On donne dans cet article une interprétation géométrique des « douze surfaces de Darboux », qui apparaissent en appliquant façon répétée transformation simple à déformation isométrique infinitésimale d’une surface l’espace euclidien dimension trois. Cette est version différentielle la trialité, concernant les immersions totalement isotropes quadrique projective réelle 6 définie par forme quadra...

One of the important components of the theory of nonlinear S-integrable differential equations [1] is Darboux transformations, the story of which starts with the paper by Moutard [2] and therefore, contrary to the common tendency, we will call them Moutard-Darboux transformations. An example of an equation which admits Moutard-Darboux transformations is the second order linear differential equa...

Abstract We prove that (1) diagonal systems of hydrodynamic type are Darboux integrable if and only the corresponding for commuting flows integrable, (2) Laplace transformation sequences terminate, (3) necessarily semihamiltonian. give geometric interpretation integrability such in terms congruences lines solution orbits with respect to symmetry subalgebras, discuss known new examples.

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations fo...