Iterative method and stability

With increasing velocity and energy the solitary waves become narrower and their width can be in the order of the lattice constant. In this case the QCA does not hold. And it would not help to take more terms of the Taylor expansion of ( ) in Eq. 16 into account. The resulting higher order differential equations cannot be integrated like Eq. 17. Moreover, even the infinite Taylor series cannot fully represent A q) because of the finite radius of convergence. 

Fortunately, an iterative method [18] has been developed for the monoatomic chain. Here the accuracy of taking into account the discreteness effects is increased systematically. In the case of the Toda lattice the iteration converges to the exact one-soliton solution. 

This method can also he applied to the pulse-type solitary waves of our diatomic model. Our basic Eq. 15 can he written in the form 

Similar to the monoatomic case [18], an iteration for Eq. 37 would converge only to the trivial solution q) z) = 0. This can be prevented by keeping (0) constant during the iteration [18]. This condition leads to the elimination of c from Eq. 37 and thus to a new iteration procedure 

This implies c, c , i.e. the same condition as for the QCA. Each iteration consists of four steps  

---