Nonlinear Sciences
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Showing new listings for Friday, 17 January 2025
- [1] arXiv:2501.09313 [pdf, html, other]
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Title: Invariant Reduction for Partial Differential Equations. II: The General MechanismComments: 27 pagesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Differential Geometry (math.DG)
A mechanism of reduction of symmetry-invariant conservation laws, presymplectic structures, and variational principles of partial differential equations (PDEs) is proposed. The mechanism applies for an arbitrary PDE system that admits a local (point, contact, or higher) symmetry, and relates symmetry-invariant conservation laws, as well as presymplectic structures, variational principles, etc., to their analogs for systems that describe the corresponding invariant solutions. A version of Noether's theorem for the PDE system satisfied by symmetry-invariant solutions is presented. Several detailed examples, including cases of point and higher symmetry invariance, are considered.
- [2] arXiv:2501.09340 [pdf, html, other]
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Title: Definition and data-driven reconstruction of asymptotic phase and amplitudes of stochastic oscillators via Koopman operator theoryComments: 14 pages, 2 figuresJournal-ref: Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Design of Mechanical Systems Across Different Length/Time Scales (31 July - 4 August, 2023). IUTAM Bookseries, vol 43. Springer, Cham.page 141-153Subjects: Adaptation and Self-Organizing Systems (nlin.AO)
Asymptotic phase and amplitudes are fundamental concepts in the analysis of limit-cycle oscillators. In this paper, we briefly review the definition of these quantities, particularly a generalization to stochastic oscillatory systems from the viewpoint of Koopman operator theory, and discuss a data-driven approach to estimate the asymptotic phase and amplitude functions from time-series data of stochastic oscillatory systems. We demonstrate that the standard Extended dynamic mode decomposition (EDMD) can successfully reconstruct the phase and amplitude functions of the noisy FitzHugh-Nagumo neuron model only from the time-series data.
- [3] arXiv:2501.09415 [pdf, html, other]
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Title: Slowly decaying strain solitons in nonlinear viscoelastic waveguidesSubjects: Pattern Formation and Solitons (nlin.PS); Materials Science (cond-mat.mtrl-sci); Soft Condensed Matter (cond-mat.soft)
This paper is devoted to the modeling of longitudinal strain waves in a rod composed of a nonlinear viscoelastic material characterized by frequency-dependent second- and third-order elastic constants. We demonstrate that long waves in such a material can be effectively described by a damped Boussinesq-type equation for the longitudinal strain, incorporating dissipation through retarded operators. Using the existing theory of solitary wave solutions in nearly integrable systems, we derive a slowly-decaying strain soliton solution to this equation. The derived soliton characteristics are shown to be in a good agreement with results from full 3D simulations. We demonstrate the importance of taking into account the frequency dependence of third-order elastic constants for the description of strain solitons.
- [4] arXiv:2501.09463 [pdf, other]
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Title: Some properties of the simple nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ with $p$ an arbitrary positive integerSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Classical Analysis and ODEs (math.CA)
It is shown that the behavior of the solutions of the nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ -- where the dependent variable $y(l)$ is a real number, $\ell= 0; 1; 2...$ is the independent variable, and $p$ is an arbitrary positive integer--is easily ascertainable.
New submissions (showing 4 of 4 entries)
- [5] arXiv:2501.08250 (cross-list from hep-th) [pdf, html, other]
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Title: Soliton Resonances in Four Dimensional Wess-Zumino-Witten ModelComments: 35 pages, 16 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We present two kinds of resonance soliton solutions on the Ultrahyperbolic space $\mathbb{U}$ for the G=U(2) Yang equation which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. We reveal and illustrate the solitonic behaviors in four-dimensional Wess-Zumino-Witten (WZW$_4$) model through the sigma model action densities. The WZW$_4$ model on $\mathbb{U}$ describes a string field theory action of open N=2 string theories and hence our solutions suggest the existence of the corresponding classical objects in the N=2 string theories. Our solutions include multiple-pole solutions and V-shape soliton solutions. The V-shape solitons suggest annihilation and creation process of two solitons and would be building blocks to classify the ASDYM solitons, like the role of Y-shape solitons in classification of the KP (line) solitons.
