In this study, the underlying traits of the new wave equation in extended (3+1) dimensions, utilized in the field of plasma physics and fluids to comprehend nonlinear wave scenarios in various physical systems, were explored. Furthermore, this investigation enhanced comprehension of the characteristics of nonlinear waves present in seas and oceans. The analytical solutions of models under consideration were retrieved using the sub-equation approach and Sardar sub-equation approach. A diverse range of solitons, including bright, dark, combined dark-bright, and periodic singular solitons, was made available through the proposed methods. These solutions were illustrated through visual depictions utilizing 2D, 3D, and density plots with carefully chosen parameters. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation analysis was conducted via phase portraits at critical points, revealing the system's transition dynamics. Introducing an external periodic force induced chaotic phenomena in the dynamical system, which were visualized through time plots, two-dimensional plots, three-dimensional plots, and the presentation of Lyapunov exponents. Furthermore, the sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. The obtained findings indicated the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems.

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.

This work studies the behavior of electrical signals in resonant tunneling diodes through the application of the Lonngren wave equation. Utilizing the method of Lie symmetries, we have identified optimal systems and found symmetry reductions; we have also found soliton wave solutions by applying the tanh technique. The bifurcation and Galilean transformation are found to determine the model implications and convert the system into a planar dynamical system. In this experiment, the equilibrium state, sensitivity, and chaos are investigated and numerical simulations are conducted to show how the frequency and amplitude of alterations affect the system. Furthermore, local conservation rules are demonstrated in more detail to unveil the whole system of movements.

We present a new concept of fractional curvature invariant for regular curves in the Lorentz plane by generalizing the Caputo‐fractional curvature from Euclidean geometry to the pseudo‐Riemannian setting. Our construction projects the integer-order derivative of the Caputo vector of fractional-order derivatives onto the Lorentzian normal direction, yielding a curvature measure that naturally distinguishes timelike and spacelike curves. Explicit formulas for representative model curves are derived, and we illustrate how the Lorentzian metric signature fundamentally changes fractional curvature behavior. This framework extends fractional‐order geometric analysis into relativity, providing new tools for studying memory effects and nonlocal dynamics along curves in relativistic contexts.

The (2+1)-dimensional Chaffee-Infante equation (CIE) is a significant model of the ion-acoustic waves in plasma. The primary objective of this paper was to establish and examine closed-form soliton solutions to the CIE using the modified extended direct algebraic method (m-EDAM), a mathematical technique. By using a variable transformation to convert CIE into a nonlinear ordinary differential equation (NODE), which was then reduced to a system of nonlinear algebraic equations with the assumption of a closed-form solution, the strategic m-EDAM was implemented. When the resulting problem was solved using the Maple tool, many soliton solutions in the shapes of rational, exponential, trigonometric, and hyperbolic functions were produced. By using illustrated 3D and density plots to evaluate several soliton solutions for the provided definite values of the parameters, it was possible to determine if the soliton solutions produced for CIE are cuspon or kink solitons. Additionally, it has been shown that the m-EDAM is a robust, useful, and user-friendly instrument that provides extra generic wave solutions for nonlinear models in mathematical physics and engineering.