Pub Date : 2025-04-11DOI: 10.1016/j.physd.2025.134652
Elie Abdo , Ruimeng Hu , Quyuan Lin
Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.
{"title":"Neural network solutions to the critical SQG equations via approximating nonlocal periodic operators","authors":"Elie Abdo , Ruimeng Hu , Quyuan Lin","doi":"10.1016/j.physd.2025.134652","DOIUrl":"10.1016/j.physd.2025.134652","url":null,"abstract":"<div><div>Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134652"},"PeriodicalIF":2.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824171","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}
Pub Date : 2025-04-10DOI: 10.1016/j.physd.2025.134669
Boumediene Hamzi , Marcus Hutter , Houman Owhadi
Machine Learning (ML) and Algorithmic Information Theory (AIT) offer distinct yet complementary approaches to understanding and addressing complexity. This paper investigates the synergy between these disciplines in two directions: AIT for Kernel Methods and Kernel Methods for AIT. In the former, we explore how AIT concepts inspire the design of kernels that integrate principles like relative Kolmogorov complexity and normalized compression distance (NCD). We propose a novel clustering method utilizing the Minimum Description Length principle, implemented via K-means and Kernel Mean Embedding (KME). Additionally, we apply the Loss Rank Principle (LoRP) to learn optimal kernel parameters in the context of Kernel Density Estimation (KDE), thereby extending the applicability of AIT-inspired techniques to flexible, nonparametric models. In the latter, we show how kernel methods can be used to approximate measures such as NCD and Algorithmic Mutual Information (AMI), providing new tools for compression-based analysis. Furthermore, we demonstrate that the Hilbert–Schmidt Independence Criterion (HSIC) approximates AMI, offering a robust theoretical foundation for clustering and other dependence-measurement tasks. Building on our previous work introducing Sparse Kernel Flows from an AIT perspective, we extend these ideas to unsupervised learning, enhancing the theoretical robustness and interpretability of ML algorithms. Our results demonstrate that kernel methods are not only versatile tools for ML but also crucial for bridging AIT and ML, enabling more principled approaches to unsupervised learning tasks.
{"title":"Bridging Algorithmic Information Theory and Machine Learning: Clustering, density estimation, Kolmogorov complexity-based kernels, and kernel learning in unsupervised learning","authors":"Boumediene Hamzi , Marcus Hutter , Houman Owhadi","doi":"10.1016/j.physd.2025.134669","DOIUrl":"10.1016/j.physd.2025.134669","url":null,"abstract":"<div><div>Machine Learning (ML) and Algorithmic Information Theory (AIT) offer distinct yet complementary approaches to understanding and addressing complexity. This paper investigates the synergy between these disciplines in two directions: <em>AIT for Kernel Methods</em> and <em>Kernel Methods for AIT</em>. In the former, we explore how AIT concepts inspire the design of kernels that integrate principles like relative Kolmogorov complexity and normalized compression distance (NCD). We propose a novel clustering method utilizing the Minimum Description Length principle, implemented via K-means and Kernel Mean Embedding (KME). Additionally, we apply the Loss Rank Principle (LoRP) to learn optimal kernel parameters in the context of Kernel Density Estimation (KDE), thereby extending the applicability of AIT-inspired techniques to flexible, nonparametric models. In the latter, we show how kernel methods can be used to approximate measures such as NCD and Algorithmic Mutual Information (AMI), providing new tools for compression-based analysis. Furthermore, we demonstrate that the Hilbert–Schmidt Independence Criterion (HSIC) approximates AMI, offering a robust theoretical foundation for clustering and other dependence-measurement tasks. Building on our previous work introducing Sparse Kernel Flows from an AIT perspective, we extend these ideas to unsupervised learning, enhancing the theoretical robustness and interpretability of ML algorithms. Our results demonstrate that kernel methods are not only versatile tools for ML but also crucial for bridging AIT and ML, enabling more principled approaches to unsupervised learning tasks.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134669"},"PeriodicalIF":2.7,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821114","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}
Pub Date : 2025-04-08DOI: 10.1016/j.physd.2025.134653
Ali Rehman, Ranchao Wu
