Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead system is extended to an enlarged system with the auxiliary variable. Then the finite symmetry transformation for the prolonged system is found by solving the initial-value problems. Furthermore, by applying symmetry reduction method to the enlarged system, two kinds of the group invariant solutions are given.

This study constructed the explicit N -fold Darboux transformation (DT) for the variable-coefficient nonlocal Gerdjikov–Ivanov equation based on a higher-order polynomial matrix, thereby generating various localized waves. Moreover, the generation mechanism of rogue waves (RWs) was analyzed by combining modulational instability. First, starting from a zero seed solution, N solitons and multiple solitons on the periodic background were derived using the N -fold classical DT. Analytical expressions for the amplitudes at soliton collision centers were obtained, showing that the peak amplitudes are determined only by the real parts of the spectral parameters and are independent of the variable coefficient δ ( t ) . Second, starting from a nonzero seed solution, breathers were investigated, and the first-order RW was derived by taking the limit of the first-order breather. Furthermore, the second-order RW was studied via the degenerate DT. With appropriate parameter choices, a second-order RW can split into three first-order RWs arranged in a triangular pattern. Finally, numerical simulations visually illustrated the effects of δ ( t ) on soliton interactions, soliton evolution on periodic backgrounds, breather oscillation patterns, and the spatial structures and dynamical behaviors of RWs of different orders.