Nonlinear Sciences
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Showing new listings for Friday, 25 April 2025
- [1] arXiv:2504.17291 [pdf, html, other]
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Title: Top on a smooth planeJournal-ref: Maria Przybylska and Andrzej Maciejewski, Top on a smooth plane, Chaos 34, 043144 (2024)Subjects: Chaotic Dynamics (nlin.CD); Exactly Solvable and Integrable Systems (nlin.SI)
We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration $g\neq0$ and without external field $g=0$ are considered. The non-integrability proof for $g\neq0$ based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of constant gravitational field $g=0$ the integrability is much more difficult. At first, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied one of the Ziglin theorem concerning the splitting of separatrices phenomenon. Then we prove the non-integrability of the symmetric sliding top using differential Galois group of variational equations except two the same as for $g\neq0$ cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.
- [2] arXiv:2504.17302 [pdf, html, other]
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Title: Non-integrability of charged three-body problemJournal-ref: Przybylska, M., Maciejewski, A.J. Non-integrability of charged three-body problem. Celest Mech Dyn Astron 137, 8 (2025)Subjects: Chaotic Dynamics (nlin.CD)
We consider the problem of $n$ points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational $n$-body problem. Under this general form of interaction, we investigate the integrability problem of three bodies. We show that the system is not integrable except in one case when two among three interaction constants vanish. In our investigation, we used the Morales-Ramis theorem concerning the integrability of a natural Hamiltonian system with a homogeneous potential and its generalization.
- [3] arXiv:2504.17400 [pdf, html, other]
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Title: Selectivity filter conductance, rectification and fluctuations of subdomains - how can this all relate to the value of Hurst exponent in the dwell-times of ion channels states?Przemysław Borys, Paulina Trybek, Beata Dworakowska, Anna Sekrecka-Belniak, Ewa Nurowska, Piotr Bednarczyk, Agata Wawrzkiewicz-JałowieckaSubjects: Chaotic Dynamics (nlin.CD)
The Hurst effect in the signals describing ion channels' activity has been known for many years. This effect is present in the experimental recordings of single-channel currents, but not only. The sequences of dwell times of functionally different channel states also exhibit long-range correlations. We have found that the memory effect within the dwell-time series is related to the coupling between the channel's activation gate (AG) and selectivity filter (SF), which controls the ion conduction. In this work, we analyzed both the experimental data describing the activity of potassium channels of different types (e.g., BK, mitoBK, mitoTASK-3, mitoKv1.3, TREK-2-like channels) and the series generated according to our previously proposed Hurst effect model. The obtained results suggest that the strength of the allosteric cooperation between the AG and SF determines not only the conductance of the channel - which governs how often ions in SF move or remain blocked - but also modulates the correlations present in the dwell times when sampled with a suitably high sampling rate. Moreover, we found that rectification can interfere with this process, contributing to additional changes in correlations within the channel's sojourns in subsequent states. Similarly, the correlations may be affected by processes proceeding at longer time scales, like interactions with the channel's auxiliary domains or lipid surroundings.
- [4] arXiv:2504.17537 [pdf, other]
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Title: Long-time asymptotics of the Sawada-Kotera equation on the lineSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Analysis of PDEs (math.AP)
The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlevé region, and the rapid decay region. Notably, the Painlevé region is governed by the F-XVIII equation in the Painlevé classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlevé transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlevé regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.
New submissions (showing 4 of 4 entries)
- [5] arXiv:2504.17294 (cross-list from hep-th) [pdf, html, other]
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Title: Higher-Spin Currents and Flows in Auxiliary Field Sigma ModelsComments: 42 pages + appendicesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We study local, higher-spin conserved currents in integrable $2d$ sigma models that have been deformed via coupling to auxiliary fields. These currents generate integrability-preserving flows introduced by Smirnov and Zamolodchikov. For auxiliary field (AF) deformations of a free boson, we prove that local spin-$n$ currents exist for all $n$ and give recursion relations that characterize Smirnov-Zamolodchikov (SZ) flows driven by these currents. We then show how to construct spin-$2n$ currents in a unified class of auxiliary field sigma models with common structure -- including AF theories based on the principal chiral model (PCM), its non-Abelian T-dual, (bi-)Yang-Baxter deformations of the PCM, and symmetric space models -- for interaction functions of one variable, and describe SZ flows driven by any function of the stress tensor in these cases. Finally, we give perturbative solutions for spin-$3$ SZ flows in any member of our unified class of AF models with underlying $\mathfrak{su}(3)$ algebra. Part of our analysis shows that the class of AF deformations can be extended by allowing the interaction function to depend on a larger set of variables than has previously been considered.