Bacterial infections challenge the immune system, causing inflammation in which leukocytes play an important role in identifying and combating harmful bacteria. These white blood cells, leukocytes, navigate to infection sites by chemotaxis, which is guided by chemical cues from bacteria. The leukocytes then either engulf and destroy the harmful bacteria or release enzymes to neutralize the infection. This phenomenon is crucial for controlling infections and preventing their spread. However, this process is influenced by memory effects, which cause their movement to be affected by previous signals, as well as reaction delays. These variables complicate immune responses, thus understanding their impact on infection dynamics and inflammation is critical for developing better treatments. In this paper, we analyze a diffusive bacterial infection model with a spatial memory effect, taking into account the impact of delay on the movement of leukocytes. Through stability and bifurcation analysis, we obtain the sufficient and necessary conditions for the Hopf bifurcation and stability switches. It is found that in the absence of delay the system remains stable under certain conditions. However, in the presence of time delay, the system will undergo the Hopf bifurcation, when the time delay exceeds a critical threshold, and the stability of the equilibrium point is affected by the memory delay, leading to inhomogeneous spatially periodic oscillations. Moreover, we explore the occurrence of Hopf–Hopf bifurcation and the stability switches. The induced Hopf–Hopf bifurcation is further studied in detail based on normal form theory and the center manifold theorem. Finally, numerical simulations are provided to validate our theoretical findings.
{"title":"Hopf–Hopf bifurcation of the memory-based diffusive bacterial infection model","authors":"Ali Rehman, Ranchao Wu","doi":"10.1016/j.physd.2025.134653","DOIUrl":"10.1016/j.physd.2025.134653","url":null,"abstract":"<div><div>Bacterial infections challenge the immune system, causing inflammation in which leukocytes play an important role in identifying and combating harmful bacteria. These white blood cells, leukocytes, navigate to infection sites by chemotaxis, which is guided by chemical cues from bacteria. The leukocytes then either engulf and destroy the harmful bacteria or release enzymes to neutralize the infection. This phenomenon is crucial for controlling infections and preventing their spread. However, this process is influenced by memory effects, which cause their movement to be affected by previous signals, as well as reaction delays. These variables complicate immune responses, thus understanding their impact on infection dynamics and inflammation is critical for developing better treatments. In this paper, we analyze a diffusive bacterial infection model with a spatial memory effect, taking into account the impact of delay on the movement of leukocytes. Through stability and bifurcation analysis, we obtain the sufficient and necessary conditions for the Hopf bifurcation and stability switches. It is found that in the absence of delay the system remains stable under certain conditions. However, in the presence of time delay, the system will undergo the Hopf bifurcation, when the time delay exceeds a critical threshold, and the stability of the equilibrium point is affected by the memory delay, leading to inhomogeneous spatially periodic oscillations. Moreover, we explore the occurrence of Hopf–Hopf bifurcation and the stability switches. The induced Hopf–Hopf bifurcation is further studied in detail based on normal form theory and the center manifold theorem. Finally, numerical simulations are provided to validate our theoretical findings.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134653"},"PeriodicalIF":2.7,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799803","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}
Recognition of dynamic complexity changes in spontaneous speech signals can be regarded as a significant criterion for the early diagnosis of Alzheimer's disease (AD). Using the information embedded in spontaneous speech signals, in the framework of computational geometry; this paper introduces a new method for classifying speech diversity differences of healthy subjects compared to those with three stages of AD. Due to the dynamic and nonlinear nature of the speech signals, a weighted visibility graph (WVG) is proposed as a quantitative approach based on the concept of strength between nodes. The differential complexities of the network among the people of the four groups are analyzed using two criteria: average weighted degree and modularity. A long short-term memory (LSTM) network-based deep architecture is used to classify AD stages allied to its performance dealing with WVG-based features. The results show that the proposed algorithm has outstanding accuracy compared to its rivals in detecting the early stages of AD. It can classify speech signals into four groups with a high accuracy of 99.75%. In addition, the proposed approach has the potential to make it much easier to adopt the running state of the speech generation system and the central nervous system disorders affecting language skills by revealing significant differences between the speech reactions of the four mentioned groups. Therefore, it can be a valuable tool for evaluating AD in its preclinical stages.
