Dynamics of discrete breathers in the integrable model of the 1D crystal

In the frame of the exactly integrable model of the 1D crystal — Hirota lattice model — the dynamics and interaction of the discrete breathers has been considered. These high-frequency localized nonlinear excitations elastically interact with each other and with such excitations as shock and linear waves. Using the nonlinear superposition formula the pair collision processes of the excitations are analytically described and explicit expressions for center-of-mass shifts of shock waves (kinks) and breathers, and phase shifts of oscillations of breathers and linear waves are discussed. The dynamics of the discrete breathers and kinks as the particle-like excitations of the Hirota lattice is described using the Hamiltonian formalism. The exact nonlinear periodic solutions describing the breathers and solitons superlattices in the Hirota lattice are analysed, and their stability boundaries are determined. The analogue of the discrete breather for the finite-size system is presented in terms of the elliptic Jacobi functions and it is shown that the excitation is detached from the branch of nonlinear homogenous antiphase oscillations in the bifurcation manner.