# American Institute of Mathematical Sciences

September  2008, 22(3): 465-508. doi: 10.3934/dcds.2008.22.465

## Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain

 1 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 2 Department of Mathematics, Shenzhen University, Shenzhen, China

Received  July 2007 Revised  March 2008 Published  August 2008

We consider the equation $\varepsilon^2\Delta$ ũ-ũ+ũ$^p =0$ in a bounded, smooth domain $\Omega$ in $\R^2$ under homogeneous Neumann boundary conditions. Let $\Gamma$ be a segment contained in $\Omega$, connecting orthogonally the boundary, non-degenerate and non-minimal with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.
Citation: Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465
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