Abstract
This article discusses the dynamical structures of unique traveling wave solutions to the unidirectional time-fractional Dullin–Gottwald–Holm equation, as well as the modulation instability analysis of solitary wave prorogations in shallow water mechanics. Different sorts of explicit solutions are obtained which are expressed as the kink, singular kink, singular soliton, multiple solitons and other forms of soliton with specified parameters by utilizing the unified method. Three-dimensional plots, density plots, and their two-dimensional combined line plots of the obtained novel solutions satisfying the corresponding equation of interest are given to comprehend the underlying mechanisms of the identified family.The obtained novel wave solutions can motivate applied scientists to refine their theories to the best of their abilities and can be utilized to verify the outcomes of numerical simulations for wave propagation in shallow water and other nonlinear cases. The implemented methods are shown to be straightforward and effective for approximating the considered equation, and it may be utilized to resolve numerous classes of nonlinear partial differential equations that arise in engineering and mathematical physics.
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