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Smoothness property for bifurcation diagrams.

Robert Roussarie (1997)

Publicacions Matemàtiques

Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to subanalytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line...

Solitons and large time behavior of solutions of a multidimensional integrable equation

Anna Kazeykina (2013)

Journées Équations aux dérivées partielles

Novikov-Veselov equation is a (2+1)-dimensional analog of the classic Korteweg-de Vries equation integrable via the inverse scattering translform for the 2-dimensional stationary Schrödinger equation. In this talk we present some recent results on existence and absence of algebraically localized solitons for the Novikov-Veselov equation as well as some results on the large time behavior of the “inverse scattering solutions” for this equation.

Solitons of the sine-Gordon equation coming in clusters.

Cornelia Schiebold (2002)

Revista Matemática Complutense

In the present paper, we construct a particular class of solutions of the sine-Gordon equation, which is the exact analogue of the so-called negatons, a solution class of the Korteweg-de Vries equation discussed by Matveev [17] and Rasinariu et al. [21]. Their characteristic properties are: Each solution consists of a finite number of clusters. Roughly speaking, in such a cluster solitons are grouped around a center, and the distance between two of them grows logarithmically. The clusters themselves...

Solution of the 1 : −2 resonant center problem in the quadratic case

Alexandra Fronville, Anton Sadovski, Henryk Żołądek (1998)

Fundamenta Mathematicae

The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field ( x + A 1 x 2 + B 1 x y + C y 2 ) x + ( - 2 y + D x 2 + A 2 x y + B 2 y 2 ) y . There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...

Solution to the gradient problem of C.E. Weil.

Zoltán Buczolich (2005)

Revista Matemática Iberoamericana

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G ⊂ R2 we construct a differentiable function f: G → R for which there exists an open set Ω1 ⊂ R2 such that ∇f(p) ∈ Ω1 for a p ∈ G but ∇f(q) ∉ Ω1 for almost every q ∈ G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

Solutions canards en des points tournants dégénérés

Thomas Forget (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Nous étudions un opérateur défini à partir d’une classe générale d’équations différentielles singulièrement perturbées dans le champ réel ; son caractère contractant permet de conclure à l’existence de solutions canard dans le cas où l’on a un point tournant dégénéré.

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