The time-fractional (2+1)-dimensional Hirota–Satsuma–Ito equations: Lie symmetries, power series solutions and conservation laws

Abstract

In this paper, we concentrate on the Lie symmetries and conservation laws of the time-fractional (2+1)-dimensional Hirota–Satsuma–Ito equations which contain both the integer-order partial differential equations (PDEs) and the fractional PDEs with the mixed derivative of Riemann–Liouville fractional derivative and second order x-derivative. Therefore, based on a new prolongation formula of the infinitesimal generator in the case of mixed derivatives, we find the admitted Lie symmetries and reduce the equations to time-fractional (1+1)-dimensional PDEs by means of an optimal system of one-dimensional Lie subalgebras. In particular, we construct an explicit power series solution with fractional degrees and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Moreover, we give a conservation law formula using the idea of nonlinear self-adjointness for the fractional case and obtain several nontrivial conservation laws by means of the obtained Lie symmetries.