Pub Date : 2024-06-24DOI: 10.1007/s00332-024-10053-3
Christian Bick, Tobias Böhle, Christian Kuehn
Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.
{"title":"Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude","authors":"Christian Bick, Tobias Böhle, Christian Kuehn","doi":"10.1007/s00332-024-10053-3","DOIUrl":"https://doi.org/10.1007/s00332-024-10053-3","url":null,"abstract":"<p>Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.\u0000</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503469","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: 10.1007/s00332-024-10056-0
Rafael López
In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its potential energy is determined by the distance to a given geodesic of the space. Several characterizations of the catenary are established in terms of the curvature of the curve and of the angle that its unit normal makes with a vector field of the ambient space. Furthermore, in the hyperbolic plane, we extend the concept of catenary substituting the reference geodesic by a horocycle or the hyperbolic distance by the horocycle distance.
{"title":"The Hanging Chain Problem in the Sphere and in the Hyperbolic Plane","authors":"Rafael López","doi":"10.1007/s00332-024-10056-0","DOIUrl":"https://doi.org/10.1007/s00332-024-10056-0","url":null,"abstract":"<p>In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its potential energy is determined by the distance to a given geodesic of the space. Several characterizations of the catenary are established in terms of the curvature of the curve and of the angle that its unit normal makes with a vector field of the ambient space. Furthermore, in the hyperbolic plane, we extend the concept of catenary substituting the reference geodesic by a horocycle or the hyperbolic distance by the horocycle distance.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503470","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 : 2024-06-13DOI: 10.1007/s00332-024-10055-1
Daniele Avitabile, N. Chemetov, Pedro M. Lima
{"title":"Well-Posedness and Regularity of Solutions to Neural Field Problems with Dendritic Processing","authors":"Daniele Avitabile, N. Chemetov, Pedro M. Lima","doi":"10.1007/s00332-024-10055-1","DOIUrl":"https://doi.org/10.1007/s00332-024-10055-1","url":null,"abstract":"","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141345535","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 : 2024-06-13DOI: 10.1007/s00332-024-10047-1
Sean P. McGowan, William S. P. Robertson, Chantelle Blachut, Sanjeeva Balasuriya
Predicting the evolution of dynamics from a given trajectory history of an unknown system is an important and challenging problem. This paper presents a model-free method of forecasting unknown chaotic systems through reconstructing vector fields from noisy measured data via an adaptation of optimal control methods. This technique is also applicable to partially observed systems using a Takens delay embedding approach. The algorithms are validated on the Lorenz system and the four-dimensional hyperchaotic Rössler system, and demonstrate successful predictions well beyond the Lyapunov timescale. It is found that for small datasets or datasets with large levels of noise, the prediction accuracy of partially observed systems approaches that of fully observed systems. The presented approach also allows the model-free assessment of local predictability on the attractor by evolving initial condition density through the reconstructed vector fields via estimation of the transfer operator. The method is compared to predictions made by an imperfect model which highlights the utility of model-free approaches when the only available models have significant model error. The capability of this method for reconstruction of continuous and global vector fields may be applied to model validation, forecasting of initial conditions not in the training set, and model-free filtering.
{"title":"Optimal Reconstruction of Vector Fields from Data for Prediction and Uncertainty Quantification","authors":"Sean P. McGowan, William S. P. Robertson, Chantelle Blachut, Sanjeeva Balasuriya","doi":"10.1007/s00332-024-10047-1","DOIUrl":"https://doi.org/10.1007/s00332-024-10047-1","url":null,"abstract":"<p>Predicting the evolution of dynamics from a given trajectory history of an unknown system is an important and challenging problem. This paper presents a model-free method of forecasting unknown chaotic systems through reconstructing vector fields from noisy measured data via an adaptation of optimal control methods. This technique is also applicable to partially observed systems using a Takens delay embedding approach. The algorithms are validated on the Lorenz system and the four-dimensional hyperchaotic Rössler system, and demonstrate successful predictions well beyond the Lyapunov timescale. It is found that for small datasets or datasets with large levels of noise, the prediction accuracy of partially observed systems approaches that of fully observed systems. The presented approach also allows the model-free assessment of local predictability on the attractor by evolving initial condition density through the reconstructed vector fields via estimation of the transfer operator. The method is compared to predictions made by an imperfect model which highlights the utility of model-free approaches when the only available models have significant model error. The capability of this method for reconstruction of continuous and global vector fields may be applied to model validation, forecasting of initial conditions not in the training set, and model-free filtering. </p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503471","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 : 2024-06-11DOI: 10.1007/s00332-024-10050-6
G. Fotopoulos, N. I. Karachalios, V. Koukouloyannis, P. Kyriazopoulos, K. Vetas
The study of nonlinear Schrödinger-type equations with nonzero boundary conditions introduces challenging problems both for the continuous (partial differential equation) and the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions, which describe solutions decaying on the top of a finite background, we provide a rigorous proof that for the corresponding initial boundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value (A_*) defined by the gain-loss parameters. We argue that this essential property of this infinite lattice cannot be captured by finite lattice approximations of the problem. Commonly, such approximations are provided by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite-dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude (A_*). Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude (A_*) in the long-time evolution of the system.
