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This work investigates the time–space fractional coupled nonlinear Schrödinger equation. By applying an appropriate wave transformation, this equation is converted into a fourth-order system of ordinary differential equations, equivalent to a Hamiltonian system with two degrees of freedom. The integrability of the Hamiltonian system is examined using Painlevé analysis. We demonstrate that the Hamiltonian system is completely integrable in the Liouville sense in two cases, wherein the Hamilton–Jacobi equation is also separable, and we introduce the non-integrability conditions. The first integrals of motion for the separable cases are provided. Utilizing bifurcation analysis, we depict the phase portrait and subsequently integrate the separable Hamiltonian system and construct new solutions. Some of solutions are illustrated graphically, and it is shown that the wave solutions profiles are sensitive to temporal and spatial fractional derivatives.

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.