This article presents a new thermoelastic model that incorporates fractional-order derivatives of two-phase heat transfer as well as a two-temperature concept. The objective of this model is to improve comprehension and forecasting of heat transport processes in two-phase-lag systems by employing fractional calculus. This model suggests a new generalized fractional derivative that can make different kinds of singular and non-singular fractional derivatives, depending on the kernels that are used. The non-singular kernels of the normalized sinc function and the Rabotnov fractional–exponential function are used to create the two new fractional derivatives. The thermoelastic responses of a solid cylinder with a restricted surface and exposed to a moving heat flux were examined in order to assess the correctness of the suggested model. It was considered that the cylinder’s thermal characteristics are dependent on the linear temperature change and that it is submerged in a continuous magnetic field. To solve the set of equations controlling the suggested issue, Laplace transforms were used. In addition to the reliance of thermal characteristics on temperature change, the influence of derivatives and fractional order was also studied by providing numerical values for the temperature, displacement, and stress components. This study found that the speed of the heat source and variable properties significantly impact the behavior of the variables under investigation. Meanwhile, the fractional parameter has a slight effect on non-dimensional temperature changes but plays a crucial role in altering the peak value of non-dimensional displacement and pressure.

The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the $\left(\frac{{\mathsf{\Omega }}^{\prime }}{\mathsf{\Omega }}, \frac{1}{\mathsf{\Omega }}\right)$ method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields.

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful ($G^{\prime }/G$, $1/G$)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters $\lambda$, $\tau$, $c$, $\mathcal{B}$, and $\mathsf{{\rm K}}$ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter $\lambda$ exhibits consistently high sensitivity across all scenarios, while parameters $\tau$ and $c$ demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis.