Time-fractional Davey–Stewartson equation: Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions

Abstract As a celebrated nonlinear water wave equation, the Davey–Stewartson equation is widely studied by researchers, especially in the field of mathematical physics. On the basis of the Riemann–Liouville fractional derivative, the time-fractional Davey–Stewartson equation is investigated in this paper. By application of the Lie symmetry analysis approach, the Lie point symmetries and symmetry groups are obtained. At the same time, the similarity reductions are derived. Furthermore, the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative. By virtue of the symmetry corresponding to the scalar transformation, the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators. By using Noether’s theorem and Ibragimov’s new conservation theorem, the conserved vectors and the conservation laws are derived. Finally, the traveling wave solutions are achieved and plotted.