{"title":"Weighted Visibility Graph-based Deep Complex Network Features: New Diagnostic Spontaneous Speech Markers of Alzheimer's Disease","authors":"Mahda Nasrolahzadeh , Zeynab Mohammadpoory , Javad Haddadnia","doi":"10.1016/j.physd.2025.134681","DOIUrl":"10.1016/j.physd.2025.134681","url":null,"abstract":"<div><div>Recognition of dynamic complexity changes in spontaneous speech signals can be regarded as a significant criterion for the early diagnosis of Alzheimer's disease (AD). Using the information embedded in spontaneous speech signals, in the framework of computational geometry; this paper introduces a new method for classifying speech diversity differences of healthy subjects compared to those with three stages of AD. Due to the dynamic and nonlinear nature of the speech signals, a weighted visibility graph (WVG) is proposed as a quantitative approach based on the concept of strength between nodes. The differential complexities of the network among the people of the four groups are analyzed using two criteria: average weighted degree and modularity. A long short-term memory (LSTM) network-based deep architecture is used to classify AD stages allied to its performance dealing with WVG-based features. The results show that the proposed algorithm has outstanding accuracy compared to its rivals in detecting the early stages of AD. It can classify speech signals into four groups with a high accuracy of 99.75%. In addition, the proposed approach has the potential to make it much easier to adopt the running state of the speech generation system and the central nervous system disorders affecting language skills by revealing significant differences between the speech reactions of the four mentioned groups. Therefore, it can be a valuable tool for evaluating AD in its preclinical stages.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134681"},"PeriodicalIF":2.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821115","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}
Pub Date : 2025-04-05DOI: 10.1016/j.physd.2025.134673
Qi-xiang Li (李玘祥) , You-sheng Zhang (张又升)
Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) turbulent mixing occur frequently in various natural phenomena and practical engineering applications. Accurate prediction of the evolution of mixing width, which comprises the bubble mixing width (BMW) and spike mixing width (SMW), holds significant scientific and engineering importance. Over the past several decades, buoyancy-drag models have been widely used to predict this evolution, but these models exhibit several limitations. In this paper, we propose a new buoyancy-drag model that incorporates additional physical constraints. The new model posits that the evolutions of the BMW and SMW are interrelated and mutually dependent. Consequently, we innovatively introduce the principle of mass conservation to link the evolution of SMW to that of BMW. The BMW is modeled using an ordinary differential equation (ODE) that includes inertial force, drag, and buoyancy terms. Accurate modeling of the inertial force term requires knowledge of the mean density profile, which we analytically derived for any density ratio by improving the previous density-ratio-invariant mean-species profile theory. The form and coefficient of the drag term were determined inversely by imposing the constraint that the ODE must predict the physical evolution of RM mixing. For the buoyancy term, we accounted for the entrainment phenomenon by introducing a density-ratio-dependent buoyancy coefficient additionally. The specific form of this coefficient was derived by requiring that the ODE predict the physical evolution of RT mixing as the density ratio approaches 1. Using a single set of coefficients, the new model successfully predicted the physical evolution of the mixing width across different density ratios and acceleration histories. This study enhances both the accuracy and robustness of the buoyancy-drag model.
