Variational principle for the dislocation equation in lattice theory

In the lattice theory of dislocation, the dislocation equation including the discreteness effect is a nonlinear integro-differential equation. It can hardly be solved. In this paper, the variational principle for the dislocation equation is presented. By using the Ritz approximation method, the variational solution is obtained, which is the same as the Peierls-Nabarro solution when the discreteness effect terms were ignored and the accurate value of the variational parameter is obtained when the discreteness effect terms were taken into account. As a natural generalization, a modified form of the sinusoidal law of force is proposed. It is found that the fine structure of the dislocation core can be predicted theoretically as long as the parameter in the modified law of force is selected properly.