Analytic solutions of the generalized water wave dynamical equations based on time-space symmetric differential operator

It is well known that there is a deep connection between the symmetric and traveling wave solutions. It has been shown that all symmetric waves are traveling waves. In this paper, we establish new analytic solution collections of nonlinear conformable time-fractional water wave dynamical equation in a complex domain. For this purpose, we construct a new definition of a symmetric conformable differential operator (SCDO). The operator has a symmetric representation in the open unit disk. By using SCDO, we generalize a class of water wave dynamical equation type time-space fractional complex Ginzburg–Landau equation. The results show that the obtainable approaches are powerful, dependable and prepared to apply to all classes of complex differential equations.