On a method for constructing the Lax pairs for nonlinear integrable equations

The Laplace cascade integration method is known to be a powerful tool for solving linear hyperbolic type partial differential equations with two independent variables. About two decades ago a deep connection was established between Laplace invariants and the Liouville type nonlinear integrable equations. The present article is devoted to investigation of the Laplace sequence for the sine-Gordon type integrable models. It is observed that in this case the Laplace sequence admits a finite reduction. We demonstrate that the finite reductions of the Laplace sequence allow one to construct the Lax pairs for the hyperbolic soliton equations. The reductions generate invariant manifolds for the linearized equation. Due to this circumstance we reformulate the algorithm of searching the L-A pairs and adopt it to the case of the evolution type integrable equations as well. Examples of continuous and discrete equations are discussed. Efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs has not been found earlier.