New wave dynamics of the time-fractional Kaup–Kupershmidt model of seventh-order arises in shallow water waves

In this paper, a new variety of solitary wave patterns to the time-fractional seventh-order Kaup–Kupershmidt equation is studied. This model is important because of its nonlinear effects on the propagation of different water waves. For this study, we have considered the beta-fractional derivative form of the model. To derive the required exact solutions, we have used two analytical methods, specifically the new Kudryashov (nK) and modified Khater (mK) methods. Different types of wave patterns are produced from the solutions for distinct fractional and unidentified parameter values. These solutions include bright, two-soliton propagation, combined bright-dark, w-shaped pattern, combined dark-bright, m-shape wave, l-shaped bright wave, w-shaped periodic, u-shaped wave-form, and grey-type w-shaped periodic wave solutions. These dynamics of different wave natures are analyzed thoroughly by the graphical depiction of the solutions. Additionally, the characteristics of water waves and their many application areas can benefit greatly from these solutions such as surface waves in deep water, the dynamics of liquid-vapour interfaces, and many more. These solutions help us to understand how nonlinearity can affect the system during wave propagation. The novel aspect of this work is that the investigated model of the beta fractional form has never been solved before.