Influenza continues to pose a significant global health challenge due to recurrent seasonal outbreaks and the emergence of antigenically evolving strains that reduce vaccine effectiveness. In this study, a nonlinear influenza transmission model incorporating a hospitalized compartment is proposed to provide a more realistic representation of disease progression and healthcare burden. The well-posedness of the model is established by proving the positivity and boundedness of solutions within a biologically feasible region. The influenza-free and endemic equilibria are derived, and the basic reproduction number R_inf is computed using the next-generation matrix approach. Sensitivity analysis is performed to identify the key parameters influencing disease transmission dynamics. Local and global stability analyses are carried out using linearization techniques and a suitable Lyapunov function, showing that the influenza-free equilibrium is stable when R_inf<1 , whereas the endemic equilibrium becomes stable when R_inf>1 . To support the theoretical findings, a nonstandard finite difference (NSFD) scheme is employed for numerical simulations, preserving essential qualitative properties of the model. Furthermore, an integral sliding mode control (ISMC) strategy is developed to reduce infection levels and enhance system robustness under parametric uncertainties and external disturbances. The proposed modeling and control framework provides a comprehensive analytical and computational approach for understanding influenza dynamics and designing effective intervention strategies.

In this article, we investigate a generalized Fornberg-Whitham (gFW) equation. Based on the Lax pair and Darboux transformation, one-soliton solution, breather solutions, lump solution, lump-1-strip, lump-periodic, manifold periodic, and rogue wave solutions for the gFW equation are obtained by choosing appropriate transformations and symbolic computation. In addition, we evaluate Ma breather, Kuznetsov-Ma breather and their corresponding rogue waves, generalized breather, and Akhmediev breathers. The obtained solutions reveal rich nonlinear wave phenomena such as localized structures, recurrence patterns, and energy concentration effects. These findings have direct applications in fluid dynamics, optical fiber communications, and plasma physics, where such nonlinear behaviors are prevalent. Moreover, the explicit construction of rogue waves and breather interactions provides insight into the prediction and control of extreme wave events in oceanography and nonlinear optics. The results further demonstrate the utility of analytical methods in exploring complex wave structures in integrable and near-integrable systems.

This paper presents a bilinear neural network architecture (BNNA) to analyze the (2+1) -dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation. The suggested framework integrates the Hirota bilinear formulation (HBF) and a neural network (NN) representation in which the bilinear structure is maintained and the neural weights are adjustable symbolic coefficients. One hidden-layer architecture is used, namely, [3-3-1] . With this method, a number of closed-form wave solutions are found, such as lump, lump-kink, interaction, three-soliton, and breather-like interaction structures. The outcomes are able to capture important nonlinear characteristics like localization, interaction dynamics, and periodic behavior. These solutions have dynamical properties that are depicted using three-dimensional (3D) surface and contour plots. In general, the suggested BNNA offers a straightforward, understandable, and effective model for building analytical solutions of nonlinear partial differential equations (NLPDEs) through the combination of HBF and NN-based representation.

Electromagnetic heating Processes (microwave and radio-frequency (RF) heating) are strongly nonlinear because of temperature-dependent dielectric characteristics and result in very non-uniform temperature distributions. The current literature mostly depends on the formulations of analytical studies or traditional numerical approaches that usually involve simplification of the assumptions or time-stepping schemes to address such nonlinear coupling. In this study, we develop a physics-informed deep learning framework to directly solve the fully coupled nonlinear electromagnetic-thermal system in a three-dimensional domain with temperature-dependent material properties. In contrast to the traditional method, the proposed approach does not require time discretization and offers a mesh-free solution by incorporating both the transient heat equation and the nonlinear electromagnetic power absorption term into the learning process. Three architectures, including standard physics-informed neural networks (PINN), gradient-enhanced PINN (gPINN) and extended PINN (XPINN) are designed and tested in the same physical conditions. The findings indicate that gPINN is more effective in sharpening thermal gradients and local non-uniformities whereas XPINN is more effective in converging and precision in the representation of multi-scale temperature fields. This work emphasizes how highly nonlinear coupled problems can be addressed with advanced physics-informed learning methods and offers a practical set of guidelines to choosing appropriate architectures in electromagnetic bio-thermal contexts.

