The Helmholtz equation (HE) is comprised of two partial differential equations that illustrated wave propagation in a medium, especially for situations where the wavelength is much smaller than the spatial extent of the system. In this paper, we consider a coupled nonlinear HE to study its complex dynamics and analytical solutions. We portray some qualitative features such as: Bifurcation analysis, complex chaos dynamics, return map, recurrence plots, synchronization analysis and Sparse identification of nonlinear dynamics via applying sparse regression techniques. For analytical solutions, we use newly introduced method, i.e, Kumar-Malik approach to study soliton solutions. All results are depicted via 3D and 2D plots to showcase their behaviors. The integration of analytical, numerical, and data-driven techniques highlights the versatility and depth of the coupled HE system, with implications for nonlinear optics, wave dynamics, and computational modeling.
Short-wave rail irregularities pose significant risks to high-speed rail safety, as their high-frequency shock features can be effectively captured by axle-box acceleration (ABA) signals. However, visualizing features embedded these signals remains challenging. This paper proposes an improved method for intelligently identifying short-wave rail diseases by combining Adaptive Chirp Mode Decomposition (ACMD), Recurrence Plot (RP), and Convolutional Neural Network (CNN). The method was tested on ABA signals contaminated with rail corrugation and rail impact. In the proposed approach, an ABA signal is first decomposed through ACMD to eliminate noise. Then, the signal is converted to a two-dimensional phase space trajectory image using a recurrence plot. The experimental results demonstrate that RPs of the same rail disease exhibit significant visual similarities, while those of different diseases show distinct patterns. Specifically, RPs of normal signals appear uniform and stochastic with low complexity, RPs of rail corrugation display a periodic checkerboard pattern, and RPs of rail impact feature a prominent cross-shaped void. Finally, a two-dimensional CNN model was developed to adaptively extract irregularity features. The CNN model achieved an accuracy of 96.3%, thus validating its effectiveness in accurately identifying rail impact and rail corrugation, two critical types of short-wave rail irregularities.
Time series similarity matrices (informally, recurrence plots or dot-plots), are useful tools for time series data mining. They can be used to guide data exploration, and various useful features can be derived from them and then fed into downstream analytics. However, time series similarity matrices suffer from very poor scalability, taxing both time and memory requirements. In this work, we introduce novel ideas that allow us to scale the largest time series similarity matrices that can be examined by several orders of magnitude. The first idea is a novel algorithm to compute the matrices in a way that removes dependency on the subsequence length. This algorithm is so fast that it allows us to now address datasets where the memory limitations begin to dominate. Our second novel contribution is a multiscale algorithm that computes an approximation of the matrix appropriate for the limitations of the userโs memory/screen-resolution, then performs a local, just-in-time recomputation of any region that the user wishes to zoom-in on. Given that this largely removes time and space barriers, human visual attention then becomes the bottleneck. We further introduce algorithms that search massive matrices with quadrillions of cells and then prioritize regions for later examination by either humans or algorithms. We will demonstrate the utility of our ideas for data exploration, segmentation, and classification in domains as diverse as astronomy, bioinformatics, entomology, and wildlife monitoring.
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