Physica D: Nonlinear Phenomena最新文献

Predictions of complex systems ranging from natural language processing to weather forecasting have benefited from advances in Recurrent Neural Networks (RNNs). RNNs are typically trained using techniques like Backpropagation Through Time (BPTT) to minimize one-step-ahead prediction loss. During testing, RNNs often operate in an auto-regressive mode, with the output of the network fed back into its input. However, this process can eventually result in exposure bias since the network has been trained to process ”ground-truth” data rather than its own predictions. This inconsistency causes errors that compound over time, indicating that the distribution of data used for evaluating losses differs from the actual operating conditions encountered by the model during training. Inspired by the solution to this challenge in language processing networks we propose the Scheduled Autoregressive Truncated Backpropagation Through Time (BPTT-SA) algorithm for predicting complex dynamical systems using RNNs. We find that BPTT-SA effectively reduces iterative error propagation in Convolutional and Convolutional Autoencoder RNNs and demonstrates its capabilities in the long-term prediction of high-dimensional fluid flows.

Taking the nonlinear Schrödinger equation (NLSE) as an example, we provide from a mathematical viewpoint, rigorous evidence that numerical noise of a chaotic system as tiny artificial stochastic disturbances can increase exponentially to a macro-level. As a result, numerical simulations given by traditional algorithms in double precision may rapidly become badly polluted leading to huge deviations from the ‘true’ solution not only in trajectory but also, sometimes, even in statistics and/or some qualitative properties. Small physical disturbances in time and space are unavoidable in practice, which are often much larger than artificial numerical noise. So, from a physical viewpoint, it is wrong to neglect small spatio-temporal disturbances of a chaotic system: chaos should not be described by deterministic equations.

This manuscript explores the extensions and classifications of the bosonic supersymmetric systems. For the third order bosonic superfield equations, four types of integrable supersymmetric extensions are identified, including the B-type (trivial) supersymmetric modified Korteweg–de Vries equation, the supersymmetric Sharma–Tasso–Olver equation, and an A-type (non-trivial) supersymmetric potential Korteweg–de Vries equation. In the case of the fifth order bosonic supersymmetric systems, nine kinds of extensions are discovered, with six being B-type and three being A-type. Notably, several equations such as the supersymmetric Sawada–Kotera equation, the supersymmetric Kaup–Kupershmidt equation and the supersymmetric Fordy–Gibbons equation are classified as B-type extensions. Despite this classification, these supersymmetric systems are shown to be connected to linear integrable couplings. The findings have implications for various fields including string theory and dark matter and highlight the importance of understanding bosonic supersymmetric systems. The obtained supersymmetric systems are solved via bosonization method. Applying the bosonization procedure to every one of supersymmetric systems, one can find various dark equation systems. These dark equation systems can be solved by means of the solutions of the classical equations and some graded linear couplings including homogeneous and nonhomogeneous symmetry equations.

Topological data analysis (TDA) is a versatile tool that can be used to extract scientific knowledge from complex pattern formation processes. However, the physics correspondence between the features obtained from TDA and pattern dynamics does not agree one-to-one, and the physical interpretation of the TDA features needs to be set appropriately according to the phenomenon to be analyzed. In this study, we propose an analytical procedure to physically interpret pattern dynamics through TDA and machine learning techniques. The proposed procedure was applied to the process of magnetic domain pattern formation to quantify non-trivial domain pattern classifications and reveal the nature of the underlying dynamics. On the basis of these findings, we also propose a candidate reduction model to understand the nature of magnetic domain formation.

We consider systems of conservation laws derived from coupled fluid-kinetic equations intended to describe particle-laden flows. By means of Chapman–Enskog type expansion, we determine second order corrections and we discuss the existence and stability of shock profiles. Entropy plays a central role in this analysis.

This approach is implemented on a simplified model, restricting the fluid description to the Burgers equation, and a more realistic model based on the Euler equations. The comparison between the two systems gives the opportunity to bring out the role of certain structural properties, like the Galilean invariance, which is satisfied only by the Euler-based system.

We justify existence and stability of small amplitude shock profiles for both systems. For the Euler-based model, we also employ a geometric singular perturbation approach in view of passing from small- to large-amplitude shock profiles, considering temperature as small parameter. This program, fully achieved for the zero-temperature regime, is extended on numerical grounds to small positive temperatures.

Multipole solitons in higher-dimensional nonlinear Schrödinger equation with fractional diffraction are of high current interest. This paper studies multipole gap solitons in parity-time (PT)-symmetric lattices with fractional diffraction. The results obtained demonstrate that both on-site and off-site eight-pole solitons with fractional-order diffraction can be stabilized in a two-dimensional (2D) PT-symmetric optical lattice with defocusing Kerr nonlinearity. These solitons are in-phase and centrosymmetric. On-site eight-pole solitons propagate in a square formation, while off-site solitons propagate in a two-by-four formation. Both on-site and off-site solitons are found to be stable within a low-power range in the first band gap. As the Lévy index decreases, the stability regions of both on-site and off-site solitons narrow. Off-site eight-pole solitons can approach the lower edge of the first Bloch band, whereas on-site eight-pole solitons cannot. Additionally, we investigate the transverse power flow vector of these multipole gap solitons, illustrating the transverse energy flow from gain to loss regions.

Nudging induced neural networks (NINNs) algorithms are introduced to control and improve the accuracy of deep neural networks (DNNs). The NINNs framework can be applied to almost all pre-existing DNNs, with forward propagation, with costs comparable to existing DNNs. NINNs work by adding a feedback control term to the forward propagation of the network. The feedback term nudges the neural network towards a desired quantity of interest. NINNs offer multiple advantages, for instance, they lead to higher accuracy when compared with existing data assimilation algorithms such as nudging. Rigorous convergence analysis is established for NINNs. The algorithmic and theoretical findings are illustrated on examples from data assimilation and chemically reacting flows.

In this study, it was investigated numerically how boundary conditions influence the formation of Turing-like patterns under various diffusion conditions in complex media. It was found that Dirichlet boundary conditions can induce their symmetry in the patterns once the boundary concentrations of morphogens reach critical thresholds that depend on the diffusion regime and the domain size. We find that anomalous diffusion, characterized in our model by the parameter $\lambda$, can expand or contract the Turing instability region. Then, since superdiffusive conditions lead to a larger instability window, we conjecture that a possible explanation for the emergence of self-similarity in our system may be associated with the excitation of different scales. Our findings generally offer insights into reaction–diffusion systems’ pattern orientation and selection mechanisms.

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order $O({\tau }^2+h^2)$ in the discrete $L^{\infty }$-norm, where $h$ denotes the spatial step size and $\tau$ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.