On the local fractional variable‐coefficient Ablowitz–Kaup–Newell–Segur hierarchy: Hamiltonian structure, localization of nonlocal symmetries, and exact solution of reduced equations

摘要

Starting from a local fractional variable‐coefficient Ablowitz–Kaup–Newell –Segur (AKNS) spectral problem involving an arbitrary function et0It(α)a(ξ), we derive a hierarchy of fractional variable‐coefficient nonlinear evolution equations (NLEEs), which possesses Hamiltonian structure. It is interesting that the hierarchy is explicitly related to many important equations such as the fractional variable‐coefficient nonlinear Schr ödinger equation, the fractional variable‐coefficient Burgers equation, and the fractional variable‐coefficient mKdV equation involving time‐dependent damping and dispersion. Then, the truncated Painlevé method is generalized to find the multiple nonlocal symmetries of the fractional variable‐coefficient Burgers equation. Then, starting from the initial value problem of the localized symmetries for the expanded system, we derive the auto‐B äcklund transformation of the fractional variable‐coefficient Burgers equation. Furthermore, exact solutions of the fractional variable‐coefficient mKdV equation is obtained by the improved solitary ansatze methods.