Nonlinear vector waves of a flexural mode in a chain model of atomic particles

Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the ‘second order cubic Benjamin–Ono (socBO) equation’ and ‘nonlinear pseudo-diffusion equation’. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.