On the Construction of Solutions of the Davey – Stewartson I Equation via Dressing Chain

An original effective method for constructing explicit solutions of integrable Davey – Stewartson type equations is proposed, based on the use of dressing chains. The main difficulty arising when using the symmetry approach in 3D is associated with nonlocal variables entering the equation. To solve the nonlocality problem, it is proposed to replace the infinite dressing chain with its finite-field reductions preserving the integrability property. The application of the method is illustrated by the DS I equation, for which a new class of explicit solutions is constructed that depend on two arbitrary functions. In this example, the dressing chain is replaced by a finite-field reduction of the Toda lattice corresponding to a simple Lie algebra $A_2$.