We investigate four analytical families of vector quartic solitons (VQS) propagating on a continuous-wave background (CWB) in weakly birefringent optical fibers with 2^nd , 3^rd , and 4^th -order dispersion (OD). The resulting solutions, constructed from smooth combinations of hyperbolic functions, describe localized pulses as well as complementary bright–dark vector pairs. Their formation arises from the joint action of self-phase modulation (SPM), cross-phase modulation (CPM), four-wave mixing (FWM), and higher-order dispersion. Although the four subclasses differ in amplitude, wavenumber, and temporal width, they share a common frequency shift and propagation velocity, both fixed solely by the dispersion parameters. Closed-form algebraic constraints determine the existence ranges, while direct numerical simulations of the extended coupled nonlinear Schrödinger (NLSE) model demonstrate that all four families remain stable and preserve their profiles under additive white-noise perturbations. Altogether, the results provide a unified analytical description of CWB-supported vector quartic solitons and point to new opportunities for dispersion-managed slow-light and ultrafast photonic applications.

The integrable Kuralay equations (IKEs) represent a new class of nonlinear partial differential equations (NLPDEs) that capture the delicate balance between dispersion and nonlinearity in complex physical systems such as nonlinear optical fibers, plasmas, and ferromagnetic spin chains. From a geometric standpoint, these equations correspond to the motion of space curves, where soliton solutions describe the curvature and torsion of evolving wave structures. In this work, we employ the Complete Discrimination System for Polynomial Method (CDSPM) to systematically construct and classify a rich family of optical soliton solutions, including Jacobian elliptic, double-hyperbolic (DH), double-periodic (DP), and trigonometric forms. By exploring the limiting behaviors of elliptic solutions, we also derive solitary wave (SW) structures that exhibit kink-type, periodic, and localized pulse geometries. Furthermore, the bifurcation and stability analyses reveal intricate quasi-periodic and transitional behaviors, highlighting the strong coupling between the system’s geometry and its nonlinear dynamics. The findings not only deepen the theoretical understanding of the Kuralay framework but also offer practical applications in optical communication, plasma energy transport, and nonlinear circuit design. Overall, the study presents a unified analytical and geometrical framework that bridges mathematical theory with real-world physical phenomena.

In this study, we investigate the stochastic Kakutani-Matsuuchi model (SKMM) with multiplicative noise in the It & ocirc; sense, which describes the propagation of internal gravity waves in stratified fluids such as the Earth's atmosphere and ocean. These waves, generated by density or temperature variations, play a fundamental role in transferring energy and momentum across the system. The multiplicative noise term accounts for random fluctuations whose intensity depends on wave amplitude, thereby providing a realistic description of noise-wave interactions in geophysical environments, while the It & ocirc; framework ensures a rigorous mathematical treatment of such randomness and its cumulative effect on system evolution. By applying the Sub-ODE method, we derive a broad spectrum of exact analytical solutions, including bright soliton, periodic wave, rational-type, hyperbolic-type, and singular structures. Their geometrical characteristics are explored through 3D graphical representations obtained under different values of the random noise parameter, which reveal distinctive behaviors such as localization, periodic modulation, algebraic decay, and blow-up dynamics. These findings deepen the understanding of nonlinear wave phenomena governed by the SKMM and demonstrate the versatility of the Sub-ODE approach in capturing the impact of stochastic influences. The results are expected to provide a valuable reference for modeling wave propagation in oceanic and atmospheric systems where stochastic effects cannot be ignored.

