Quantum breathers and two-breathers in the β-Fermi-Pasta-Ulam chain with the second-neighbor coupling

A theoretical study on quantum breathers and two-breathers in a β-Fermi-Pasta-Ulam chain with the second-neighbor coupling is reported. In the Hartree approximation, the equation of motion for the single-vibron wave functions is obtained analytically. By the discrete modulational instability analysis, we analyze the influence of the second-neighbor coupling on modulational instability areas and predict the appearance condition for the stationary localized solution. What is more, we obtain analytical forms of the stationary localized solution and discuss the influence of the next-nearest-neighbor coupling on their existence conditions. With stationary localized single-vibron wave functions obtained, the quantum breather state is constructed. The results show that varying the relative strength of the second-neighbor coupling can change the wave number corresponding to the appearance of the quantum breathers and the degree of the localization of quantum breathers. In addition, we successfully obtain the analytical form of quantum two-breathers, which have significant quantum properties.