  Using the continuous limit approximation in the dynamical system we study a
nonlinear partial differential equation which corresponds to the generalization
of both the Fermi-Pasta-Ulam and the Frenkel-Kontorova models. This generalized
model can be considered as a model for the description nonlinear dislocation
waves in the crystal lattice. We obtain the nonlinear partial differential
equation for the description of dislocations. Taking into account the wave
moving in one direction we obtain another nonlinear evolution equation. Using
the Painlev\'e test we analyze the integrability of this equation. We find that
there exist an integrable case of the partial differential equation for
nonlinear dislocations. The Lax pair for the solution of the Cauchy problem is
found. The solution of the Cauchy problem for nonlinear evolution equation is
discussed. One and the two-soliton solutions for the nonlinear evolution
equation are presented. The influence of parameters on the propagation of one
and the two-soliton solutions is analyzed and demonstrated.

Exactly Solvable and Integrable Systems