In this study, accurate solitary wave solutions of the generalized (3+1)-dimensional Sasa-Satsuma equation are investigated using the modified F-expansion approach and the extended hyperbolic function method. The model under consideration describes the propagation of ultrashort optical pulses in nonlinear optical fibers, where nonlinear derivative terms, self-steepening and third-order dispersion are important higher-order effects. A wide range of analytical solutions, including solitonic, trigonometric, rational and exponential wave shapes, are derived using the modified F-expansion method. Additionally, the extended hyperbolic function method is used to generate bright, dark and singular soliton solutions as well as periodic wave solutions under appropriate parametric restrictions. Furthermore, modulation instability is examined using linear stability theory, and the parameter regimes in which the solutions become unstable are determined using the associated dispersion relation. The acquired solutions are unique and have not been documented in the current literature, as far as we are aware. The suggested techniques are useful tools for the analysis of a broad class of nonlinear evolution equations because they are simple, effective and widely applicable. Graphical representations are used to help clarify the physical properties of the generated solutions.
The cornerstone of this paper is the developing of novel analytical travelling wave solutions for the AizhanGudekli-Nurshuak-Zhanbota equation (AGNZE), which was recently introduced as an integrable model arising in the study of space curves and surfaces. Therefore, its analytical exploration is vital for understanding its physical applications. We will explore solitons and various solitary wave solutions with diverse physical structures of this model. Significantly, this model characterizes the generalized Heisenberg ferromagnetic model. The innovation of this work lies in the enriched and distinct soliton solutions obtained, and in performing a comparative analysis of the proposed method, which has not been previously addressed. These achieved solutions correspond to the spin waves in the context of the ferromagnetic model. The construction of these new travelling wave solutions is carried out within the framework of notable schemes, the modified exp [-xi]-expansion method and the modified simple equation method which are governed by the balance rule. Furthermore, the numerical solutions for all obtained analytical travelling wave solutions have been implemented by using the differential transform method (DTM), which is considered one of the most effective semianalytical and numerical methods. The extracted solutions display distinct physical structures, including forms like bright solitons, dark solitons, combination of bright solitons and dark solitons, peakon-type bright and dark waves, anti-kink wave structures, W-like solitons, M-like solitons, hyperbolic solitons, singular solitons, and other rational solitons forms. The obtained solutions may be applied to model ultra-short pulse propagation in some nonlinear fields such as optical fibers, photonic crystals, waveguides, and solitary waves in shallow water. Additionally, we will use 2D and 3D graphical simulations to illustrate the new dynamic properties of our obtained solutions by using a mathematical tool.
The efficiency of microsprinkler irrigation depends on water distribution uniformity, which is influenced by the interaction between operating pressure and spacing. This study aimed to model and optimize the hydraulic performance of the Hadar 7110 microsprinkler through the integration of Principal Component Analysis (PCA) and Response Surface Methodology (RSM). The experiment was conducted under pressures ranging from 0.5 to 2.5 bar and areas from 9 to 36 m². PCA was employed to synthesize the Christiansen Uniformity Coefficient (CUC) and the Distribution Uniformity Coefficient (DUC) into the first principal component (PC1), which accounted for the majority of the total variance. A second-order polynomial model was fitted to PC1, exhibiting a high coefficient of determination (R² = 0.8134; p < 0.001). Response surface analysis identified a stationary point classified as a saddle point, located at the operational coordinates of 1.46 bar and 31.15 m². The nature of this point indicates that the optimal conditions for integrated uniformity do not occur in a central region of the domain, but rather at the experimental boundaries. The multivariate approach proved to be a robust tool for simplifying performance analysis, identifying that uniformity is maximized under combinations of high pressures with larger areas, or moderate pressures with smaller areas per emitter. These results provide technical support for the design and strategic management of microsprinkler irrigation systems.
The aim of this paper is to investigate soliton solutions of the extended (2+1)-dimensional nonlinear Schr & ouml;dinger equation incorporating third-order dispersion and nonlinearity, which models the propagation of ultra-short optical pulses in nonlinear dispersive media. By employing the modified extended direct algebraic method, we successfully derive a rich variety of exact solutions, including bright, dark, and singular solitons. Moreover, we obtain periodic, singular periodic, hyperbolic, exponential, and Jacobi elliptic function solutions, as well as doubly periodic solutions expressed in terms of Weierstrass elliptic functions. In addition, we perform a bifurcation analysis to examine the phase space dynamics and identify the stability characteristics of the system's fixed points. To illustrate the physical behavior of the derived solitons, 2D and 3D graphical simulations are presented, showcasing their evolution over time.
In this research, exact traveling wave solutions of nonlinear Kairat-X model are derived using modified F-expansion method and extended hyperbolic function method. Different solutions to the proposed model has been constructed using these methods. Trigonometric function solutions, soliton solutions, rational solutions and exponential solutions are obtained using modified F-expansion method. The solutions obtained by extended hyperbolic function method are periodic, singular, bright, dark and periodic singular soliton solutions. The obtained results are explained by plotting some graphsin3D, 2D (line plots) and contour plots using Mathematica which demonstrates the structure of solutions.
