Search Results (161)

This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position $\widehat{x}$ and a momentum operator $\widehat{p}$ (called the deformed momentum), which becomes a function F of the same operators: $\left[\widehat{x},\widehat{p}\right]=F(\widehat{x},\widehat{p})$, that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra $\left[\widehat{x},{\widehat{p}}_0\right]=i\hslash \mathbb{I}$. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian $\widehat{x}$ coordinate and its non-deformed conjugate momentum ${\widehat{p}}_0$, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum $\widehat{p}$ and a new position coordinate $\widehat{y}$, which is its canonical conjugate variable, so $\widehat{y}$ and $\widehat{p}$ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation. Full article

This study investigates lump wave structures that arise from the interplay of dispersion and nonlinearity in a generalized Calogero–Bogoyavlenskii–Schiff-like model with spatially symmetric nonlinearity in (2+1) dimensions. A generalized bilinear representation of the governing equation is formulated using extended bilinear derivatives of the fourth order, providing a convenient framework for analytic treatment. Through symbolic computation, we construct positive quadratic wave solutions, which give rise to rationally localized lump wave tructures that decay algebraically in all spatial directions at fixed time. Analysis shows that the critical points of these quadratic waves lie along a straight line in the spatial plane and propagate at a constant velocity. Along this characteristic trajectory, the amplitudes of the lump waves remain essentially unchanged, reflecting the stability of these coherent structures. The emergence of these lumps is primarily driven by the combined influence of five dispersive terms in the model, highlighting the crucial role of higher-order dispersion in balancing the nonlinear interactions and shaping the resulting localized waveforms. Full article

Traditional transient stability-constrained optimal power flow (TSCOPF) methods rely on solving complex nonlinear differential equations, resulting in high computational demands and lengthy processing times. To address these issues, this paper proposes a TSCOPF model based on a cascaded CatBoost model (CatBoost-DF) and an improved seagull optimization algorithm (ISOA). First, a TSCOPF model is constructed. Second, the CatBoost-DF model is developed to establish a mapping relationship between the dynamic characteristics of the power system and the power angle of generators. The trained CatBoost-DF model is then employed as a surrogate model to handle transient stability constraints, thereby avoiding the computation of complex differential-algebraic equations traditionally required in transient stability constraint analysis. Then, the ISOA is employed to iteratively solve the TSCOPF model. This enables timely adjustment of generator output when transient instability risks arise, preventing accidents while maintaining system economic efficiency. Finally, simulations conducted on the IEEE 39 bus system demonstrate that this method effectively safeguards both system security and economic performance. Full article

This study explores the influence of topography on Rossby solitary waves by analyzing the dynamics in both inner and outer regions, with solutions matched at the boundary. Using perturbation methods, the Benjamin–Davis–Ono (BDO) equation is derived from the shallow-water equations to characterize the amplitude evolution of Rossby algebraic solitary waves. The Physics-Informed Neural Networks (PINNs) method is applied to numerically solve the BDO equation, effectively capturing its nonlinear behavior and simulating the propagation of Rossby algebraic solitary waves under diverse topographic and external conditions. The results demonstrate that variations in key parameters significantly alter topographic effects, thereby impacting the amplitude of solitary waves. These findings provide valuable insights into the potential consequences of global warming and other external disturbances on the behavior of Rossby waves. Full article

This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than $0.5\%$ in the Burgers limit and $3.2\%$ in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article

We study stochastic variants of the Kairat-II and Kairat-X equations in (3 + 1) dimensions, two canonical models in soliton theory. Random fluctuations are incorporated through a Wiener process, yielding a multiplicative stochastic embedding of the wave fields. By combining the enhanced direct algebraic technique with the new projective Riccati equation approach, we obtain closed-form stochastic soliton solutions and analyze how noise modulates their amplitude and localization. The solutions are illustrated with consistent 3D surface plots (mean field vs. sample paths) and 2D time traces to highlight wave geometry and variability. In addition, we employ the energy balance approach to separate kinetic and potential contributions and to verify an energy balance relation for the derived solutions, thereby clarifying their physical plausibility and stability under noise. The results provide exact, easily verifiable benchmarks for stochastic nonlinear wave models and a practical template for incorporating randomness into nonlinear dispersive systems. Full article

Three-wave resonant interaction equations can model nonlinear dynamics in many fields, e.g., fluids, optics, and plasma. Rogue waves, i.e., modes algebraically localized in both space and time, are obtained analytically. The aim of this paper is to study degenerate three-wave resonant interaction equations, where two out of the three interacting wave packets have identical group velocities. Physically, degenerate resonance typically occurs for dispersion relation, possessing many branches, e.g., internal waves in a continuously stratified fluid. Here, the Nth-order rogue wave solutions for this dynamical model are presented. Based on these solutions, we examine the effects of the group velocity on the width and structural profiles of the rogue waves. The width of the rogue waves exhibit a linear increase as the group velocity increases, a feature well-correlated with the prediction made using modulation instability. In terms of structural profiles, first-order rogue waves display ‘four-petal’ and ‘eye-shaped’ patterns. Second-order rogue waves can reveal intriguing configurations, e.g., ‘butterfly’ patterns and triplets. To ascertain the robustness of these modes, numerical simulations with random initial conditions were performed. Sequences of localized modes resembling these analytical rogue waves were observed. A spectral jump was observed, with the jump broadening in the case of rogue wave triplets. Furthermore, we predict new rogue waves based on information from two existing ones obtained using the deep learning technique in the context of rogue wave triplets. This predictive model holds potential applications in ocean engineering. Full article