We also clarify the relationship between the Cauchy matrix approach and the binary Darboux transformation in terms of quasideterminants. Our formalism can start with a simpler input data for the soliton solutions and hence might give a suitable framework of the classification of the ASDYM solitons. - [6] arXiv:2501.09546 (cross-list from q-bio.PE) [pdf, other]
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Title: Bacterial proliferation pattern formationComments: 12 pages (+ 20 SI text pages), 6 figures (+ 16 SI figures). To be published in PRX LifeSubjects: Populations and Evolution (q-bio.PE); Adaptation and Self-Organizing Systems (nlin.AO); Pattern Formation and Solitons (nlin.PS); Biological Physics (physics.bio-ph); Cell Behavior (q-bio.CB)
Bacteria can form a great variety of spatially heterogeneous cell density patterns, ranging from simple concentric rings to dynamical spiral waves appearing in growing colonies. These pattern formation phenomena are important as they reflect how cellular processes such as metabolism operate in heterogeneous chemical environments. In the laboratory, they can be studied in simplified set-ups, where spatial gradients of oxygen and nutrients are externally imposed, and cells are immobilized in a gel matrix. An intriguing example, observed in such set-ups over 80 years ago, is the sequential formation of narrow bands of high cell density, taking place even for a clonal population. However, key aspects of the dynamics of band formation remained obscure. Using time-lapse imaging of replicate transparent columns in simplified growth media, we first quantify the precision of the positioning and timing of band formation. We also show that the appearance and position of different bands can be modulated independently. This "modularity" is suggested by the observation that different bands differ in their gene expression, and it is reproduced by a theoretical model based on the existence of internal metabolic states and the induction of a pH gradient. Finally, we can also modify the observed pattern formation by introducing genetic modifications that impair selected metabolic pathways. In our opinion, the possibility of precise measurements and controls, together with the simplicity and richness of the "proliferation pattern formation" phenomenon, can make it a model system to study the response of cellular processes to heterogeneous environments.
- [7] arXiv:2501.09629 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: Optimal paths and dynamical symmetry breaking in the current fluctuations of driven diffusive mediaComments: 74 pages, 9 figures, lecture notes for the 2024 Les Houches Summer School on the Theory of Large Deviations and ApplicationsSubjects: Statistical Mechanics (cond-mat.stat-mech); Soft Condensed Matter (cond-mat.soft); Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS); Fluid Dynamics (physics.flu-dyn)
Large deviation theory provides a framework to understand macroscopic fluctuations and collective phenomena in many-body nonequilibrium systems in terms of microscopic dynamics. In these lecture notes we discuss the large deviation statistics of the current, a central observable out of equilibrium, using mostly macroscopic fluctuation theory (MFT) but also microscopic spectral methods. Special emphasis is put on describing the optimal path leading to a rare fluctuation, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level. We start with an overview of trajectory statistics in driven diffusive systems as described by MFT. We discuss the additivity principle, a simplifying conjecture to compute the current distribution in one-dimensional nonequilibrium systems, and extend this idea to higher dimensions, where the nonlocal structure of the optimal current vector field becomes crucial. Next we explore dynamical phase transitions (DPTs) in current fluctuations, which manifest as symmetry-breaking events in trajectory statistics. These include particle-hole symmetry-breaking DPTs in open channels, for which we work out a Landau-like theory as well as the joint statistics of the current and the order parameter. Time-translation symmetry-breaking DPTs in periodic systems are also discussed, where coherent traveling condensates emerge to facilitate current deviations. We also discuss the microscopic spectral mechanism leading to these DPTs, which is linked to an emerging degeneracy of the leading eigenspace. Using this spectral perspective, we find the signatures of the recently discovered time-crystal phases of matter in traveling-wave DPTs, and use Doob's transform to propose a packing-field mechanism to create programmable time-crystals in driven systems. Finally, we address open challenges and future directions in this rapidly evolving field.
