The Jacobi-Toda system and foliated interfaces

Pages: 975 - 1006, Volume 28, Issue 3, November 2010

doi:10.3934/dcds.2010.28.975       Abstract        Full Text (396.8K)       Related Articles

Manuel del Pino - Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (email)
Michal Kowalczyk - Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (email)
Juncheng Wei - Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China (email)

Abstract: Let (M ,ğ) be an $N$-dimensional smooth (compact or noncompact) Riemannian manifold. We introduce the elliptic Jacobi-Toda system on (M ,ğ). We review various recent results on its role in the construction of solutions with multiple interfaces of the Allen-Cahn equation on compact manifolds and entire space, as well as multiple-front traveling waves for its parabolic counterpart.

Keywords:  Jacobi operator, Toda system, Concentration phenomena, multiple transition layers, positive Gauss curvature.
Mathematics Subject Classification:  35J60, 58J05, 58J37, 53C21, 53C22.

Received: March 2010;      Revised: April 2010;      Published: April 2010.