Physica D: Nonlinear Phenomena最新文献

{"title":"Four-fifths laws in incompressible and magnetized fluids: Helicity, energy and cross-helicity","authors":"Yulin Ye ,&nbsp;Yanqing Wang ,&nbsp;Otto Chkhetiani","doi":"10.1016/j.physd.2025.134655","DOIUrl":"10.1016/j.physd.2025.134655","url":null,"abstract":"<div><div>In this paper, we are concerned with the Kolmogorov’s scaling laws of conserved quantities in incompressible fluids. By means of Eyink’s longitudinal structure functions and the analysis of interaction of different physical quantities, we extend celebrated four-fifths laws from energy to helicity in incompressible fluid and, total energy and cross-helicity in magnetohydrodynamic flow. In contrast to pervious 4/5 laws of energy and cross-helicity in magnetized fluids obtained by Politano and Pouquet, they are in terms of the mixed third-order structure functions rather than the structure coupling correlation functions. New insights to the cascade rate of inviscid invariants are provided in incompressible turbulence.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134655"},"PeriodicalIF":2.7,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kernel methods for the approximation of the eigenfunctions of the Koopman operator","authors":"Jonghyeon Lee ,&nbsp;Boumediene Hamzi ,&nbsp;Boya Hou ,&nbsp;Houman Owhadi ,&nbsp;Gabriele Santin ,&nbsp;Umesh Vaidya","doi":"10.1016/j.physd.2025.134662","DOIUrl":"10.1016/j.physd.2025.134662","url":null,"abstract":"<div><div>The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct computations. In this paper, we introduce a kernel-based method to construct the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. These principal eigenfunctions are associated with the equilibrium dynamics, and their eigenvalues match those of the linearization of the nonlinear system at the equilibrium point. We exploit the structure of the principal eigenfunctions by decomposing them into linear and nonlinear components. The linear part corresponds to the left eigenvector of the system’s linearization at the equilibrium, while the nonlinear part is obtained by solving a partial differential equation (PDE) using kernel methods. Our approach avoids common issues such as spectral pollution and spurious eigenvalues, which can arise in previous methods. We demonstrate the effectiveness of our algorithm through numerical examples.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134662"},"PeriodicalIF":2.7,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bifurcations patterns and heat transmissions in couple-stress fluid layer with isothermal rigid boundaries","authors":"Purbasha Deb,&nbsp;G.C. Layek","doi":"10.1016/j.physd.2025.134672","DOIUrl":"10.1016/j.physd.2025.134672","url":null,"abstract":"<div><div>In this work, we investigate the dynamical patterns of thermo-convective loops in a horizontal shallow layer of couple-stress fluid confined between isothermal rigid boundaries and heated from beneath. The novelty of this study lies in examining the influences of couple-stresses on the dynamical patterns of fluid convection in the presence of rigid boundaries. Both the linear and non-linear stability analyses are performed. It is found that the critical Rayleigh number for the onset of convection increases significantly with the couple-stress parameter (<span><math><mi>C</mi></math></span>). Using low-order Galerkin approximations within the framework of non-linear stability analysis, a three-dimensional, non-linear, dissipative system governed by four control parameters is derived. Studies reveal that the transitions to the stationary and oscillatory convections (through pitchfork and Hopf bifurcations, respectively) detain with the enhancement of <span><math><mi>C</mi></math></span>, consistent with observations for free-isothermal boundaries. Notably, a striking outcome of this research is that, unlike the stress-free case, the subcritical Hopf bifurcation evolves to supercritical one as <span><math><mi>C</mi></math></span> exceeds a critical threshold of approximately 0.1022708, and it fundamentally alters the flow dynamics. At this threshold value of the parameter <span><math><mi>C</mi></math></span>, the system experiences a codimension-2 Bautin bifurcation, which is not likely to appear in the classical Lorenz system for realistic parameters values. Furthermore, the mode of heat transport stabilizes from convection to conduction with increasing <span><math><mi>C</mi></math></span>. Variations in the stream function and isotherm function with respect to <span><math><mi>C</mi></math></span> are analyzed and depicted. Additionally, the effect of couple-stresses on the chaotic regime at a high reduced normalized Rayleigh number (<span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>200</mn></mrow></math></span>) exhibits intermittent behavior, and chaos is entirely suppressed for a suitable value of <span><math><mi>C</mi></math></span>, indicating the stabilization of the system.