Nonlinear internal ocean waves are important in stratified marine environments because they are linked to energy transport, momentum transfer, and mixing processes. In this study, the (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama (pYTSF) equation is used as an idealized mathematical model to study wave propagation, energy localization, and nonlinear interactions in multidimensional dispersive media. A multivariate bilinear neural network method (mBNNM), which serves as a hybrid neural-symbolic framework, is applied to obtain exact solutions of the governing equation. The developed [4 2 2 1] neural architecture is used to derive lump wave (LW), one-wave (OW), new lump wave (NLW), and triangular periodic wave (TPW) solutions. The [4 3 2 1] architecture is then used to obtain lump-kink wave (LKW), lump-periodic-kink interaction wave (INT3-W), and lump-periodic interaction wave (INT2-W) solutions. These solutions show localization, nonlinear deformation, periodic behavior, and multidimensional wave interactions. Their main features are illustrated using three-dimensional surface and contour plots. The results provide exact analytical benchmarks for multidimensional nonlinear-wave studies and for validating numerical and physics-informed models. The derived solutions provide qualitative benchmark structures that may be incorporated into future studies of ocean mixing, climate variability, sea-surface roughness, and electromagnetic scattering when combined with realistic physical data and coupled models.

Dengue fever continues to pose a serious and persistent threat to public health and remains endemic in more than fifty countries worldwide. The disease is primarily transmitted to humans through the bite of infected Aedes mosquitoes, whose wide geographic distribution, adaptation to urban settings, and sensitivity to climatic variability contribute to sustained outbreaks. Although several prevention measures (such as vector control, environmental management, and community awareness) are widely implemented, dengue transmission remains difficult to suppress due to complex human-vector interactions, heterogeneous exposure, and limitations in controlling mosquito breeding habitats. For these reasons, epidemiological and mathematical modeling has become an essential tool for systematically describing transmission mechanisms, identifying dominant risk determinants, and assessing the expected impact of intervention strategies, thereby supporting evidence-based planning for prevention and control. In this work, we formulate a host-vector mathematical model for dengue virus transmission by coupling a mosquito Susceptible-Infected (SI) subsystem with a human Susceptible-Infected-Recovered (SIR) subsystem. The model captures the exchange of infection between vector and human populations and enables a quantitative investigation of threshold dynamics. In particular, the basic reproduction number R_0 is derived using the next-generation matrix technique, providing a key threshold quantity that characterizes whether an initial infection can invade the population. We then analyze the stability properties of the equilibria: the local stability analysis shows that the disease-free equilibrium is asymptotically stable when R_0<1 , implying eventual elimination of dengue, whereas dengue persists and an endemic equilibrium may arise when R_0>1 . Furthermore, global stability results are established using a suitable Lyapunov function, which strengthens the conclusions beyond local behavior. To validate the theoretical outcomes and to explore the effect of the principal parameters that promote or mitigate dengue spread, numerical simulations are carried out in MATLAB for the relevant compartments. These simulations illustrate the temporal evolution of the human and mosquito populations under different epidemiological scenarios and parameter settings. In addition, sensitivity analysis is performed to quantify how variations in model parameters influence R_0 and the overall transmission dynamics. This analysis highlights the most influential factors and provides insight into which control measures (e.g., reducing mosquito biting rate, lowering vector density, or increasing recovery) may be most effective for limiting dengue transmission.

