Symmetry analysis, power series solutions and conservation laws of fractional nonlinear generalized Schrödinger equation with Caputo derivative

Nonlinear fractional Schrödinger equations play an important role in quantum mechanics, one of which with a variable coefficient is studied by Lie symmetry analysis method in this paper. We obtained two Lie symmetries and the admissible coefficient [Formula: see text], which maintain the governing equation invariant, and used one of the obtained symmetries to reduce nonlinear fractional partial differential equations to nonlinear fractional ordinary differential equations in the sense of the Caputo fractional derivative. Then we applied the power series method to obtain the power series solutions of the reduced equations, thereby proving their convergence and providing the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators developed by Ibragimov are used to construct the conservation laws for the equations studied.