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November  2011, 10(6): 1779-1790. doi: 10.3934/cpaa.2011.10.1779

## Even solutions of the Toda system with prescribed asymptotic behavior

 1 Departamento de Ingeniería Matemática and CMM, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Received  May 2010 Revised  March 2011 Published  May 2011

Every solution of the Toda system, describing the behavior of a finite number of mass points on the line, each one interacting with its neighbors, is asymptotically linear at infinity. We show the existence and uniqueness of even solution with suitable prescribed asymptotic behavior, by analyzing a system of algebraic equations derived from the relation between the slopes and the intercepts of the asymptotic lines.
Citation: Yong Liu. Even solutions of the Toda system with prescribed asymptotic behavior. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1779-1790. doi: 10.3934/cpaa.2011.10.1779
##### References:
 [1] M. Del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $R^2$, J. Funct. Anal., 258 (2010), 458-503. doi: 10.1016/j.jfa.2009.04.020. [2] M. Del Pino, M. Kowalczyk, F. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516. arXiv:0710.0640. doi: 10.1016/j.aim.2010.01.003. [3] M. Del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187. doi: 10.1007/s00205-008-0143-3. [4] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. Math., 34 (1979), 195-338. doi: 10.1016/0001-8708(79)90057-4. [5] J. Moser, Finitely many mass points on the line under the influence of an exponential potential-an integrable system, in "Dynamical Systems, Theory and Applications," Lecture Notes in Phys., Vol. 38, Springer, (1975), 467-497. doi: 10.1007/3-540-07171-7_12.

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##### References:
 [1] M. Del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $R^2$, J. Funct. Anal., 258 (2010), 458-503. doi: 10.1016/j.jfa.2009.04.020. [2] M. Del Pino, M. Kowalczyk, F. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516. arXiv:0710.0640. doi: 10.1016/j.aim.2010.01.003. [3] M. Del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187. doi: 10.1007/s00205-008-0143-3. [4] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. Math., 34 (1979), 195-338. doi: 10.1016/0001-8708(79)90057-4. [5] J. Moser, Finitely many mass points on the line under the influence of an exponential potential-an integrable system, in "Dynamical Systems, Theory and Applications," Lecture Notes in Phys., Vol. 38, Springer, (1975), 467-497. doi: 10.1007/3-540-07171-7_12.
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