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This research paper's primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing. [ABSTRACT FROM AUTHOR]

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota's bilinear technique to get multiple regular and singular kink solutions. [ABSTRACT FROM AUTHOR]

The modified exponential function method is used to obtain some new analytical solutions of the nonlinear Gerdjikov–Ivanov equation with the M-fractional operator. The novel obtained solutions are expressed in hyperbolic, trigonometric, and exponential function forms. Moreover, the instability modulation and gain spectra of the Gerdjikov–Ivanov equation are also analyzed. Constraints conditions are utilized to verify the existence of the solutions. Presented solutions are novel, satisfy and verify the M-fractional Gerdjikov–Ivanov equation. [ABSTRACT FROM AUTHOR]

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]