Flexural transverse waves in the anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes is derived in the long-wave approximation, when the dispersion effects are relatively weak. Depending on the relative strength of the coupling constants characterising the chain different model equations can be obtained from the basic equation. In one particular case the equation derived can be reduced to the non-integrable vector mKdV equation earlier obtained in the paper by Gorbacheva and Ostrovsky (Physica D, 1983). In another limiting case two new model equations can be obtained. One of them is the vector two-wave equation dubbed here the ?second-order cubic Benjamin?Ono (socBO) equation?, and another one is its one-wave version. The last two equations describe weakly nonlinear transverse phonon modes in a chain with the quadratic dispersion law. Another interesting particular case takes place for the specific intermediate value of the coupling constant, when the nonlinear term in the equation derived is balanced by the sixth-order dispersion.
Stationary solutions to the derived equation are studied. Conditions of existence of physically reasonable periodic and solitary type solutions are found. It is shown that among solitary solutions there are plane vector solitons and helical solitons. Interactions of plane solitary waves with different polarisations as well as helical solitons are studied by means of numerical simulation. It is shown that two plane solitons with the same polarisation interact elastically, whereas the interaction of solitons lying initially in the different planes is inelastic. Interesting features of interaction of helical solitons are presented.