New soliton, kink and periodic solutions for fractional space–time coupled Schrödinger equation

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.