Cross submissions (showing 3 of 3 entries)
- [8] arXiv:2412.18782 (replaced) [pdf, html, other]
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Title: The discrete Painlev\'{e} XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensemblesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
We consider the symmetric gap probability distributions of certain Freud unitary ensembles. This problem is related to the Hankel determinants generated by the Freud weights supported on the complement of a symmetric interval. By using Chen and Ismail's ladder operator approach, we obtain the difference equations satisfied by the recurrence coefficients for the orthogonal polynomials with the discontinuous Freud weights. We find that these equations, with a minor change of variables, are the discrete Painlevé XXXIV hierarchy proposed by Cresswell and Joshi [{\em J. Phys. A: Math. Gen.} {\bf 32} ({1999}) {655--669}]. This is the first time that the discrete Painlevé XXXIV hierarchy appears in the study of Random Matrix Theory. We also derive the differential-difference equations for the recurrence coefficients and show the relationship between the logarithmic derivative of the gap probabilities, the nontrivial leading coefficients of the monic orthogonal polynomials and the recurrence coefficients.
- [9] arXiv:2204.08955 (replaced) [pdf, html, other]
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Title: Chaotic Behaviour of the Earth System in the AnthropoceneComments: 18 pages, 6 figuresSubjects: Earth and Planetary Astrophysics (astro-ph.EP); Chaotic Dynamics (nlin.CD); Atmospheric and Oceanic Physics (physics.ao-ph)
It is shown that the Earth System (ES) can, due to the impact of human activities, exhibit chaotic behaviour. Our arguments are based on the assumption that the ES can be described by a Landau-Ginzburg model, which, in itself, predicts that the ES evolves through regular trajectories in phase space towards a Hothouse Earth scenario under a finite amount of human-driven impact. Furthermore, we find that the equilibrium point for temperature fluctuations can exhibit bifurcations and a chaotic pattern if human impact follows a logistic map. Our final analysis includes interactions between different terms of the planetary boundaries in order to gauge the predictability of our model.
- [10] arXiv:2406.08190 (replaced) [pdf, other]
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Title: CrowdEgress: A Multi-Agent Simulation Platform for Pedestrian CrowdComments: 27 pages, 21 figuresSubjects: Physics and Society (physics.soc-ph); Adaptation and Self-Organizing Systems (nlin.AO)
This article introduces a simulation platform to study complex crowd behavior in social context. The agent-based model is extended based on the social force model, and it mainly describes how agents interact with each other, and also with surrounding facilities such as walls, doors and exits. The simulation platform is compatible to FDS+Evac, and the input data in FDS+Evac could be imported into our simulation platform to create single-floor compartment geometry, and a flow solver is used to generate the roadmap towards exits. Most importantly, we plan to integrate advanced social and psychological theory into our simulation platform, especially investigating human behavior in emergency evacuation,such as pre-evacuation behavior, exit-selection activities, social group and herding effect and so forth.
- [11] arXiv:2409.07546 (replaced) [pdf, html, other]
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Title: Localized synchronous patterns in weakly coupled bistable oscillatorsComments: 17 pages, 7 figuresSubjects: Dynamical Systems (math.DS); Pattern Formation and Solitons (nlin.PS)
Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with nonlinearities of Lambda-Omega type and establish the existence of localized synchrony patterns in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Depending on the coupling, localized synchrony patterns lie on a discrete stack of isola branches or on a single connected snaking branch.
- [12] arXiv:2412.14691 (replaced) [pdf, html, other]
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Title: Mathematical analysis of a flux-jump model in superconductivitySubjects: Superconductivity (cond-mat.supr-con); Pattern Formation and Solitons (nlin.PS)
We analyzed mathematically a model describing flux jumps in superconductivity in a 1D configuration. Three effects occur from fastest to slowest: Joule heating, magnetic relaxation and temperature diffusion. Adimensionalising the equations showed that magnetic field fronts penetrate the material as inhomogeneous Burgers fronts. An additional nonlinear term affects the magnetic field and is responsible for flux jumps. We considered a medium temperature for which the heat capacity of a sample can be assumed constant and a low temperature where heat capacity depends on temperature causing a nonlinear temperature evolution. We found that flux jumps occur for pulses of duration close to the magnetic relaxation time and mostly at low temperature. Flux trapping is maximal for medium amplitude long duration pulses and low to medium temperatures.