A variety of novel closed‐form soliton solutions to the family of Boussinesq‐like equations with different types

This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion. The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary, kink, anti-kink, combo, and singular-periodic wave solutions. The attained solutions are expressed by the trigonometric and hyperbolic functions including $\tan$, $\sec$, $\cot$, $\csc$, $\tanh$, $\mathrm{sech}$, $\coth$, $\mathrm{csch}$, and of their combination. In addition, the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family. For the bright wave solutions in terms of Atangana’s conformable derivative, the amplitudes of the bright wave gradually decrease, but the smoothness increases when the fractional parameters $\alpha$ and $\beta$ increase. On the other hand, the periodicities of periodic waves increase. The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases. The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.