Comptes Rendus
no. 12
Mathematical Problems in Mechanics
Equivariant cosymmetry and front solutions of the Dubreil–Jacotin–Long equation. Part 2: Exact solutions
[Cosymétrie équivariante et solutions « fronts » de l'équation de Dubreil–Jacotin–Long. 2ème partie : Solutions exactes]

Présenté par : Gérard Iooss

Nikolai Makarenko1
1 Lavrentyev Institute of Hydrodynamics, Lavrentyev av., 15, 630090 Novosibirsk, Russia
Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 815-818.

Cette Note traite de l'existence de solutions exactes de l'équation de Dubreil–Jacotin–Long (DJL), qui décrivent les fronts internes de grande amplitude dans un fluide continûment stratifié. La démonstration utilise une variante du théorème des fonctions implicites en présence d'une cosymétrie, basée sur le groupe d'invariance de la fonctionnelle variationnelle de l'opérateur de DJL. Une bifurcation supercritique a lieu au bord du spectre continu du problème linéarisé au voisinage de l'écoulement primaire.

The paper concerns existence of exact solutions of the Dubreil–Jacotin–Long equation describing large amplitude internal fronts in a continuously stratified fluid. The proof uses cosymmetric variant of the implicit function theorem based on the group invariance of the variational functional for DJL operator. Supercritical branching occurs near approximate front solutions at the boundary of continuous spectrum of the problem linearized with respect to the basic uniform flow.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.09.036
Nikolai Makarenko 1

1 Lavrentyev Institute of Hydrodynamics, Lavrentyev av., 15, 630090 Novosibirsk, Russia 
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Nikolai Makarenko. Equivariant cosymmetry and front solutions of the Dubreil–Jacotin–Long equation. Part 2: Exact solutions. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 815-818. doi : 10.1016/j.crma.2003.09.036. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.036/

[1] J.T. Beale Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math., Volume XLIV (1991), pp. 211-257

[2] N.I. Makarenko On the bifurcation of solution to invariant variational equations, Dokl. Math., Volume 53 (1996) no. 3, pp. 369-371

[3] V.I. Yudovich Cosymmetry, degeneration of solutions of operator equations, and origin of a filtration convection, Math. Notes, Volume 49 (1991) no. 5, pp. 540-545

[4] V.I. Yudovich Implicit function theorem for cosymmetric equations, Math. Notes, Volume 60 (1996) no. 2, pp. 235-238

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