In this paper, the Lie symmetry analysis method is extended to investigate the space–time fractional Gerdjikov–Ivanov equation, a generalized derivative nonlinear Schrödinger equation that arises in various branches of nonlinear physics, including nonlinear optics, fluid mechanics, and plasma physics. By applying the Lie symmetry approach, we derive the corresponding symmetry groups for both the space–time fractional equation and its special cases—the time-fractional and the integer-order Gerdjikov–Ivanov equations. The obtained Lie symmetries are then utilized to reduce the fractional partial differential equations with Riemann–Liouville fractional derivative to systems of fractional ordinary differential equations involving the Erdélyi–Kober fractional derivative. Subsequently, the power series method is employed to construct explicit power series solutions for the reduced systems, and the convergence of these solutions is rigorously analyzed with numerical simulations provided to support the analytical results. Furthermore, the conservation laws for the studied equations are systematically established by developing a generalized version of Noether's operators and applying the new conservation theorem. The results presented in this work not only extend the existing Lie symmetry analysis of derivative nonlinear Schrödinger-type equations but also provide a comprehensive analytical framework for studying fractional nonlinear evolution equations.
Citation: Jicheng Yu. Invariant analysis, power series solutions, and conservation laws of the space–time fractional Gerdjikov–Ivanov equation[J]. AIMS Mathematics, 2026, 11(10): 34546-34569. doi: 10.3934/math.20261346
In this paper, the Lie symmetry analysis method is extended to investigate the space–time fractional Gerdjikov–Ivanov equation, a generalized derivative nonlinear Schrödinger equation that arises in various branches of nonlinear physics, including nonlinear optics, fluid mechanics, and plasma physics. By applying the Lie symmetry approach, we derive the corresponding symmetry groups for both the space–time fractional equation and its special cases—the time-fractional and the integer-order Gerdjikov–Ivanov equations. The obtained Lie symmetries are then utilized to reduce the fractional partial differential equations with Riemann–Liouville fractional derivative to systems of fractional ordinary differential equations involving the Erdélyi–Kober fractional derivative. Subsequently, the power series method is employed to construct explicit power series solutions for the reduced systems, and the convergence of these solutions is rigorously analyzed with numerical simulations provided to support the analytical results. Furthermore, the conservation laws for the studied equations are systematically established by developing a generalized version of Noether's operators and applying the new conservation theorem. The results presented in this work not only extend the existing Lie symmetry analysis of derivative nonlinear Schrödinger-type equations but also provide a comprehensive analytical framework for studying fractional nonlinear evolution equations.
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Jicheng Yu. Invariant analysis, power series solutions, and conservation laws of the space–time fractional Gerdjikov–Ivanov equation[J]. AIMS Mathematics, 2026, 11(10): 34546-34569. doi: 10.3934/math.20261346
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