{"title":"The Discrete Nonlinear Schrödinger Equation with Linear Gain and Nonlinear Loss: The Infinite Lattice with Nonzero Boundary Conditions and Its Finite-Dimensional Approximations","authors":"G. Fotopoulos, N. I. Karachalios, V. Koukouloyannis, P. Kyriazopoulos, K. Vetas","doi":"10.1007/s00332-024-10050-6","DOIUrl":"https://doi.org/10.1007/s00332-024-10050-6","url":null,"abstract":"<p>The study of nonlinear Schrödinger-type equations with nonzero boundary conditions introduces challenging problems both for the continuous (partial differential equation) and the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions, which describe solutions decaying on the top of a finite background, we provide a rigorous proof that for the corresponding initial boundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value <span>(A_*)</span> defined by the gain-loss parameters. We argue that this essential property of this infinite lattice cannot be captured by finite lattice approximations of the problem. Commonly, such approximations are provided by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite-dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude <span>(A_*)</span>. Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude <span>(A_*)</span> in the long-time evolution of the system.\u0000</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503472","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 : 2024-06-10DOI: 10.1007/s00332-024-10045-3
Chunhua Jin
The occurrence of finite time blow-up phenomenon in the Keller–Segel (KS) model has always been a significant area of interest for mathematicians. Despite extensive research on the blow-up phenomenon in KS models with signal production, Understanding of this phenomenon in models with signal consumption mechanisms has been scarce.This paper marks a preliminary investigation into this unexplored field. In this study, we employ a backward self-similar solution to demonstrate that the finite time blowup indeed occurs within this model. More precisely, in one-dimensional space, finite time blowing up corresponding to the chemotactic collapse phenomenon (the formation of Dirac (delta )-singularity ) happens; in high-dimensional space, the self-similar solution will blow up everywhere. Finally, we also consider the special cases where the diffusion coefficient of bacteria or oxygen is 0. For these cases, chemotactic collapse phenomenon occurs in both one-dimensional and two-dimensional spaces.
{"title":"Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption","authors":"Chunhua Jin","doi":"10.1007/s00332-024-10045-3","DOIUrl":"https://doi.org/10.1007/s00332-024-10045-3","url":null,"abstract":"<p>The occurrence of finite time blow-up phenomenon in the Keller–Segel (KS) model has always been a significant area of interest for mathematicians. Despite extensive research on the blow-up phenomenon in KS models with signal production, Understanding of this phenomenon in models with signal consumption mechanisms has been scarce.This paper marks a preliminary investigation into this unexplored field. In this study, we employ a backward self-similar solution to demonstrate that the finite time blowup indeed occurs within this model. More precisely, in one-dimensional space, finite time blowing up corresponding to the chemotactic collapse phenomenon (the formation of Dirac <span>(delta )</span>-singularity ) happens; in high-dimensional space, the self-similar solution will blow up everywhere. Finally, we also consider the special cases where the diffusion coefficient of bacteria or oxygen is 0. For these cases, chemotactic collapse phenomenon occurs in both one-dimensional and two-dimensional spaces.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503522","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 : 2024-06-09DOI: 10.1007/s00332-024-10052-4
Tiago Carvalho, Douglas D. Novaes, Durval J. Tonon
We consider piecewise smooth vector fields (Z=(Z_+, Z_-)) defined in ({mathbb {R}}^n) where both vector fields are tangent to the switching manifold (Sigma ) along a submanifold (Msubset Sigma ). We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on M, governed by what we call the tangential sliding vector field. Here, we will provide the necessary and sufficient conditions for characterizing such a vector field. Additionally, we prove that the tangential sliding vector field is conjugated to the reduced dynamics of a singular perturbation problem arising from the Sotomayor–Teixeira regularization of Z around M. Finally, we analyze several examples where tangential sliding vector fields can be observed, including a model for intermittent treatment of HIV.