Lump and breather-type solutions have recently gained wide interest in nonlinear optics because of their fundamental importance in photonic systems and wave propagation studies. A lump soliton (LS) refers to a rationally localized wave that decays algebraically in every spatial direction, representing stable energy packets that remain confined without dispersing in optical media. In contrast, breather solutions (BS) describe waves that are localized in both space and time but display oscillatory behavior, either temporally or spatially, and they often serve as prototypes for nonlinear modulations and the generation of rogue waves. In this study, we consider a generalized wave propagation equation known as generalized Longitudinal Lugiato Lefever equation (gLLLE) which is capable of modeling both passive and active cavities, as well as hybrid structures such as semiconductor ring lasers subject to external optical driving. By employing suitable transformation techniques, we construct a variety of nonlinear excitations, including lump, lump–one strip (LoS), lump–two strip (LtS), lump–periodic (LP), and rogue wave solutions, together with several interaction patterns involving lumps, periodic waves, and kink waves. Moreover, we compute several families of breather solutions, such as Ma breathers (MBs), Kuznetsov-Ma breathers (KMBs), generalized breathers (GBs), and Akhmediev breathers (ABs), together with their associated rogue waves, and depict their structures using 3D, 2D, and contour plots. The outcomes of this study carry strong practical relevance, as lump and breather dynamics underpin applications in optical communication, all-optical switching, ultra-fast pulse generation, and secure information transfer. Furthermore, understanding their interactions sheds light on controlling extreme optical events like rogue waves, which is crucial for the stability of high-power laser systems and fiber-optic networks. Overall, our results offer new insights into the interplay of localized and periodic wave structures, paving the way for advances in integrated photonics, nonlinear optical systems, and next-generation telecommunication technologies.

This paper reports six novel families of vector quartic pulse (VQP) trains admitted by weakly birefringent optical fibers, described through a coupled system of extended nonlinear Schrödinger equations (NLSEs) incorporating quartic nonlinear interactions. The obtained VQP trains display rich and nontrivial intra-period structures in both polarization components, including single-hump, double-hump, and hybrid profiles, thereby revealing a level of waveform complexity that extends beyond conventional vector soliton and pulse-train dynamics in birefringent fibers. To capture realistic inhomogeneous propagation effects, a similarity-transformation framework is developed, which enables the analytical construction of self-similar vector pulse trains in the presence of longitudinally varying dispersion, nonlinearity, and gain/loss modulation. This approach provides a unified and systematic mechanism for embedding fiber-management effects directly into the analytical solutions, representing a significant methodological advancement over fixed-coefficient treatments. By applying the framework to periodically modulated systems, the propagation dynamics of the constructed pulse trains are investigated, demonstrating that careful engineering of dispersion and gain/loss coefficients can induce distinct evolution regimes, including shape-preserving propagation, periodic breathing, and robust oscillatory behavior. The results establish the existence, controllability, and stability-relevant features of quartic vector pulse trains (QVPT) in weakly birefringent fibers (WBF), thereby offering new physical insight and analytical tools for the design and control of complex periodic waveforms in advanced optical fiber systems.

We studied traveling-wave solutions of the Dullin–Gottwald–Holm (DGH) equation via a sub-ODE construction. Under explicit algebraic constraints, the approach yielded closed-form families—bell-shaped, hyperbolic (sech/tanh), Jacobi-elliptic function (JEF), Weierstrass-elliptic function (WEF), periodic, and rational—and classified their symmetry properties. Optical solitons (bright and dark) arose as limiting cases of the elliptic solutions. We specified the parameter regimes that produced each profile and illustrated representative solutions with 2D/3D plots to highlight symmetry. The results provide a unified, reproducible procedure for generating solitary and periodic DGH waves and expand the catalog of exact solutions for this model.

We study some optical pulses including dark solitary waves (DSWs), moving front solitons (MFSs) or optical shock type soliton, and periodic wave solution (PWSs) and some other solution for generalized Kundu-Eckhaus equation (KEE) and generalized nonlinear Schrödinger equation (GNLSE) using ansatz transformations. We derived the propagation dynamics for short light pulses in optical fiber for our governing models. The generalized KEE describes the propagation of ultra short femto pulses in optical fiber, while GNLSE discuses the picosecond pulses in optical fiber. The MFSs are a type of soliton that possess a sharp transition or discontinuity in the optical field. The DSWs represent a fascinating phenomenon in nonlinear wave propagation, offering unique opportunities for controlling and manipulating wave fields in various physical systems. The PWSs in optical fibers show potential applications in optical communications, where they can be utilized for generating multiplexing signals and frequency combs. We also illustrate our solutions using graphs in different dimensions.

