Optimal system of Lie subalgebra for symmetry reductions, group invariant solutions and exact solutions to the coupled Hirota–Maccari system driving pulse propagation in optical fiber

In this paper, Lie symmetry analysis has been proposed by utilizing the Lie group of continuous point transformation for obtaining the new exact soliton solutions of the (2+1)-dimensional Hirota–Maccari system. Lie infinitesimals and possible geometric vector fields are obtained by applying the third-order prolongation on this system. Also, their commutative product and adjoint relations have been presented in Tables 1 and 2. By considering the resulting symmetries, one-dimensional optimal system of Lie subalgebra is obtained. Meanwhile, the Hirota–Maccari system is reduced to a system of ordinary differential equations with the help of optimal subalgebras. Furthermore, the simplest equation method has been used to obtain the abundant exact soliton solutions of the reduced system. At last, conservation laws of Hirota–Maccari system have been extracted by utilizing the generalized “new conservation theorem” invoked by Ibragimov.