Progresses on the dynamical system theory of 2D Euler equation are reported. An open problem is presented. The isospectral theory on Lax pairs of Euler equations is initiated.

The dependence of the sixth equation of Painlevé on its four parameters (2α,−2β, 2γ, 1− 2δ) = (θ ∞ , θ 0 , θ 2 1 , θ 2 x) is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar “Lax” pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on θ∞ (and a holomorphic dep...

Some two-component generalizations of the Novikov equation, except Geng-Xue are presented, as well their Lax pairs and bi-Hamiltonian structures. Furthermore, we study Hamiltonians equation discuss homogeneous local properties them.

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...

The theory of commuting linear differential expressions has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). Gesztesy and the present author have established a relationship of this circle of ideas with the property that all solutions of the differential equations Ly = zy, z ∈ C, are meromorphic. In this paper this relationship is explored ...

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...

A reduced-order method based on Approximated Lax Pairs (ALP) is applied to the integration of electrophysiology models. These are often high-dimensional parametric equation systems, challenging from a model reduction standpoint. The method is tested on two and three dimensional test-cases, of increasing complexity. The solutions are compared to the ones obtained by a finite element. The reduced...

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on finite interval with 3×3 Lax pairs. The solut...