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Two new applications of Λ-representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane Λ. 2. Employing Λ-representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method. The geometric interpretation of PDEs linking their investigation to analysis of certain objects on the hyperbolic plane (Lobachevsky plane Λ) was introduced in [3–5]. According to this interpretation to each differential equation f [u(x, t)] = 0 from a certain class under investigation (Λ-class or Lobachevsky class) one can associate a pseudospherical metric (of Gaussian curvature K ≡ −1) defined on any regular solution u(x, t): {f [u(x, t)] = 0, u ∈ C(R)} ↔ {ds[u(x, t)], K ≡ −1}. The set of coefficients {E[u(x, t)], F [u(x, t)], G[u(x, t)]} of the pseudospherical metric ds[u(x, t)] is called the Λ-representation for the given equation and is denoted by Λ[f [u(x, t)] = 0] = {E[u(x, t)], F [u(x, t)], G[u(x, t)]}. The existence of the Λ-representation for PDEs leads to the possibility of using the Lobachevsky geometry methods for their investigation. In this paper two different applications of Λ-representations for the study of PDEs are presented: 2000 Mathematics Subject Classification: Primary 35Q53.

Some constructive applications of $Λ^{2}$-representations to integration of PDEs