{"title":"Improved buoyancy-drag model based on mean density profile and mass conservation principle","authors":"Qi-xiang Li (李玘祥) , You-sheng Zhang (张又升)","doi":"10.1016/j.physd.2025.134673","DOIUrl":"10.1016/j.physd.2025.134673","url":null,"abstract":"<div><div>Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) turbulent mixing occur frequently in various natural phenomena and practical engineering applications. Accurate prediction of the evolution of mixing width, which comprises the bubble mixing width (BMW) and spike mixing width (SMW), holds significant scientific and engineering importance. Over the past several decades, buoyancy-drag models have been widely used to predict this evolution, but these models exhibit several limitations. In this paper, we propose a new buoyancy-drag model that incorporates additional physical constraints. The new model posits that the evolutions of the BMW and SMW are interrelated and mutually dependent. Consequently, we innovatively introduce the principle of mass conservation to link the evolution of SMW to that of BMW. The BMW is modeled using an ordinary differential equation (ODE) that includes inertial force, drag, and buoyancy terms. Accurate modeling of the inertial force term requires knowledge of the mean density profile, which we analytically derived for any density ratio by improving the previous density-ratio-invariant mean-species profile theory. The form and coefficient of the drag term were determined inversely by imposing the constraint that the ODE must predict the physical evolution of RM mixing. For the buoyancy term, we accounted for the entrainment phenomenon by introducing a density-ratio-dependent buoyancy coefficient additionally. The specific form of this coefficient was derived by requiring that the ODE predict the physical evolution of RT mixing as the density ratio approaches 1. Using a single set of coefficients, the new model successfully predicted the physical evolution of the mixing width across different density ratios and acceleration histories. This study enhances both the accuracy and robustness of the buoyancy-drag model.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134673"},"PeriodicalIF":2.7,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815969","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}
Pub Date : 2025-04-04DOI: 10.1016/j.physd.2025.134648
Kathinka Frieswijk , Lorenzo Zino , A. Stephen Morse , Ming Cao
Climate change is the biggest global threat facing humanity in the coming decades. The scientific community agrees that human activity has been responsible for virtually all global heating over the past two centuries, emphasising the urgent need for the collective adoption of environmentally responsible behaviour. In this paper, we propose a novel behavioural–environmental mathematical model that explores the complex and nonlinear co-evolution of human environmental behaviour and anthropogenic environmental degradation. Our model considers a population of individuals, which includes climate change deniers, interacting on a polarised population structure. In addition to addressing climate change denial, our framework captures other key aspects of the climate crisis by modelling human behaviour through a social learning mechanism inspired by game theory that accounts for social influence, environmental sensitivity, government policies, and the costs associated with environmental-friendly actions. By employing a mean-field approach in the limit of large populations, we derive an analytically tractable set of equations that is easy to simulate. By analysing this set of equations, we shed light into the emergent behaviour of the system. Under reasonable assumptions, we demonstrate global convergence to a periodic solution, with oscillations influenced by climate change deniers and polarisation in a non-trivial manner, as discussed via a campaign of numerical simulations.