This study investigates the dynamics of optical solitons for the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE) enriched with $\beta$-derivatives. By employing an extended direct algebraic method (EDAM), we successfully derive explicit soliton solutions that illustrate the intricate interplay between nonlinearities and variable coefficients. Our approach facilitates the transformation of the complex NLS into a more manageable form, allowing for the systematic exploration of diverse solitonic structures, including bright, dark, and singular solitons, as well as exponential, polynomial, hyperbolic, rational, and Jacobi elliptic solutions. This diverse family of solutions substantially expands beyond the limited soliton interactions studied in conventional approaches, demonstrating the superior capability of our method in unraveling new wave phenomena. Furthermore, we rigorously demonstrate the robustness of these soliton solutions against various perturbations through comprehensive stability analysis and numerical simulations under parameter variations. The practical significance of this work lies in its potential applications in advanced optical communication systems. The derived soliton solutions and the analysis of their dynamics provide crucial insights for designing robust signal carriers in nonlinear optical media. Specifically, the management of variable coefficients and fractional-order effects can be leveraged to model and engineer sophisticated dispersion-managed optical fibers, tunable photonic devices, and ultrafast laser systems, where controlling pulse propagation and stability is paramount. The presence of $\beta$-fractional derivatives introduces additional complexity to the wave propagation behaviors, leading to novel dynamics that we analyze through numerical simulations and graphical representations. The findings highlight the potential of the proposed methodology to uncover rich patterns in soliton dynamics, offering insights into their robustness and stability under varying conditions. This work not only contributes to the theoretical foundation of nonlinear optics but also provides a framework for practical applications in optical fiber communications and other fields involving nonlinear wave phenomena. Full article

In this article, we present an efficient and highly accurate numerical scheme that achieves exponential convergence for solving nonlinear Riesz distributed-order fractional differential equations (RDFDEs) in one- and two-dimensional initial–boundary value problems. The proposed method is based on a two-stage collocation framework. In the first stage, spatial discretization is performed using the shifted Legendre–Gauss–Lobatto (SL-G-L) collocation method, where the approximate solutions and spatial derivatives are expressed in terms of shifted Legendre polynomial expansions. This reduces the original problem to a system of fractional differential equations (FDEs) for the expansion coefficients. Then, the temporal discretization is achieved in the second stage via Romanovski–Gauss–Radau collocation approach, which converts the system into a system of algebraic equations that can be solved efficiently. The method is applied to one- and two-dimensional nonlinear RDFDEs, and numerical experiments confirm its spectral accuracy, computational efficiency, and reliability. Existing numerical approaches to distributed-order fractional models often suffer from poor accuracy, instability in nonlinear settings, and high computational costs. By combining the efficiency of Legendre polynomials for bounded spatial domains with the stability of Romanovski polynomials for temporal discretization, the proposed two-stage framework effectively overcomes these limitations and achieves superior accuracy and stability. Full article

It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article

This study focuses on the nonlinear partial differential equation known as the Lonngren wave equation, which plays a significant role in plasma physics, nonlinear wave propagation, and astrophysical research. By applying a suitable wave transformation, the nonlinear model is reduced to an ordinary differential equation. Analytical wave solutions of the Lonngren wave equation are then derived using the extended direct algebraic method. The physical behavior of these solutions is illustrated through 2D, 3D, and contour plots generated in Mathematica. Finally, the stability analysis of the Lonngren wave equation is discussed. Full article

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies. Full article

The convective Cahn–Hilliard–Oono equation is analyzed under the conditions ${\mu }_1\ge 0$ and ${\mu }_3+{\mu }_4\le 0$. The Lie invariance criteria are examined through symmetry generators, leading to the identification of Lie algebra, where translation symmetries exist in both space and time variables. By employing Lie group methods, the equation is transformed into a system of highly nonlinear ordinary differential equations using appropriate similarity transformations. The extended direct algebraic method are utilized to derive various soliton solutions, including kink, anti-kink, singular soliton, bright, dark, periodic, mixed periodic, mixed trigonometric, trigonometric, peakon soliton, anti-peaked with decay, shock, mixed shock-singular, mixed singular, complex solitary shock, singular, and shock wave solutions. The characteristics of selected solutions are illustrated in 3D, 2D, and contour plots for specific wave number effects. Additionally, the model’s stability is examined. These results contribute to advancing research by deepening the understanding of nonlinear wave structures and broadening the scope of knowledge in the field. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article