Analytical solutions of the time-fractional symmetric regularized long wave equation using the $$\phi ^{6}$$ model expansion method

In this paper, we introduce a new analytical technique to study the time-fractional symmetric regularized long wave (SRLW) equation, which is an important model for nonlinear wave phenomena in dispersive media. Combining the new [Formula: see text] model expansion technique with a conformable fractional derivative provides a systematic means of finding a wide class of exact traveling wave solutions, such as bright solitons, kink solitons, singular periodic solitons, and periodic solitons. which are crucial in optical and fluid systems, and their localized singularities, indicating wave-breaking or energy concentration effects, and their real-world implications. The solutions have been successfully shown and illustrated in 2D and 3D graphics. We then consider the effects of specific memory effects that are characteristic of fractional derivatives and expose that they are the key in regulating the amplitude and the phase shift of the waves and their stability. Our research not only enhances the mathematical resources available for fractional nonlinear systems but also establishes a solid foundation for modeling intricate wave phenomena in fluid mechanics, plasma physics, and advanced materials. This work links theoretical analysis with practical applications, emphasizing the transformative potential of fractional calculus in understanding real-world nonlinear phenomena.