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In this article, we investigated the solitary wave solutions of the KdV–mKdV equation using Hirota’s bilinear method. Closed-form analytical single and multiple solitary wave solutions were obtained. Through qualitative methods and the analysis of solitary waveforms, we discovered that in addition to sech-type solitary waves, the system also contains Sech 2 -type solitary waves. By employing the trial functions method, we obtained a single Sech 2 -type solitary wave and verified its existence and stability using the split-Step Fourier Transform method. Furthermore, we use the collision of two Sech 2 -type single solitary waves to excite a stable Sech 2 -type double solitary wave. Similarly, we excite a stable triple solitary wave with three Sech 2 -type single solitary waves. This method can also be used to excite stable multiple solitary waves. It is shown that these solitary wave solutions enrich the dynamic behavior of the KdV–mKdV equation and provide methods for solving Sech 2 -type solitary waves, which hold significant theoretical value. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel formulation of the modified Atangana–Baleanu (AB) Fractional Operators (FrOs). The paper begins by discussing the boundedness of the novel fractional derivative operator. Some fractional differential equations corresponding to different choices of functions as well as comparative graphical representations of a function and its derivative are provided. Furthermore, the paper investigates the generalized Laplace transform for this newly introduced operator. By employing the generalized Laplace transform, a wide range of fractional differential equations can be effectively solved. Additionally, the paper establishes the corresponding form of the AB Caputo fractional integral operator, examines its boundedness and obtains its Laplace transform. It is worth noting that the FrOs previously documented in the existing literature can be derived as special cases of these recently explored FrOs. [ABSTRACT FROM AUTHOR]