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134672"},"PeriodicalIF":2.7,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Euler equation for incoherent fluid in curved spaces","authors":"B.G. Konopelchenko ,&nbsp;G. Ortenzi","doi":"10.1016/j.physd.2025.134667","DOIUrl":"10.1016/j.physd.2025.134667","url":null,"abstract":"<div><div>Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph equations. These hodograph equations provide us with various classes of solutions of the Euler equation, including stationary solutions. Particular cases of cone and sphere in the 3-dimensional Euclidean space are analysed in detail. Euler equation on the sphere in the 4-dimensional Euclidean space is considered too.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134667"},"PeriodicalIF":2.7,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact expression for the propagating front velocity in nonlinear discrete systems under nonreciprocal coupling","authors":"David Pinto-Ramos","doi":"10.1016/j.physd.2025.134665","DOIUrl":"10.1016/j.physd.2025.134665","url":null,"abstract":"<div><div>Nonlinear waves are a robust phenomenon observed in complex systems ranging from mechanics to ecology. Fronts into the stable state, nonlinear waves appearing in bistable and multistable systems, are fundamental due to their robustness against perturbations and capacity to propagate one state over another. Controlling and understanding these waves is then crucial to make use of their properties. Their velocity is one of the most important features, which can be analytically computed only for specific dynamical systems and under restricted conditions on the parameters, and it becomes more elusive in the presence of spatial discreteness and nonreciprocal coupling. A key difficulty in developing expressions for the front velocity is the lack of a front rigid shape exploiting translational invariance, a property that is broken in discrete systems with a finite number of elements. This work reveals that fronts in discrete systems can be treated as rigid objects when analyzing their whole trajectory by collecting the system state at each time step instead of just observing the instantaneous, current state. Then, a relationship between the front velocity and its reconstructed rigid shape is found. Applying this method to a generic model for nonreciprocally coupled bistable systems reveals that the derived velocity formula provides insight into fronts’ long-observed properties, such as the oscillatory trajectory for the front position and the pinning–depinning transition, and agrees with the approximative and parameterized methods described in the literature. Furthermore, it reveals an explicit linear relationship between the velocity and the nonreciprocal coupling constant. Numerical simulations show perfect agreement with the theory.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134665"},"PeriodicalIF":2.7,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Action of viscous stresses on the Young–Laplace equation in Hele-Shaw flows: A gap-averaged calculation","authors":"Eduardo O. Dias,&nbsp;Írio M. Coutinho,&nbsp;José A. Miranda","doi":"10.1016/j.physd.2025.134663","DOIUrl":"10.1016/j.physd.2025.134663","url":null,"abstract":"<div><div>The Saffman–Taylor (or, viscous fingering) instability arises when a less viscous fluid displaces a more viscous one in the narrow gap of a Hele-Shaw cell. The dynamics of the fluid–fluid interface is usually described by a set of gap-averaged equations, including Darcy’s law and fluid incompressibily, supported by the pressure jump (Young–Laplace equation) and kinematic boundary conditions. Over the past two decades, various research groups have studied the influence of viscous normal stresses on the Young–Laplace equation on the development of radial fingering. However, in these works, the contribution of viscous normal stresses is included through the insertion of a legitimately two-dimensional term into the pressure jump condition. As a result, the significant variation of the fluid velocity along the direction perpendicular to the Hele-Shaw plates is neglected. In line with Hele-Shaw flow approximations, and analogous to the derivation of Darcy’s law, we introduce viscous stresses in the Young–Laplace equation through a gap-averaged calculation of the three-dimensional viscous stress tensor. We then compute the contributions of these gap-averaged viscous stresses in both the linear stability and early nonlinear analyses of the interface perturbation evolution. Our findings indicate that this approach leads to a slowdown in finger growth, and an intensification of typical tip-splitting events.