The Date–Jimbo–Kashiwara–Miwa equation describes the wave diffusion of a physical quantity and plays a central role in modeling nonlinear wave propagation in higher‑dimensional integrable systems. It has important applications in plasma physics, fluid mechanics and nonlinear wave theory, where it governs the dynamics of soliton structures, coherent wave packets and other collective excitations. In this study, we investigate the (2+1)‑dimensional Date–Jimbo–Kashiwara–Miwa equation, which governs important classes of integrable nonlinear waves in (2+1) dimensions. Using an extended modified auxiliary equation mapping technique, we construct a wide variety of exact traveling wave solutions. These include solitary‑wave solutions, kink‑type (or shock‑type) waves, periodic wave patterns and rational solutions, all expressed in terms of suitable auxiliary‑equation basis functions. The resulting solutions provide new explicit forms that enrich the known solution space of the Date–Jimbo–Kashiwara–Miwa equation and generalize or complement previously reported results. To highlight the physical structure and spatial behavior of these solutions, we include corresponding 3D plots that clearly illustrate their wave profiles, amplitude modulation, and localization properties. The combination of analytical construction and graphical visualization demonstrates the effectiveness of the approach in capturing rich wave structures in higher‑dimensional integrable equations and contributes to a deeper understanding of exact nonlinear wave propagation in (2+1)-dimensional physical systems.

In this paper, we investigate the existence and uniqueness of fixed points for a class of self-mappings defined on complete metric spaces by employing the concept of altering distance functions. Altering distance functions provide a flexible framework that allows the extension of classical contraction-type conditions while preserving convergence properties of iterative sequences. By introducing a generalized contractive inequality involving an altering distance function together with an auxiliary control function, we establish sufficient conditions ensuring the existence of a unique fixed point for the considered mappings. The proposed results generalize several well-known fixed point theorems, including the Banach contraction principle and its subsequent extensions based on nonlinear contractive conditions.

The main theorems are proved without imposing restrictive assumptions on the mapping, thereby broadening the applicability of the results to a wider class of problems. Furthermore, the convergence of Picard iteration sequences generated by the mappings under consideration is discussed in detail, and it is shown that the iterates converge strongly to the unique fixed point. To demonstrate the effectiveness and novelty of the obtained results, illustrative examples are provided, showing that the introduced conditions are genuinely weaker than many existing contraction conditions in the literature.

The results presented in this work contribute to the ongoing development of fixed point theory and may be useful in applications to nonlinear analysis, differential equations, and integral equations, where generalized contractive mappings naturally arise.

The (3+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff fluid equation with variable coefficients serves as a mathematical model for nonlinear wave propagation in non-homogeneous media characterized by spatially and temporally varying physical parameters. This formulation addresses limitations inherent in constant-coefficient models by incorporating realistic variations encountered in complex fluid environments. The present investigation applies the Variable Coefficient Generalized Abel Equation Method, a specialized analytical technique designed for handling variable-coefficient nonlinear partial differential equations of high dimension. We construct exact traveling-wave solutions including kink, lump soliton, breather and periodic waves. Each solution class manifests characteristic physical behaviors typical of nonlinear wave interactions in inhomogeneous media. Also the qualitative analysis examines phase transitions between different solution types, stability properties under parameter perturbations and the spectrum of accessible dynamical regimes. This analysis delineates critical parameter thresholds governing qualitative changes in wave propagation characteristics. Corresponding three-dimensional surface plots and phase portraits provide comprehensive visualization of spatio-temporal evolution patterns, amplitude profiles, localization features and structural diversity exhibited by these analytical solutions. The graphical representations underscore practical relevance to applications including turbulent flow modeling, oceanic surface wave dynamics and plasma wave instabilities occurring within non-uniform physical environments. This systematic study extends the catalog of known analytical solutions for variable-coefficient CBS models while establishing foundational analytical tools essential for investigating nonlinear wave phenomena in realistic, spatially-varying physical systems.

