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Algebras of differentiable functions in the plane

Karel De Leeuw, H. Mirkil (1963)

Annales de l'institut Fourier

Soit A un ensemble quelconque d’opérateurs différentiels en deux variables à coefficients complexes constants. Soit C 0 l’espace des fonctions continues complexes tendant vers zéro à l’infini dans le plan euclidien. Soit C 0 ( A ) l’espace { f : f C 0 , A f C 0 , tout a A } . Classifier ces espaces équivaut à trouver des conditions nécessaires et suffisantes sur des opérateurs différentiels P 1 , ... , P n pour que P 1 φ K ( P 2 φ + + P n φ ) . Il paraît que ce problème général est bien difficile. Nous présentons ici la solution complète dans le cas spécial des C 0 ( A ) stables...

Algebro-geometric approach to the Ernst equation I. Mathematical Preliminaries

O. Richter, C. Klein (1997)

Banach Center Publications

1. Introduction. It is well known that methods of algebraic geometry and, in particular, Riemann surface techniques are well suited for the solution of nonlinear integrable equations. For instance, for nonlinear evolution equations, so called 'finite gap' solutions have been found by the help of these methods. In 1989 Korotkin [9] succeeded in applying these techniques to the Ernst equation, which is equivalent to Einstein's vacuum equation for axisymmetric stationary fields. But, the Ernst equation...

Algebro-geometric solutions of the Camassa-Holm hierarchy.

Fritz Gesztesy, Helge Holden (2003)

Revista Matemática Iberoamericana

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial...

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