Even solutions of the Toda system with prescribed asymptotic behavior

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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

Abstract
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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.

Keywords: asymptotic behavior., Toda system, even solution.

Mathematics Subject Classification: Primary: 34B30, 34E99, 35J6.

Citation: Yong Liu. Even solutions of the Toda system with prescribed asymptotic behavior. Communications on Pure & 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. 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. 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. 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. 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, (1975), 467. 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. 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. 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. 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. 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, (1975), 467. doi: 10.1007/3-540-07171-7_12.

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