In this study, the traveling wave solutions for the time-fractional shallow water wave equation system, whose physical application is defined as the dynamics of water bodies in the ocean or seas, are investigated by [Formula: see text]-expansion method. The nonlinear fractional partial differential equation is transformed to the non-fractional ordinary differential equation with the use of a special wave transformation. In this special wave transformation, we consider the conformable fractional derivative operator to which the chain rule is applied. We obtain complex hyperbolic and complex trigonometric functions for the time-fractional shallow water wave equation system with the help of this technique. New traveling wave solutions are obtained for the special values given to the parameters in these complex hyperbolic and complex trigonometric functions, and the behavior of these solutions is examined with the help of 3D and 2D graphics.
BIFURCATION ANALYSIS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERIC TWO-DIMENSIONAL SINE-GORDON EQUATION IN NONLINEAR OPTICS