Population decline has become a defining feature of many regional systems, with Japan’s total population entering a sustained decline from 2009 onward, yet limited attention has been paid to how diffusion-based migration models perform under prolonged demographic contraction and systemic shocks. This study examines internal migration dynamics in Japan over a 22-year period (2004–2025) by modeling inter-prefectural population flows across the country’s 47 prefectures as a spatial network governed by diffusion processes with long-range interaction effects. Annual prefecture-to-prefecture migration matrices are constructed and analyzed within a graph-based framework using a discrete Laplacian formulation augmented with gravity-type interaction terms. To account for structural changes over time, the analysis adopts a regime-based approach that distinguishes periods of demographic stability, sustained population decline, exogenous shock, associated with the COVID-19 period (2020–2021), and post-shock reconfiguration (2022–2025). Regimes are defined based on demographic and systemic transitions rather than equal temporal lengths, while model estimation is conducted on annual transitions to maintain consistency across periods. This framework allows for a systematic comparison of migration dynamics across contrasting demographic conditions. The results indicate that diffusion-based dynamics provide a reasonable representation of migration adjustment during periods of relative demographic stability, but exhibit reduced explanatory capacity during prolonged population decline and systemic shocks. In contrast, hybrid diffusion–gravity formulations are more robust across regimes by incorporating both local spatial interactions and long-distance hierarchical linkages. Post-shock periods display mixed migration behavior, suggesting partial re-stabilization without a full return to pre-decline diffusion patterns. By framing internal migration as a regime-dependent dynamical system, this study contributes to applied mathematics by clarifying the conditions under which diffusion operators remain suitable in declining population systems and when extended formulations are required. Although Japan serves as the empirical case, the proposed framework also applies to other regions experiencing demographic decline and spatial reorganization.

Chaotic behavior is a common feature of nonlinear dynamics.

The subject of our discussion today is the stability of hyperchaos in high-dimensional systems. This study serves as an introductory guide to a discrete fractional four-dimensional hyperchaotic Rössler system with a Caputo-like operator is a complex system that can be used to study the chaos of discrete fractional nonlinear dynamics. Our results demonstrate the existence of a hyperchaotic invariant set in these systems, which leads to extended hyperchaotic transient behaviors. The coexistence of chaos and hyperchaos is evident in the numerical results, which are presented in phase plots and bifurcation diagrams for various fractional orders and different parameters and specific initial conditions. These diagrams provide a comprehensive explanation of the dynamics of the proposed discrete system. This research substantiates the presence of chaos in discrete fractional hyperchaotic Rössler systems that are reminiscent of Caputo-like discrete systems. Low control is offered to display synchronization of coupled Caputo-like discrete fractional hyperchaotic Rössler systems and to force the states of the proposed system to converge asymptotically to zero. The findings of the study are demonstrated through the following implementation of numerical simulations, which has been a significant development in our research.

The Gilson-Pickering equation is a nonlinear partial differential equation that models complex wave propagation, incorporating nonlinear steepening, higher-order dispersion, and mixed derivatives. This paper constructs new exact solutions using analytical methods, focusing on the Hirota bilinear method, which is used to bilinearize the equation, facilitating the derivation of explicit solutions. Rational lump solutions are derived, examining their localized and algebraically decaying properties, providing insight into wave interaction dynamics. Breather solutions are also obtained, capturing oscillatory and time-periodic behavior, highlighting synchronization effects, bound-state formation, and complex collision phenomena. The solutions are analyzed within their mathematical framework, emphasizing distinctive features and physical significance. The results enhance the understanding of nonlinear wave dynamics governed by the Gilson-Pickering equation, offering a foundation for further theoretical and applied investigations of higher-order multidimensional nonlinear models. The study sheds light on intricate dynamics of nonlinear waves, enabling better comprehension of complex wave propagation phenomena in various physical systems, with potential applications in fields like fluid dynamics, optics, and plasma physics. This research opens up new directions in nonlinear science and applied mathematics, enhancing our understanding of complex nonlinear systems and their behavior, and creating many opportunities for future studies and exploration across diverse scientific and engineering applications worldwide.

Fish populations in natural ecosystems are influenced not only by biological interactions and fishing activities, but also by random fluctuations in the environment and natural populations. Deterministic models, though useful for modelling, sometimes fail to capture the complexity of uncertainty in fish populations. Motivated by this limitation, this study introduces and examines a stochastic model of the interaction between two competing fish species with random birth and death rates. Within the proposed framework, the classical model of a two-species competing population is extended to include a harvesting term and stochastic fluctuations in the reproduction and death rates. This is achieved by formulating continuous-time stochastic differential equations in which the intensities of the stochastic fluctuations depend on population densities in a biologically realistic manner. Analytical techniques have been employed to investigate the equilibria of stochastic populations, as well as the conditions for their persistence and extinction. The impacts of harvesting and stochasticity on system dynamics are investigated numerically using the Euler–Maruyama method. In contrast to the deterministic situation, stochastic impacts can significantly change the system's behavior. In particular, random perturbations may increase the risk of extinction even at harvesting levels considered sustainable in a deterministic framework. We observed that stochasticity, together with suitably designed harvesting strategies, affects the two competing fish species. Overall, the results of this study demonstrate the significance of random births and deaths in fishery models and provide insights for developing harvesting policies that ensure sustainability even with uncertainty, emphasising the importance of stochastic approaches for interacting species, specifically competitive fish populations, in fluctuating environments.