- [6] arXiv:2504.17411 (cross-list from math-ph) [pdf, html, other]
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Title: The KP equation of plane elastodynamicsSubjects: Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS)
The propagation of nonlinear and dispersive waves in various materials can be described by the well-known Kadomtsev-Petviashvili (KP) equation, which is a (2+1)-dimensional partial differential equation. In this paper, we show that the KP equation can be used to describe the in-plane motion of compressible elastic solids with dispersion. Furthermore, a modified KP equation with cubic nonlinearity is obtained in the case of incompressible solids with dispersion. Then, several solutions of these partial differential equations are discussed and computed using a Fourier spectral method. In particular, both equations admit solitary wave solutions.
- [7] arXiv:2504.17465 (cross-list from math.AP) [pdf, html, other]
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Title: On soliton resolution to Cauchy problem of the spin-1 Gross-Pitaevskii equationComments: 51 pages, 6 figuresSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We investigate the Cauchy problem for the spin-1 Gross-Pitaevskii(GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng $etal.$ (Commun. Math. Phys. 382, 585-611 (2021)) reported the long-time asymptotic result with error $\mathcal{O}(\frac{\log t}t)$ for the spin-1 GP equation that only exists in the continuous spectrum. The main purpose of our work is to further generalize and improve Geng's work. Compared with the previous work, our asymptotic error accuracy has been improved from $\mathcal{O}(\frac{\log t}t)$ to $\mathcal{O}(t^{-3/4})$. More importantly, by establishing two matrix valued functions, we obtained effective asymptotic errors and successfully constructed asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including two cases: (i)coexistence of discrete and continuous spectrum; (ii)only continuous spectrum which considered by Geng's work with error $\mathcal{O}(\frac{\log t}t)$. For the case (i), the corresponding asymptotic approximations can be characterized with an $N$-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. For the case (ii), the corresponding asymptotic approximations can be characterized with the leading term on the continuous spectrum and the residual error order $\mathcal{O}(t^{-3/4})$. Finally, our results confirm the soliton resolution conjecture for the spin-1 GP equation.
- [8] arXiv:2504.17491 (cross-list from q-bio.NC) [pdf, html, other]
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Title: On the robustness of the emergent spatiotemporal dynamics in biophysically realistic and phenomenological whole-brain models at multiple network resolutionsComments: 51 pages in total, main manuscript until page 27, references, and supplementary after that. Main manuscript has 12 Figures and 3 tables, supplementary has 19 figures and 9 tablesSubjects: Neurons and Cognition (q-bio.NC); Adaptation and Self-Organizing Systems (nlin.AO)
The human brain is a complex dynamical system which displays a wide range of macroscopic and mesoscopic patterns of neural activity, whose mechanistic origin remains poorly understood. Whole-brain modelling allows us to explore candidate mechanisms causing the observed patterns. However, it is not fully established how the choice of model type and the networks' resolution influence the simulation results, hence, it remains unclear, to which extent conclusions drawn from these results are limited by modelling artefacts. Here, we compare the dynamics of a biophysically realistic, linear-nonlinear cascade model of whole-brain activity with a phenomenological Wilson-Cowan model using three structural connectomes based on the Schaefer parcellation scheme with 100, 200, and 500 nodes. Both neural mass models implement the same mechanistic hypotheses, which specifically address the interaction between excitation, inhibition, and a slow adaptation current, which affects the excitatory populations. We quantify the emerging dynamical states in detail and investigate how consistent results are across the different model variants. Then we apply both model types to the specific phenomenon of slow oscillations, which are a prevalent brain rhythm during deep sleep. We investigate the consistency of model predictions when exploring specific mechanistic hypotheses about the effects of both short- and long-range connections and of the antero-posterior structural connectivity gradient on key properties of these oscillations. Overall, our results demonstrate that the coarse-grained dynamics are robust to changes in both model type and network resolution. In some cases, however, model predictions do not generalize. Thus, some care must be taken when interpreting model results.
- [9] arXiv:2504.17503 (cross-list from cs.LG) [pdf, html, other]
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Title: Tailored minimal reservoir computing: on the bidirectional connection between nonlinearities in the reservoir and in dataComments: 13 pages, 11 figuresSubjects: Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD)
We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.
