Exploring the new soliton solutions to the nonlinear M-fractional evolution equations in shallow water by three analytical techniques

This paper explores the new soliton solutions of the evolution equations named as truncated M-fractional (1+1)-dimensional non-linear Kaup-Boussinesq system by utilizing the expa function, modified simplest equation and Sardar sub-equation techniques. This system is used in the analysis of long waves in shallow water. The attained results involving trigonometric, hyperbolic and exponential functions. The effect of fractional order derivative is also discussed. Obtained results are very close to the approximate results due to the use of M-fractional derivative. Achieved results are verified by Mathematica tool. Few of the gained results are also explained through 2-D, 3-D and contour graphs. At the end, these techniques are straight forward, useful and effective to deal with non-linear FPDEs. •We conduct an investigation of the truncated M-fractional (1+1)-D Kaup-Boussinesq system.•Distinct kinds of occurring phenomena are expressed in the form of nonlinear evolution models.•The achieved solutions having dark, bright, dark-bright, periodic and other solitons.•The solutions are verified as well as explained through 2-D, 3-D and contour plots.