{"title":"A behavioural–environmental model to study the impact of climate change denial on environmental degradation","authors":"Kathinka Frieswijk , Lorenzo Zino , A. Stephen Morse , Ming Cao","doi":"10.1016/j.physd.2025.134648","DOIUrl":"10.1016/j.physd.2025.134648","url":null,"abstract":"<div><div>Climate change is the biggest global threat facing humanity in the coming decades. The scientific community agrees that human activity has been responsible for virtually all global heating over the past two centuries, emphasising the urgent need for the collective adoption of environmentally responsible behaviour. In this paper, we propose a novel behavioural–environmental mathematical model that explores the complex and nonlinear co-evolution of human environmental behaviour and anthropogenic environmental degradation. Our model considers a population of individuals, which includes climate change deniers, interacting on a polarised population structure. In addition to addressing climate change denial, our framework captures other key aspects of the climate crisis by modelling human behaviour through a social learning mechanism inspired by game theory that accounts for social influence, environmental sensitivity, government policies, and the costs associated with environmental-friendly actions. By employing a mean-field approach in the limit of large populations, we derive an analytically tractable set of equations that is easy to simulate. By analysing this set of equations, we shed light into the emergent behaviour of the system. Under reasonable assumptions, we demonstrate global convergence to a periodic solution, with oscillations influenced by climate change deniers and polarisation in a non-trivial manner, as discussed via a campaign of numerical simulations.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134648"},"PeriodicalIF":2.7,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In previous work, we showed that learning dynamical system Hamzi and Owhadi (2021) with kernel methods can achieve state of the art, both in terms of accuracy and complexity, for predicting climate/weather time series Hamzi et al., (2021) as well as for a family of 133 chaotic systems Lu et al., (2023), Yang et al., (2024), when the kernel is also learned from data. While the kernels considered in previous work were parametric, in this follow-up paper, we test a non-parametric approach and tune warping kernels (with kernel flows, a variant of cross-validation) for learning prototypical dynamical systems. We train the kernel using the regression relative error between two interpolants (measured in the RKHS norm of the kernel) as the quantity to minimize, as well as using the Maximum Mean Discrepancy between two different samples, and that characterizes the statistical properties of the dynamical system, as a the quantity to minimize.
{"title":"Learning dynamical systems from data: A simple cross-validation perspective, part II: Nonparametric kernel flows","authors":"Matthieu Darcy , Boumediene Hamzi , Jouni Susiluoto , Amy Braverman , Houman Owhadi","doi":"10.1016/j.physd.2025.134641","DOIUrl":"10.1016/j.physd.2025.134641","url":null,"abstract":"<div><div>In previous work, we showed that learning dynamical system Hamzi and Owhadi (2021) with kernel methods can achieve state of the art, both in terms of accuracy and complexity, for predicting climate/weather time series Hamzi et al., (2021) as well as for a family of 133 chaotic systems Lu et al., (2023), Yang et al., (2024), when the kernel is also learned from data. While the kernels considered in previous work were parametric, in this follow-up paper, we test a non-parametric approach and tune warping kernels (with kernel flows, a variant of cross-validation) for learning prototypical dynamical systems. We train the kernel using the regression relative error between two interpolants (measured in the RKHS norm of the kernel) as the quantity to minimize, as well as using the Maximum Mean Discrepancy between two different samples, and that characterizes the statistical properties of the dynamical system, as a the quantity to minimize.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134641"},"PeriodicalIF":2.7,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777650","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}
Pub Date : 2025-04-03DOI: 10.1016/j.physd.2025.134656
Yanfei Dai , Changjian Liu , Yangjian Sun
In this paper, the existence and number of periodic wave solutions in a perturbed generalized KdV equation of high-order with weak backward diffusion and dissipation effects are studied. These can be converted into studying the periodic waves on a manifold via geometric singular perturbation theory. By using bifurcation theory and analyzing the number of real zeros of some linear combination of Abelian integrals whose integrand and integral curve both have two arbitrarily high-order terms, we prove the persistence of periodic waves with certain wave speeds under small perturbation. The persistence of periodic waves for any energy parameter in an open interval and sufficiently small parameter is also established. Furthermore, the monotonicity of the limit wave speed is given, and the upper and lower bounds of limit wave speed are obtained. It is the first time to prove the existence of periodic waves in this kind of equation with two arbitrarily high-order nonlinearities.
