This paper is devoted to the classification of integrable Davey-Stewartson type equations. A list of potentially deformable dispersionless systems is obtained through the requirement that such systems must be generated by a polynomial dispersionless Lax pair. A perturbative approach based on the method of hydrodynamic reductions is employed to recover the integrable systems along with their Lax...

We use Lax equations to define a scattering problem on an infinite elbow shaped line of the (x, t) plane. The evolution of scattering coefficients when the elbow is translated in the plane shows how convenient scannings may reconstruct the solution V (x, t) of the nonlinear equation associated to the Lax pair. It also helps us understanding why this can work only if strong consistency condition...

We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving va...

A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as well as their relativistic generalisations. The ansatz leads to a system of functional equations. Several new functional equations are described and the general...

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of adL. For generic L a generalized inverse (and indeed the Moore-Penrose inverse) is explicitly constructed. The R-matrices are in general momentum dependent and dynamical. The constructi...

In this paper we revisit the Adler-Bobenko-Suris H2 equation. The equation is linearly related to $S^{(0,0)}$ and $S^{(1,0)}$ variables in Cauchy matrix scheme. We elaborate coupled quad-system of terms their 3-dimensional consistency, Lax pair, bilinear form continuum limits. It shown that itself satisfies a 9-point lattice limit eigenfunction pair Korteweg-de Vries