Bifurcations of phase portraits of a Singular Nonlinear Equation of the Second Class

•We study the dynamic of the extended generalized Frenkel–Kontorova model.•In the continuum limit a Singular Nonlinear Equation of the Second Class is obtained.•Anharmonic interactions in lattice lead to a new number of interesting phenomena.•The dynamical behavior of traveling wave solutions is studied by the theory of bifurcations.•The deformability of the substrate potential plays only a minor role in the dynamic.

The soliton dynamics is studied using the Frenkel Kontorova (FK) model with non-convex interparticle interactions immersed in a parameterized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Non-convex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. In the continuum limit for such a model, the particles are governed by a Singular Nonlinear Equation of the Second Class. The dynamical behavior of traveling wave solutions is studied by using the theory of bifurcations of dynamical systems. Under different parametric situations, we give various sufficient conditions leading to the existence of propagating wave solutions or dislocation threshold, highlighting namely that the deformability of the substrate potential plays only a minor role.