Abstract : The soliton dynamics is studied using the Frenkel Kontorova (FK) model with non- convex interparticle interactions immersed in a parameter ized on-site substrate po- tential. The case of a deformable substrate potential allow s theoretical adaptation of the model to various physical situations. Non-convex inter actions in lattice systems lead to a number of interesting phenomena that cannot be prod uced with linear coupling alone. In the continuum limit for such a model, the p articles are governed by a Singular Nonlinear Equation of the Second Class. The dyn amical behavior of traveling wave solutions is studied by using the theory of bi furcations of dynamical systems. Under different parametric situations, we give vari ous sufficient conditions leading to the existence of propagating wave solutions or di slocation threshold, high- lighting namely that the deformability of the substrate pot ential plays only a minor role.