Quantum compacton and quantum peakon in Heisenberg ferromagnetic chains with Dzyaloshinsky–Moriya interaction

Abstract

The one-dimensional ferromagnet XXZ spin chain with Dzyaloshinsky–Moriya (DM) interaction, recently introduced by Djoufack and coworkers is reexplored, with a particular attention carried on their found discrete nonlinear Schrödinger (DNLS) equation, governing the quantum breathers states behaviors. This DNLS equation admits exact bright compacton and peakon-like solutions, where analytical expressions, the existence and stability criteria are found and used to obtain their quantized energy levels. Using the semi-discrete multiple-scale method, the DNLS equation is reduced to the extended nonlinear Schrödinger equation which consists of the basic NLS equation with additional nonlinear dispersive terms Via the bifurcation diagram and Liapunov exponents, we provide a summary of essential dynamics and show that the equation would admit several forms of solutions among which the localized Hartree n compacton and peakon-like boson quantum breathers states. Furthermore, we notice that on the contrary to the classical outcomes where amplitudes of both solutions are free parameters, the amplitudes for quantum states are not free since the obtained solutions need to be normalized. The stationary localization of both compacton and peakon-like states is confirmed by numerical simulations performed on the DNLS equation.