n$ 接触时无滑动滚动。 该问题考虑了四种不同的配置。 我们推导了运动方程，并证明这些系统具有不变测度。 作为主要结果，对于 $n=1$ 我们找到了两种根据欧拉-雅可比定理可进行初等积分的情况。 获得的可积非完整模型是对已知的恰普利金球可积问题的自然扩展。 此外，我们显式地整合了由$n$个半径相同但质量不同的均质球组成的平面问题，这些球在固定平面$\Sigma_0$上无滑动滚动，同时有一个平面$\Sigma$在这些球上无滑动移动。
We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping in contact to the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for $n=1$ we found two cases that are integrable in quadratures according to the Euler-Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem consisting of $n$ homogeneous balls of the same radius, but with different masses, that roll without slipping over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.
- [16] arXiv:2411.19779 (替换) [中文pdf, pdf, 其他]
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标题： 确定性多体动力学与多重分形响应标题： Deterministic many-body dynamics with multifractal response评论： 14页，15图主题： 统计力学 (cond-mat.stat-mech) ; 细胞自动机与格子气体 (nlin.CG)
动力系统可以表现出多种遍历性和遍历性破坏行为，从简单的周期性到遍历性和混沌。 在这里，我们报告了一个多体离散时间动力系统中的一种不寻常的非遍历行为，具体是在所有有理频率下平衡谱权重的多分形分布的多周期响应。 这种现象出现在新引入的所谓奇偶校验可逆细胞自动机类的一个动量守恒变体中，我们相对于任意二分格子对其进行定义。 尽管这些模型表现出配置相空间的强烈碎片化，但我们证明该效应在各个碎片化子空间中定性地持续存在，甚至在个别典型的多体轨迹中也是如此。 我们提供了在二维（六边形、正方形）和三维（立方体）晶格上的例子的详细数值分析。
Dynamical systems can display a plethora of ergodic and ergodicity breaking behaviors, ranging from simple periodicity to ergodicity and chaos. Here we report an unusual type of non-ergodic behavior in a many-body discrete-time dynamical system, specifically a multi-periodic response with multi-fractal distribution of equilibrium spectral weights at all rational frequencies. This phenomenon is observed in the momentum-conserving variant of the newly introduced class of the so-called parity check reversible cellular automata, which we define with respect to an arbitrary bi-partite lattice. Although the models display strong fragmentation of phase space of configurations, we demonstrate that the effect qualitatively persists within individual fragmented sectors, and even individual typical many-body trajectories. We provide detailed numerical analysis of examples on 2D (honeycomb, square) and 3D (cubic) lattices.
- [17] arXiv:2412.01924 (替换) [中文pdf, pdf, html, 其他]
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标题： 库隆德屏蔽在超导性存在下的情况标题： Kondo overscreening in the presence of superconductivity评论： 15+11页，5图主题： 强关联电子 (cond-mat.str-el) ; 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 精确可解与可积系统 (nlin.SI) ; 量子物理 (quant-ph)
我们考虑一个模型，描述了一个系统，其中超导性与过度屏蔽的Kondo效应竞争。该模型包括一个位于量子线边缘的单个自旋$-\frac{1}{2}$量子杂质，其中自旋$-1$的体费米子相互吸引，产生一个（超导）质量间隙。Kondo屏蔽与超导性的竞争导致了丰富的相结构。我们发现，对于强Kondo耦合，在相空间的一个区域内，Kondo相是稳定的，通过多粒子Kondo效应，杂质\textit{过度屏蔽}被屏蔽，并且动态生成了一个Kondo尺度。当体和边界相互作用强度相当的时候，我们发现谱中出现了一个中间能隙态并屏蔽了杂质，而在基态中，杂质未被屏蔽。这种中间能隙态类似于在BCS超导体整个相空间中存在的Yu-Shiba-Rusinov（YSR）态。此外，当体超导相互作用强度大于边界Kondo相互作用强度时，杂质将不再被屏蔽。进一步地，在Kondo相和YSR相之间，我们发现了一个新的相，虽然Kondo云过度屏蔽了杂质，但存在一个边界激发态，在热力学极限下其能量趋于零。类似的相图也在其他模型中发现：耗散Kondo系统，其中耗散与屏蔽竞争；耦合到自旋为1/2的吸引相互作用费米子的Kondo杂质，其中凝聚与屏蔽竞争；以及XXX-Kondo模型，其中晶格截断和体自旋相互作用与屏蔽竞争。
We consider a model describing a system where the superconductivity competes with the overscreened Kondo effect. The model consists of a single spin$-\frac{1}{2}$ quantum impurity at the edge of a quantum wire where spin$-1$ bulk fermions interact attractively, generating a (superconducting) mass gap. The competition between the Kondo screening and the superconductivity leads to a rich phase structure. We find that for strong Kondo coupling, there is a regime of phase space where the Kondo phase is stable with the impurity \textit{overscreened} by a multiparticle Kondo effect, and a Kondo scale is dynamically generated. When the bulk and boundary interaction strength are comparable, we find that a midgap state appears in the spectrum and screens the impurity, while in the ground state, the impurity is unscreened. This midgap state is akin to the Yu-Shiba-Rusinov (YSR) states that exist in the entire phase space in the BCS superconductor. Moreover, when the bulk superconducting interaction strength is stronger than the boundary Kondo interaction strength, the impurity can no longer be screened. Further, between the Kondo and YSR phases, we find a novel phase where, while the Kondo cloud overscreens the impurity, a boundary excitation exists that has vanishing energy in the thermodynamic limit. Similar phase diagrams that result from competition between different mechanisms were found for other models, too: the dissipative Kondo system, where dissipation competes with screening; the Kondo impurity coupled to spin-1/2 attractively interacting fermions where condensation competes with screening; and the XXX-Kondo model, where the lattice cutoff and the bulk spin interaction compete with screening.