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134663"},"PeriodicalIF":2.7,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A straightforward construction of Z-graded Lie algebras of full-fledged nonlocal symmetries via recursion operators","authors":"Jiřina Jahnová,&nbsp;Petr Vojčák","doi":"10.1016/j.physd.2025.134658","DOIUrl":"10.1016/j.physd.2025.134658","url":null,"abstract":"<div><div>We consider the reduced quasi-classical self-dual Yang–Mills equation (rYME) and two recently found (Jahnová and Vojčák, 2024) invertible recursion operators <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> for its full-fledged (in a given differential covering) nonlocal symmetries. We introduce a <span><math><mi>Z</mi></math></span>-grading on the Lie algebra <span><math><mrow><msubsup><mrow><mi>sym</mi></mrow><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mi>W</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> of all nonlocal Laurent polynomial symmetries of the rYME and prove that both the operators <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> are <span><math><mi>Z</mi></math></span>-graded automorphisms of the underlying vector space on the set <span><math><mrow><msubsup><mrow><mi>sym</mi></mrow><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mi>W</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>. This <em>inter alia</em> implies that all its vector subspaces formed by all homogeneous elements of a given fixed degree (i.e. a weight in the context below) are mutually isomorphic, and thus each of them can be uniquely reconstructed from the vector space of all homogeneous symmetries of the zero degree. To the best of our knowledge, such a result is unparalleled in the current body of literature. The obtained results are used for the construction of a Lie subalgebra <span><math><mi>V</mi></math></span> of <span><math><mrow><msubsup><mrow><mi>sym</mi></mrow><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mi>W</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> which contains all known to us nonlocal Laurent polynomial symmetries of the rYME. The Lie algebra <span><math><mi>V</mi></math></span> is subsequently described as the linear span of the orbits of a set of selected zero-weight symmetries — we refer to them as to the seed generators of <span><math><mi>V</mi></math></span>. Further, we study the hierarchies of symmetries related to the seed generators under the action of the group of recursion operators generated by <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>. Finally, the linear dependence/independence of the (sub)set of generators of <span><math><mi>V</mi></math></span> is discussed.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134658"},"PeriodicalIF":2.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Neural network solutions to the critical SQG equations via approximating nonlocal periodic operators","authors":"Elie Abdo ,&nbsp;Ruimeng Hu ,&nbsp;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}

{"title":"Bridging Algorithmic Information Theory and Machine Learning: Clustering, density estimation, Kolmogorov complexity-based kernels, and kernel learning in unsupervised learning","authors":"Boumediene Hamzi ,&nbsp;Marcus Hutter ,&nbsp;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}

{"title":"Analytic soliton solutions of nonlinear extensions of the Schrödinger equation","authors":"Tom Dodge ,&nbsp;Peter Schweitzer","doi":"10.1016/j.physd.2025.134666","DOIUrl":"10.1016/j.physd.2025.134666","url":null,"abstract":"<div><div>A method is presented to construct analytic solitary wave solutions in nonlinear extensions of the Schrödinger equation starting from analytic solutions of the ordinary Schrödinger equation. We provide several examples illustrating the method. We rederive three well-known soliton solutions including the <span><math><mi>N</mi></math></span>-dimensional non-relativistic Gausson as well as the one-dimensional <span><math><mrow><mn>1</mn><mo>/</mo><mo>cosh</mo></mrow></math></span>-soliton and a theory with a power-like nonlinearity proportional to <span><math><msup><mrow><mrow><mo>|</mo><mi>Ψ</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn><mi>λ</mi></mrow></msup></math></span> with <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We also find several new solutions in different nonlinear theories in various space dimensions which, to the best of our knowledge, have not yet been discussed in literature. Our method can be used to construct further nonlinear theories and generalized to relativistic soliton theories, and may have many applications.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134666"},"PeriodicalIF":2.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143830078","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}