- [10] arXiv:2504.17657 (cross-list from physics.optics) [pdf, html, other]
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Title: Fast and accurate modelling of Kerr-Brillouin combs in Fabry-Perot resonatorsSubjects: Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
We introduce a new mean-field equation for modeling Fabry-Perot resonators filled with a dispersive medium exhibiting both Brillouin and Kerr nonlinearities, e.g. an optical fiber. This model is derived from a unified framework that accounts for Brillouin scattering and four-wave mixing. It involves two coupled nonlinear Schrodinger equations for the forward and backward propagating fields, alongside a single equation governing the acoustic oscillation. Under standard assumptions for mean-field models -such as high finesse, weak nonlinearity, and weak dispersion- we demonstrate that our equation closely matches the original system. The simplified and elegant mathematical structure of our model provides valuable physical insights. As a key example, we derive an expression for the growth rate of harmonic perturbations of steady states. Additionally, our model facilitates fast and accurate numerical simulations using standard Fourier split-step methods. We highlight the effectiveness of this approach by simulating frequency comb generation in state-of-the-art high-Q fiber Fabry-Perot resonators.
- [11] arXiv:2504.17773 (cross-list from math-ph) [pdf, html, other]
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Title: Three-local Charge Conservation Implies Quantum IntegrabilitySubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Exactly Solvable and Integrable Systems (nlin.SI); Quantum Physics (quant-ph)
It is shown that the existence of a local conserved charge supported by three neighboring sites, or its local version, Reshetikhin's condition, suffices to guarantee the existence of all higher conserved charges and hence the integrability of a quantum spin chain. This explains the ``coincidence'' that no counterexample is known to Grabowski and Mathieu's long-standing conjecture despite the folklore that the conservation of local charges of order higher than 4 imposes additional constraints not implied by the conservation of the three-local charge.
Cross submissions (showing 7 of 7 entries)
- [12] arXiv:2504.14644 (replaced) [pdf, html, other]
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Title: The $q$-deformed Calogero's Goldfish SystemsComments: 14 pagesSubjects: Exactly Solvable and Integrable Systems (nlin.SI)
Searching for integrable models is a central theme in theoretical and mathematical physics, as such systems offer valuable insights into the underlying structure and symmetries of complex physical phenomena. In this work, we contribute to this pursuit by proposing a new class of one-dimensional many-body integrable systems, which we refer to as the $q$-deformed Calogero's Goldfish system. Our construction employs $q$-deformation of logarithmic and exponential functions inspired by Tsallis' formalism in non-extensive statistical mechanics. Notably, the model satisfies the double-zero condition on its solutions, underscoring its integrable nature and offering a novel perspective on deformation techniques within exactly solvable systems.
- [13] arXiv:2406.05031 (replaced) [pdf, html, other]
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Title: Unified view of scalar and vector dark matter solitonsComments: Match the published version. 18 pages, 5 figuresSubjects: High Energy Physics - Phenomenology (hep-ph); Cosmology and Nongalactic Astrophysics (astro-ph.CO); Pattern Formation and Solitons (nlin.PS)
The existence of solitons -- stable, long-lived, and localized field configurations -- is a generic prediction for ultralight dark matter. These solitons, known by various names such as boson stars, axion stars, oscillons, and Q-balls depending on the context, are typically treated as distinct entities in the literature. This study aims to provide a unified perspective on these solitonic objects for real or complex, scalar or vector dark matter, considering self-interactions and nonminimal gravitational interactions. We demonstrate that these solitons share universal nonrelativistic properties, such as conserved charges, mass-radius relations, stability and profiles. Without accounting for alternative interactions or relativistic effects, distinguishing between real and complex scalar dark matter is challenging. However, self-interactions differentiate real and complex vector dark matter due to their different dependencies on the macroscopic spin density of dark matter waves. Furthermore, gradient-dependent nonminimal gravitational interactions impose an upper bound on soliton amplitudes, influencing their mass distribution and phenomenology in the present-day universe.
- [14] arXiv:2410.11894 (replaced) [pdf, html, other]
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Title: Automated Discovery of Operable Dynamics from VideosSubjects: Systems and Control (eess.SY); Machine Learning (cs.LG); Image and Video Processing (eess.IV); Chaotic Dynamics (nlin.CD)
Dynamical systems form the foundation of scientific discovery, traditionally modeled with predefined state variables such as the angle and angular velocity, and differential equations such as the equation of motion for a single pendulum. We introduce a framework that automatically discovers a low-dimensional and operable representation of system dynamics, including a set of compact state variables that preserve the smoothness of the system dynamics and a differentiable vector field, directly from video without requiring prior domain-specific knowledge. The prominence and effectiveness of the proposed approach are demonstrated through both quantitative and qualitative analyses of a range of dynamical systems, including the identification of stable equilibria, the prediction of natural frequencies, and the detection of of chaotic and limit cycle behaviors. The results highlight the potential of our data-driven approach to advance automated scientific discovery.