- [18] arXiv:2412.09100 (替换) [中文pdf, pdf, html, 其他]
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标题： 广义Liénard系统和等时连接标题： Generalized Liénard systems and isochronous connections评论： v1：欢迎提出评论；v2：将发表于IJTP主题： 量子物理 (quant-ph) ; 数学物理 (math-ph) ; 精确可解与可积系统 (nlin.SI) ; 经典物理 (physics.class-ph)
在本文中，我们探讨了非线性Liénard方程$\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$的经典和量子方面，其中$x=x(t)$是一个实变量，$k, \omega \in \mathbb{R}$。我们证明该方程可以从Levinson-Smith类型的方程导出，其形式为$\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$，其中$z=z(t)$是一个实变量，$\{J(z), F(z), G(z)\}$是需要指定的适当函数。通过使用非局部变换，可以进一步将其映射到简谐振子，从而确立其等时性。利用Jacobi最后乘数进行计算表明，该系统表现出双哈密顿特性，即有两个不同类型的哈密顿量描述该系统。对于每一个哈密顿量，我们在动量表象中进行规范量子化，并探索束缚态的可能性。虽然其中一个哈密顿量显示出等间距谱并具有无限状态塔，另一个哈密顿量则表现出分支，但对于某些参数选择可以以闭合形式精确求解。
In this paper, we explore some classical and quantum aspects of the nonlinear Li\'enard equation $\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$, where $x=x(t)$ is a real variable and $k, \omega \in \mathbb{R}$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$, where $z=z(t)$ is a real variable and $\{J(z), F(z), G(z)\}$ are suitable functions to be specified. It can further be mapped to the harmonic oscillator by making use of a nonlocal transformation, establishing its isochronicity. Computations employing the Jacobi last multiplier reveal that the system exhibits a bi-Hamiltonian character, i.e., there are two distinct types of Hamiltonians describing the system. For each of these, we perform a canonical quantization in the momentum representation and explore the possibility of bound states. While one of the Hamiltonians is seen to exhibit an equispaced spectrum with an infinite tower of states, the other one exhibits branching but can be solved exactly in closed form for certain choices of the parameters.
- [19] arXiv:2502.19972 (替换) [中文pdf, pdf, html, 其他]
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标题： 超椭圆sigma函数与Kadomtsev-Petviashvili方程标题： Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation评论： 24页期刊参考： 物理D：非线性现象，第481卷，（2025），134819主题： 数学物理 (math-ph) ; 代数几何 (math.AG) ; 精确可解与可积系统 (nlin.SI)
在本文中，基于多维sigma函数发展了一种双椭圆函数理论，并得到了Kadomtsev-Petviashvili方程KP-I和KP-II的双椭圆解的显式公式。 描述这些解依赖于定义双椭圆曲线方程的系数变化的问题，即这些系数是方程的积分，这一长期存在的问题得到了解决。
In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The long-standing problem of describing the dependence of these solutions on the variation of the coefficients of the defining equation of a hyperelliptic curve, which are integrals of the equations, is solved.
- [20] arXiv:2504.04304 (替换) [中文pdf, pdf, html, 其他]
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标题： 多体驱动系统中脐点的普遍性标题： Universality in many-body driven systems with an umbilic point评论： v2：增加了Burgers系统的数值模拟，参考文献，与参考文献[10]中的数值数据的比较主题： 统计力学 (cond-mat.stat-mech) ; 细胞自动机与格子气体 (nlin.CG)
我们研究硬核粒子的两车道模型中守恒慢模式的稳态涨落，预计这将表现出普遍行为。 具体而言，我们关注特征速度相等的特殊脐点处的涨落特性。 在大空间和时间尺度下，涨落由最近在[13]中研究的一组随机伯格斯方程描述。 我们的数据表明耦合依赖的标度函数，甚至更令人惊讶的是，耦合依赖的动力学标度指数，与表面生长过程典型的KPZ标度指数不同。
We study stationary fluctuations of conserved slow modes in a two-lane model of hardcore particles which are expected to show universal behaviour. Specifically, we focus on the properties of fluctuations at a special umbilic point where the characteristic velocities coincide. At large space and time scales, fluctuations are described by a system of stochastic Burgers equations studied recently in [13]. Our data suggest coupling-dependent scaling functions and, even more surprisingly, coupling-dependent dynamical scaling exponents, distinct from KPZ scaling exponent typical for surface growth processes.