Over the last few decades, malaria has become a serious risk to public health, particularly in tropical and sub-tropical areas where the climate is favorable for mosquito breeding. These insects are the primary carriers of the disease, transmitting it to humans through their bites. Here, we have formulated a mathematical framework that explores malaria transmission, incorporating a structured infectious population. Numerous dynamical system methodologies are instrumentalized in studying the malaria model in human-vector interacting populations. Firstly, we have proved that the model state variables has non-negative and bounded solutions throughout time. Then, we have obtained the threshold parameter ℛ_m, by employing the next generation operator approach. We have proved that the proposed malaria model is stable locally and globally in an asymptotic manner by calculating the Jacobian matrix and Lyapunov function theory if ℛ_m<1 . The malaria model is shown to have a unique endemic equilibrium point whenever the basic reproductive number ℛ_m>1 . Consequently, the unique malaria-endemic steady point of the proposed malaria model is proven to be globally stable provided that ℛ_m>1 . Sensitivity analysis is conducted to capture the most significant parameter causing malaria transmission and controlling in the human population. Furthermore, simulations are performed to support the qualitative results of the study, and the results are graphically presented.

Nonlinear stochastic models play a crucial role in describing complex wave phenomena in multidimensional physical systems. Motivated by this, we investigated the Stochastic Nizhnik-Novikov-Veselov (SNNV) equation to explore its solitary wave dynamics, chaotic behavior, bifurcation structures, sensitivity, and stability characteristics. We used the extended modified auxiliary equation mapping (EMAEM) method, an enhanced analytical framework with greater flexibility and broader solution structures versus conventional methods. With this approach, we derived new families of exact solitary wave solutions, including single, dark, and bright singular solitons. The proposed methods explain dynamical characteristics that were previously unexplored for the SNNV equation. The stochastic model will be converted into a dynamical system using the Galilean transformation. This approach enables exploration of its dynamical behavior via stochastic processes. We used Poincare maps, phase portraits, and time-series trajectory simulations to establish its strong stochastic behavior, including chaotic behavior. To show that these methods are dynamically stable, we conducted a stability analysis using the Hamiltonian system framework. The proposed study will significantly advance the dynamic interpretation of chaos in the SNNV equation and establish the superiority of EMAEM approach for wave structure formation.

In this paper, we study variational integrators (VIs) with the help of projection technique for Korteweg–de Vries (KdV) equation. First, we use forward, backward and central difference schemes. After that, we use Lagrangian, Euler–Lagrange equation and discrete Euler–Lagrange equation to find numerical solution to KdV equation. Finally, we obtain soliton solutions like W-shape, bright and kink soliton with the help of generalized Kudryashov method (GKM). These solitons are used in optical communication, Bose–Einstein condensate, plasma physics, fiber optics sensors and so on.

Deep learning solves the Nonlinear Evolution Equations (NLEEs) with the aid of the gradient-enhanced Physics-Informed Neural Networks (gPINNs). The PINNs have successfully been used to solve the PDEs as they successfully integrate the residual of the PDE into the neural network's loss function. gPINNs use the predictions as well as the gradients of the prediction, providing information about the behavior of the solution in general. gPINNs train faster as the information of the gradients guides the neural network better. They are more effective and stronger tools for higher-dimensional nonlinear problems. The aim of this paper is to seek a soliton solution of the nonlinear PDE known as the Phi-four equation with the aid of the gPINN method. It will be applied: A Neural Network with two to six hidden layers, with 10, 20, 30, 40 and 50 neurons in one of the hidden layers for a total of five artificial NNs configurations. Time series of mean square and absolute error will also be investigated. Some special initial and boundary conditions will also be considered to train the NN for the governing model as well as to detect the equations' errors by comparing the obtained results with the analytical solution. The results show thus a relatively good correspondence in the governing model's approximate solution, the mean square error of which is of the order 10(-3) to 10(-5) for all the simulated equations. We try this method for the first time for this model; thus new and novel results are presented here.