The evolution of SARS-CoV-2 underscores the significance of immune escape and reinfection in the spread of epidemics. This study proposes a two-strain compartmental model of the Omicron and Delta variants of SARS-CoV-2 and combines it with the Physics-Informed Neural Network (PINN) method to estimate time-dependent epidemiological parameters using daily epidemiological and variant-specific COVID-19 data from the State of Tennessee. Furthermore, a methodical superiority of PINN has been revealed over traditional non-linear least squares methods. The model captures the dynamics of primary transmission among susceptible individuals and secondary transmission resulting from the reinfection of recovered individuals with the other variant, thereby providing a mechanistic understanding of partial cross-immunity.

The PINN assimilates the non-linear equations of the two-strain compartmental model into the learning process using automatic differentiation, thereby satisfying the equations with the observed infection and recovery data. Time normalization, scaling of the system states, and log parameterization are used to improve the stability of the optimization process and maintain the positivity of the estimated epidemiological parameters, which are hidden.

To quantify the effect of immune escape on the sustainability of the epidemic, we examine various reinfection cases, including no reinfection and bidirectional reinfection. The trained PINN is used to forecast epidemic trajectories over a 30-day horizon, capturing strain-specific dynamics, variant dominance, and the risk of resurgence. Overall, this work demonstrates that physics-informed neural networks provide a principled and interpretable framework for learning non-stationary multi-strain epidemic dynamics and enabling reliable short-term forecasting from real-world data.

The study of limit cycles is an important topic in the qualitative theory of planar polynomial differential systems, since it helps to understand the periodic behavior of solutions and the global dynamics of nonlinear systems. In many situations arising in applied mathematics and dynamical systems, the existence, number, and distribution of limit cycles provide essential information about the long-term behavior of trajectories. However, obtaining explicit expressions for limit cycles is usually difficult, especially for higher-degree polynomial systems, where the analytical structure of periodic orbits becomes more complicated.

In this work, we study a planar polynomial differential system of degree nine. The system is constructed in such a way that it admits four explicit invariant algebraic curves surrounding the same singular point. These invariant curves are given in explicit form and play a fundamental role in the qualitative analysis of the system. Their presence allows us to investigate the structure of the periodic solutions and the configuration of closed orbits around the singular point.

First, we prove that the considered system possesses four invariant algebraic curves and determine their explicit expressions. Then, by applying a result reported by J. Giné and M. Grau, we show that each of these invariant curves corresponds to a limit cycle of the system. Consequently, the system admits four algebraic limit cycles surrounding the same singular point, and explicit expressions for these limit cycles are obtained.

Coastal lagoons in Sri Lanka are shallow water bodies connected to the sea through narrow and often restricted inlets. These physical features strongly affect tidal water level variations and water exchange within lagoons. Understanding lagoon water level behaviour is important for studying circulation and environmental conditions. This study presents the development of a one-dimensional numerical modelling framework to simulate tidal water level variations in Sri Lankan coastal lagoons.

The model is based on simplified forms of the continuity and momentum equations, which describe the conservation of mass and momentum along the lagoon channel. The restricted water exchange at the lagoon inlet is represented using a choking model to account for the effects of narrow and shallow entrances. Observed sea level data were used as boundary conditions. The governing equations were solved numerically to obtain spatial and temporal variations of water levels inside the lagoon. The modelling framework was applied to Chilaw Lagoon as a case study.

The numerical results show how tidal water levels propagate inside a shallow lagoon with a restricted inlet. Both temporal and spatial variations of lagoon water levels are illustrated using the model simulations. These results provide an initial understanding of tidal behaviour in shallow lagoons with restricted inlets.

The developed modelling framework provides a useful starting point for future studies. It can be further improved and validated using observed lagoon data and extended to include discharge and salinity transport processes. This approach will support more detailed investigations of water circulation and environmental dynamics in the coastal lagoons of Sri Lanka.