{"title":"Periodic waves in the pgKdV equation with two arbitrarily high-order nonlinearities","authors":"Yanfei Dai , Changjian Liu , Yangjian Sun","doi":"10.1016/j.physd.2025.134656","DOIUrl":"10.1016/j.physd.2025.134656","url":null,"abstract":"<div><div>In this paper, the existence and number of periodic wave solutions in a perturbed generalized KdV equation of high-order with weak backward diffusion and dissipation effects are studied. These can be converted into studying the periodic waves on a manifold via geometric singular perturbation theory. By using bifurcation theory and analyzing the number of real zeros of some linear combination of Abelian integrals whose integrand and integral curve both have two arbitrarily high-order terms, we prove the persistence of periodic waves with certain wave speeds under small perturbation. The persistence of periodic waves for any energy parameter in an open interval and sufficiently small parameter is also established. Furthermore, the monotonicity of the limit wave speed is given, and the upper and lower bounds of limit wave speed are obtained. It is the first time to prove the existence of periodic waves in this kind of equation with two arbitrarily high-order nonlinearities.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134656"},"PeriodicalIF":2.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777100","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}
Pub Date : 2025-04-03DOI: 10.1016/j.physd.2025.134642
Alan C. Newell , Sergey V. Nazarenko
<div><div>This short paper is dedicated to Volodja Zakharov, a unique and original scientist, leader, mentor, poet and great friend for more than forty years. We will speak more of the man and his work in the body of the paper. One of his many and singular contributions to science was the development of a special and extremely relevant class of solutions for wave turbulence theory, the study of the statistical evolution of a sea of weakly nonlinear, dispersive waves (Zakharov et al., 1992). The closed kinetic equation describing the evolution of the spectral energy or, equivalently, number density, had been known for many years, and had been explicitly derived in the context of surface gravity waves by Hasselmann in 1962 (Hasselmann, 1962). Many works addressing the questions of natural closure (Benney and Saffman, 1966; Benney and Newell, 1969; Newell, 1968), other examples such as Rossby waves (Longuet-Higgins and Gill, 1967), surface tension dominated waves (Zakharov and Filonenko, 1967), plasma waves (Vedenov, 1967) soon followed. Before Zakharov, there was not much discussion of the statistical steady states, other than the equipartition spectra, to which the solutions of the kinetic equation might relax. The equipartition of conserved density solutions were obvious, readily seen by inspection. But, despite the fact that many of the western authors were familiar with the ideas of Kolmogorov in fully developed hydrodynamic turbulence, Zakharov (Zakharov, 1965; Zakharov and Filonenko 1967) was the only one who realized that there should also be statistical steady states in the wave turbulence context corresponding to the finite fluxes of the conserved densities such as energy and wave action from scales at which they were introduced to scales at which they were dissipated or absorbed. The kinetic equation has hidden symmetries that Zakharov understood should be there and he found them. They led him to solutions that are now called Kolmogorov-Zakharov (or KZ) spectra. For those insights, and in particular for the discovery of inverse fluxes, Zakharov, along with Kraichnan, was awarded the 2003 Dirac Medal. However, as recognized by the present authors (Newell et al. 2001), these solutions have limited validity in two respects. First, they are rarely universally valid throughout the whole spectrum. Second, in some cases certain integrals, associated with physically important functionals, may not converge. The term nonlocal is often used to describe such situations but quantitative definitions of local and nonlocal remain open challenges. Colloquially, local connotes that the dominant transfer is between neighboring (in scale) wavenumbers; nonlocal connotes that there is significant and direct transfer between widely separated scales. KZ solutions connote local, a cascade, a la Richardson (big whirls make little whirls that feed on their velocity …), of some conserved density. In this short paper, we discuss remedies for these two challenges. Fir
