Localized waves and modulational instability of the variable-coefficient nonlocal Gerdjikov-Ivanov equation

Abstract

This paper constructs the specific expression of the N-fold Darboux transformation (DT) for the variable-coefficient nonlocal Gerdjikov–Ivanov equation based on the higher-order polynomial matrix, thereby generating various localized waves. Moreover, the generation mechanism of rogue waves is analyzed by combining modulational instability. The details are as follows: First, starting from the zero seed solution, N solitons and multiple solitons on the periodic background are derived using the N-fold classical DT. Analytical expressions for the amplitudes at soliton collision centers are 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 the nonzero seed solution, breathers are investigated, and the first-order rogue wave is derived by taking the limit of the first-order breather. Furthermore, the second-order rogue wave is studied via the degenerate DT. With appropriate parameter choices, a second-order rogue wave can split into three first-order rogue waves arranged in a triangular pattern. Finally, numerical simulations visually illustrate the effects of δ(t) on soliton interactions, soliton evolution on periodic backgrounds, breather oscillation patterns, and the spatial structures and dynamical behaviors of rogue waves of different orders.