This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

In this study, we build multi-wave solutions of the KdV-5 model through Hirota’s bilinear method. Taking complex conjugate values of the free parameters, various colliding exact solutions in the form of rogue wave, symmetric bell soliton and rogue waves form; breather waves, the interaction of a bell and rogue wave, and two colliding rogue wave solutions are constructed. To explore the characteristics of the breather waves, localized in any direction, the higher-order KdV-5 model, which describes the promulgation of weakly nonlinear elongated waves in a narrow channel, and ion-acoustic, and acoustic emission in harmonic crystals symmetrically is analyzed. With the appropriate parameters that affect and manage phase shifts, transmission routes, as well as energies of waves, a mixed solution relating to hyperbolic and sinusoidal expression are derived and illustrated by figures. All the single and multi-soliton appeared symmetric about an axis of the wave propagation. The analyzed outcomes are functional in achieving an understanding of the nonlinear situations in the mentioned fields.

In this paper, the Gilson–Pickering (GP) equation with applications for wave propagation in plasma physics and crystal lattice theory is studied. The model with wave propagation in plasma physics and crystal lattice theory is explained. A collection of evolution equations from this model, containing the Fornberg–Whitham, Rosenau–Hyman, and Fuchssteiner–Fokas–Camassa–Holm equations is developed. The descriptions of new waves, crystal lattice theory, and plasma physics by applying the standard $tan(\varphi /2)$-expansion technique are investigated. Many alternative responses employing various formulae are achieved; each of these solutions is represented by a distinct plot. Some novel solitary wave solutions of the nonlinear GP equation are constructed utilizing the Paul–Painlevé approach. In addition, several solutions including soliton, bright soliton, and periodic wave solutions are reached using He’s variational direct technique (VDT). The superiority of the new mathematical theory over the old one is demonstrated through theorems, and an example of how to design and numerically calibrate a nonlinear model using closed-form solutions is given. In addition, the influence of changes in some important design parameters is analyzed. Our computational solutions exhibit exceptional accuracy and stability, displaying negligible errors. Furthermore, our findings unveil several unprecedented solitary wave solutions of the GP model, underscoring the significance and novelty of our study. Our research establishes a promising foundation for future investigations on incompressible fluids, facilitating the development of more efficient and accurate models for predicting fluid behavior.

The present study aims to design a mathematical system based on the Lane–Emden third-order pantograph differential model by using the general forms of the pantograph as well as the Lane–Emden models. The designed model is divided into two types along with the various singularity details at each point. The shape factors and the pantograph points are discussed for each type of the newly designed nonlinear third-order pantograph differential model. The Bernoulli collocation scheme is implemented to find the numerical results of the novel model. To show the reliability of the designed novel nonlinear model, four different variants have been solved. Moreover, the comparison of the obtained results with the exact solutions is presented to check the accuracy of the designed novel model.

This paper formulates and analyzes a modified Previte-Hoffman food web with mixed functional responses. We investigate the existence, uniqueness, positivity and boundedness of the proposed model's solutions. The asymptotic local and global stability of the steady states are discussed. Analytical study of the proposed model reveals that it can undergo supercritical Hopf bifurcation. Furthermore, analysis of Turing instability in spatiotemporal version of the model is carried out where regions of pattern creation in parameters space are obtained. Using detailed numerical simulations for the diffusive and non-diffusive cases, the theoretical findings are verified for distinct sets of parameters.