Conservation laws for nonlinear Riemann-Liouville FPDEs: A symmetry/adjoint symmetry pair method without Lagrangian requirements

This paper presents a systematic extension of the symmetry/adjoint symmetry pair (SAS) method to nonlinear Riemann-Liouville fractional partial differential equations (FPDEs) for the direct construction of conservation laws, completely bypassing the need for Lagrangian formulations. The proposed framework is applied to the nonlinear time-fractional diffusion equation (covering both subdiffusion for α ∈ (0, 1) and diffusion-wave cases for α ∈ (1, 2)), coupled fractional KdV-type system, and time-fractional b-family peakon equations with mixed derivatives. For each equation class, explicit conservation laws are successfully constructed through appropriate pairings of symmetries and adjoint symmetries, circumventing the Lagrangian mechanism entirely. Notably, we demonstrate that Ibragimov’s conservation theorem emerges as a special case within the SAS framework, thereby establishing the universality and theoretical completeness of the proposed method. This work not only provides a powerful tool for exploring conservation properties of fractional systems but also opens new avenues for extending symmetry-based methods to broader classes of non-integer order differential equations.