我们考虑在({mathbb {R}}^n) 中定义的片断光滑向量场(Z=(Z_+, Z_-)),其中两个向量场都沿着一个子流形(Msubset Sigma )切向切换流形(Sigma )。我们将看到,在合适的假设条件下,菲利波夫惯例会在 M 上产生一种独特的滑动模式,它受我们称之为切向滑动矢量场的支配。在此,我们将提供描述这种向量场的必要条件和充分条件。此外,我们还将证明切向滑动矢量场与 M 周围 Z 的索托马约尔-特谢拉正则化所产生的奇异扰动问题的还原动力学共轭。最后,我们将分析可以观察到切向滑动矢量场的几个例子,其中包括艾滋病间歇治疗模型。
{"title":"Sliding Mode on Tangential Sets of Filippov Systems","authors":"Tiago Carvalho, Douglas D. Novaes, Durval J. Tonon","doi":"10.1007/s00332-024-10052-4","DOIUrl":"https://doi.org/10.1007/s00332-024-10052-4","url":null,"abstract":"<p>We consider piecewise smooth vector fields <span>(Z=(Z_+, Z_-))</span> defined in <span>({mathbb {R}}^n)</span> where both vector fields are tangent to the switching manifold <span>(Sigma )</span> along a submanifold <span>(Msubset Sigma )</span>. We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on <i>M</i>, governed by what we call the <i>tangential sliding vector field</i>. Here, we will provide the necessary and sufficient conditions for characterizing such a vector field. Additionally, we prove that the tangential sliding vector field is conjugated to the reduced dynamics of a singular perturbation problem arising from the Sotomayor–Teixeira regularization of <i>Z</i> around <i>M</i>. Finally, we analyze several examples where tangential sliding vector fields can be observed, including a model for intermittent treatment of HIV.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527977","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 : 2024-06-06DOI: 10.1007/s00332-024-10048-0
Rong Zhou, Shi-Liang Wu, Xiong-Xiong Bao
In this paper, we study the existence of spread speeds and periodic traveling waves for a class of space–time periodic and degenerate cooperative systems with nonlocal dispersal and delays. In order to characterize the spread speeds, we first establish the principal eigenvalue theory for linear space–time periodic and degenerate systems with nonlocal dispersal and delays and give some sufficient conditions for the existence of the principal eigenvalue. Then, we prove the existence of a single spreading speed and give the computational formulae of it. Finally, we generalize the monotone iteration scheme combined with the method of sub–super-solutions to prove the existence of the space–time periodic traveling waves.
{"title":"Propagation Dynamics for a Degenerate Delayed System with Nonlocal Dispersal in Periodic Habitats","authors":"Rong Zhou, Shi-Liang Wu, Xiong-Xiong Bao","doi":"10.1007/s00332-024-10048-0","DOIUrl":"https://doi.org/10.1007/s00332-024-10048-0","url":null,"abstract":"<p>In this paper, we study the existence of spread speeds and periodic traveling waves for a class of space–time periodic and degenerate cooperative systems with nonlocal dispersal and delays. In order to characterize the spread speeds, we first establish the principal eigenvalue theory for linear space–time periodic and degenerate systems with nonlocal dispersal and delays and give some sufficient conditions for the existence of the principal eigenvalue. Then, we prove the existence of a single spreading speed and give the computational formulae of it. Finally, we generalize the monotone iteration scheme combined with the method of sub–super-solutions to prove the existence of the space–time periodic traveling waves.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527978","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 : 2024-05-27DOI: 10.1007/s00332-024-10032-8
Daniel W. Boutros, Simon Markfelder, Edriss S. Titi
We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities.
我们开发了一种凸积分方案,用于构建二维和三维静水欧拉方程(也称为海洋和大气动力学的无粘性原始方程)的非唯一弱解。我们还为构建三维粘性原始方程和二维普朗特方程的非唯一弱解开发了这样一种方案。在布特罗斯等人(Calc Var Partial Differ Equ 62(8):219, 2023)的文章中,对静水欧拉方程的经典弱解概念进行了概括,而我们在此引入了进一步的概括。对于这种广义弱解,我们证明了一大类初始数据的存在性和非唯一性。此外,我们还构建了无限多的广义弱解实例,这些广义弱解不保存能量。所构建的静力学欧拉方程的气压和气压线性解(分别是水平速度在 z 坐标上的平均值和波动值)具有不同的规律性。
{"title":"Nonuniqueness of Generalised Weak Solutions to the Primitive and Prandtl Equations","authors":"Daniel W. Boutros, Simon Markfelder, Edriss S. Titi","doi":"10.1007/s00332-024-10032-8","DOIUrl":"https://doi.org/10.1007/s00332-024-10032-8","url":null,"abstract":"<p>We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the <i>z</i>-coordinate, respectively) that are constructed have different regularities.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169384","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 : 2024-05-25DOI: 10.1007/s00332-024-10046-2
Dan J. Hill
Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as ‘fairy circles’. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction–diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction–diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.
{"title":"Predicting the Emergence of Localised Dihedral Patterns in Models for Dryland Vegetation","authors":"Dan J. Hill","doi":"10.1007/s00332-024-10046-2","DOIUrl":"https://doi.org/10.1007/s00332-024-10046-2","url":null,"abstract":"<p>Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as ‘fairy circles’. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction–diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction–diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148600","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}