In order to better understand nonlinear wave dynamics, we systematically investigate a wide variety of exact solutions for the modified equal width-Burgers equation (mEWBE), revealing new wave patterns. Lump (L), lump with one kink (L_1K) , lump with two kinks (L_2K) , multiwave (MW), rogue wave (RW), periodic wave (PW), periodic cross-lump wave (PCLW), periodic cross-kink wave (PCKW), interaction between lump, periodic, and kink waves (LPKW), and breather lump wave (BLW) are among the systematic solutions that we obtain by using the appropriate transformation method. The M-shaped rational solution (MSRS), M-shaped rational solution with one kink (MSR_1K) , M-shaped rational solution with two kinks (MSR_2K) , periodic cross-rational solution (PCRS), kink-cross rational solution (KCRS), M-shaped rational solution with periodic and kink components (MSRPK), and M-shaped rational solution with rogue and kink features (MSRRK) are among the other novel rational solutions that we thoroughly examine. These results offer novel perspectives on intricate wave interactions in nonlinear systems. We further provide a deeper understanding of the propagation and interaction of effective wave structures in the WBE framework by presenting graphical visuals in many dimensions that highlight the complex characteristics of the found solutions. These results can be used practically to comprehend complicated wave behaviors in real-world systems, including shock waves, ocean waves, and multi-wave systems in engineering.

This study explores the nonlinear conformable Gross-Pitaevskii equation (NL-CGPE), a fundamental model in Bose-Einstein condensate (BEC) theory that describes a single quantum state shared by ultra-cold bosonic particles. Beyond its quantum significance, the NL-CGPE also models optical soliton propagation with applications in data transfer, telecommunication networks, and long-distance optical fibers. By employing the complete discrimination system of polynomial method (CDSPM), we derive multiple families of exact analytical soliton solutions, including periodic, hyperbolic, rational, trigonometric, and Jacobi elliptic (JE) types. Furthermore, JE solutions are transformed into single-wave structures, enriching the solution landscape. To capture the system's nonlinear dynamics, we conduct sensitivity analysis, Poincar & eacute; mapping, time-series profiling, and identify critical conditions leading to quasi-periodic behavior. An energy balance analysis (EBA) is also performed, confirming approximate periodic oscillations under energy conservation principles. In addition, machine learning regression techniques are applied to validate the equilibrium states of the model and to determine parameter thresholds consistent with dynamical analysis. Collectively, these results highlight both the diverse soliton behaviors supported by the NL-CGPE and the utility of integrating analytical, qualitative, and computational approaches to uncover its complex dynamics.

Despite advancements in medicine and vaccination, the global death toll from dengue disease continues to rise, especially in developing countries, where modern healthcare access and prevention are limited. In the twenty-first century, dengue disease has emerged as a severe global challenge for the health sector because it severely affects more than 100 countries and causes thousands of deaths annually. This study designs an SEIR-SEI mathematical model of the dengue virus by introducing a control parameter within the mosquito population. The inclusion of this parameter enables us to determine the impact of control measures on the transmission dynamics of the dengue virus. Firstly, we ensure that all state variables are bounded and remain non-negative throughout the study. Then, we calculate the two equilibrium points, the dengue-free equilibrium (DFE) and dengue-endemic equilibrium point (DEE), of the model for further analysis. We also calculate the reproductive number ℜ , which is a crucial threshold parameter in epidemiology. The qualitative analysis shows that the model possesses local and global stability at the DFE and DEE points if ℜ<1 and ℜ>1 , respectively. We also conduct a sensitivity analysis to identify the parameter with the most significant impact on the transmission dynamics of the dengue virus. This insight assists health policy makers in optimizing their struggles to lower the infection rates between susceptible humans and infected mosquito populations. We apply the two numerical schemes, the non-standard finite difference scheme (NSFD) and the Runga-kutta (RK4) scheme, to validate the theoretical and numerical results of the proposed SEIR-SEI dengue epidemic model. Numerical simulations are also provided in support of these results.