- [21] arXiv:2507.10863 (替换) [中文pdf, pdf, 其他]
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标题： 非平衡双温度$(T_x, T_y)$诺斯-霍弗细胞模型中的混沌标题： Chaos in Nonequilibrium Two-Temperature $(T_x, T_y)$ Nosé-Hoover Cell Models主题： 统计力学 (cond-mat.stat-mech) ; 混沌动力学 (nlin.CD)
我们重新研究了一个嵌入在二维周期性2x2单元中的双温度Nosé-Hoover游动粒子，该单元在$(x,y) = (\pm 1, \pm 1)$处有四个光滑的排斥角落，以探索各向异性恒温器引起的混沌。 该模型在x和y方向上使用独立的恒温器，从而实现对平衡的受控偏离。 通过积分完整的六维运动方程并计算完整的李雅普诺夫谱，我们确认了混沌，并以高数值精度量化了相空间收缩。 总收缩率被解释为熵产生，随着恒温器各向异性非线性增长，并遵循超二次幂律$\Lambda\propto -\delta^{2.44}$，偏离了线性响应理论。 近似的Kaplan-Yorke维数揭示了一个分形吸引子，随着$|T_x - T_y|$的增加而集中。 动量统计显示在强驱动下表现出显著的非高斯行为。 尽管该模型具有耗散性，但仍保持严格的时间可逆性，为微观可逆性与宏观熵产生共存提供了一个教学丰富的例子。
We revisit a two-temperature Nos\'e-Hoover wanderer particle embedded in a two-dimensional periodic 2x2 cell with four smooth repulsive corners at $(x,y) = (\pm 1, \pm 1)$ to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction with high numerical precision. The total contraction rate, interpreted as entropy production, grows nonlinearly with the thermostat anisotropy and follows a superquadratic power law, $\Lambda\propto -\delta^{2.44}$, deviating from linear-response theory. The approximate Kaplan-Yorke dimension reveals a fractal attractor that concentrates as $|T_x - T_y|$ increases. Momentum statistics show significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.
- [22] arXiv:2507.13310 (替换) [中文pdf, pdf, 其他]
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标题： 在网络上的参与对线下抗议的溢出效应建模：随机动力学和平均场近似标题： Modelling the spillover from online engagement to offline protest: stochastic dynamics and mean-field approximations on networks评论： 44页，33图主题： 物理与社会 (physics.soc-ph) ; 社会与信息网络 (cs.SI) ; 动力系统 (math.DS) ; 适应性与自组织系统 (nlin.AO) ; 种群与进化 (q-bio.PE)
社交媒体正在改变线下生活的各个方面，从日常决策如餐饮选择到冲突的发展进程。 在本研究中，我们提出一个耦合建模框架，包含一个在线社交网络层，以分析特定主题上的参与如何溢出到线下抗议活动。 我们开发了一个随机模型，并推导出几种不同复杂度的平均场模型。 这些模型使我们能够估计繁殖数并预测活动激增可能发生的时间。 一个关键因素是在线和线下领域之间的传播率；为了出现线下爆发，这一比率必须处于一个临界范围内，既不太低也不太高。 此外，利用合成网络，我们研究了网络结构如何影响这些近似值的准确性。 我们的研究结果表明，低密度网络需要更复杂的近似，而简单的模型可以有效地表示高密度网络。 然而，在两个现实世界的网络上进行测试时，增加复杂度并未提高准确性。
Social media is transforming various aspects of offline life, from everyday decisions such as dining choices to the progression of conflicts. In this study, we propose a coupled modelling framework with an online social network layer to analyse how engagement on a specific topic spills over into offline protest activities. We develop a stochastic model and derive several mean-field models of varying complexity. These models allow us to estimate the reproductive number and anticipate when surges in activity are likely to occur. A key factor is the transmission rate between the online and offline domains; for offline outbursts to emerge, this rate must fall within a critical range, neither too low nor too high. Additionally, using synthetic networks, we examine how network structure influences the accuracy of these approximations. Our findings indicate that low-density networks need more complex approximations, whereas simpler models can effectively represent higher-density networks. When tested on two real-world networks, however, increased complexity did not enhance accuracy.