{"title":"Augmenting KZ finite flux solutions and nonlocal resonant transfer","authors":"Alan C. Newell , Sergey V. Nazarenko","doi":"10.1016/j.physd.2025.134642","DOIUrl":"10.1016/j.physd.2025.134642","url":null,"abstract":"<div><div>This short paper is dedicated to Volodja Zakharov, a unique and original scientist, leader, mentor, poet and great friend for more than forty years. We will speak more of the man and his work in the body of the paper. One of his many and singular contributions to science was the development of a special and extremely relevant class of solutions for wave turbulence theory, the study of the statistical evolution of a sea of weakly nonlinear, dispersive waves (Zakharov et al., 1992). The closed kinetic equation describing the evolution of the spectral energy or, equivalently, number density, had been known for many years, and had been explicitly derived in the context of surface gravity waves by Hasselmann in 1962 (Hasselmann, 1962). Many works addressing the questions of natural closure (Benney and Saffman, 1966; Benney and Newell, 1969; Newell, 1968), other examples such as Rossby waves (Longuet-Higgins and Gill, 1967), surface tension dominated waves (Zakharov and Filonenko, 1967), plasma waves (Vedenov, 1967) soon followed. Before Zakharov, there was not much discussion of the statistical steady states, other than the equipartition spectra, to which the solutions of the kinetic equation might relax. The equipartition of conserved density solutions were obvious, readily seen by inspection. But, despite the fact that many of the western authors were familiar with the ideas of Kolmogorov in fully developed hydrodynamic turbulence, Zakharov (Zakharov, 1965; Zakharov and Filonenko 1967) was the only one who realized that there should also be statistical steady states in the wave turbulence context corresponding to the finite fluxes of the conserved densities such as energy and wave action from scales at which they were introduced to scales at which they were dissipated or absorbed. The kinetic equation has hidden symmetries that Zakharov understood should be there and he found them. They led him to solutions that are now called Kolmogorov-Zakharov (or KZ) spectra. For those insights, and in particular for the discovery of inverse fluxes, Zakharov, along with Kraichnan, was awarded the 2003 Dirac Medal. However, as recognized by the present authors (Newell et al. 2001), these solutions have limited validity in two respects. First, they are rarely universally valid throughout the whole spectrum. Second, in some cases certain integrals, associated with physically important functionals, may not converge. The term nonlocal is often used to describe such situations but quantitative definitions of local and nonlocal remain open challenges. Colloquially, local connotes that the dominant transfer is between neighboring (in scale) wavenumbers; nonlocal connotes that there is significant and direct transfer between widely separated scales. KZ solutions connote local, a cascade, a la Richardson (big whirls make little whirls that feed on their velocity …), of some conserved density. In this short paper, we discuss remedies for these two challenges. Fir","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134642"},"PeriodicalIF":2.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.physd.2025.134654
Wenhai Shan , Guoqiang Ren
In this paper, we study a chemotaxis-based model for three-species intraguild predation model. We present global boundedness of classical solutions in any dimension without any restrictions on initial data and . Moreover, it is asserted that the considered system will approach prey-only steady state and semi-coexistence steady state in the large time limit. Finally, using the spectral analysis, we investigate the effect of Beddington-deAngelis type production term on the stability and instability of a linearized problem around the coexistence steady state, and we then extend our analysis to the nonlinear system, which makes up for the stability of coexistence steady state. Additionally, we apply our theoretical results to several concrete Holling type II functions, discuss the theoretical conditions through numerical simulations, and verify the predictions of our analysis.
{"title":"Dynamic behavior of taxis-driven intraguild predation model of three species with BD functional response","authors":"Wenhai Shan , Guoqiang Ren","doi":"10.1016/j.physd.2025.134654","DOIUrl":"10.1016/j.physd.2025.134654","url":null,"abstract":"<div><div>In this paper, we study a chemotaxis-based model for three-species intraguild predation model. We present global boundedness of classical solutions in any dimension without any restrictions on initial data and <span><math><mi>χ</mi></math></span>. Moreover, it is asserted that the considered system will approach prey-only steady state and semi-coexistence steady state in the large time limit. Finally, using the spectral analysis, we investigate the effect of Beddington-deAngelis type production term on the stability and instability of a linearized problem around the coexistence steady state, and we then extend our analysis to the nonlinear system, which makes up for the stability of coexistence steady state. Additionally, we apply our theoretical results to several concrete Holling type II functions, discuss the theoretical conditions through numerical simulations, and verify the predictions of our analysis.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134654"},"PeriodicalIF":2.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143785056","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号