July  2022, 42(7): 3119-3142. doi: 10.3934/dcds.2022011

Singular solutions of Toda system in high dimensions

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

Received  December 2021 Published  July 2022 Early access  February 2022

We construct some singular solutions to the four component Toda system. These solutions are almost split into two groups, each one modelled on an explicit solution to the two component Toda system (i.e. Liouivlle equation). These solutions are shown to be stable in high dimensions. This gives a sharp example on the partial regularity of stable solutions to Toda system.

Citation: Linlin Dou. Singular solutions of Toda system in high dimensions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3119-3142. doi: 10.3934/dcds.2022011
References:
[1]

O. AgudeloM. del Pino and J. Wei, Solutions with multiple catenoidal ends to the Allen-Cahn equation in $\mathbb{R}^3$, J. Math. Pures Appl., 103 (2015), 142-218.  doi: 10.1016/j.matpur.2014.03.010.

[2]

O. AgudeloM. del Pino and J. Wei, Higher-dimensional catenoid, Liouville equation, and Allen-Cahn equation, Int. Math. Res. Not., 2016 (2016), 7051-7102.  doi: 10.1093/imrn/rnv350.

[3]

S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.  doi: 10.1007/s005260050080.

[4]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre D'une VariétéRiemannienne, Springer-Verlag, Berlin-New York, 1971.

[5]

L. CaffarelliR. Hardt and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18.  doi: 10.1007/BF01168999.

[6]

J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.

[7]

M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.

[8]

M. del PinoM. 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.

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.

[10]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.

[11]

H. Flaschka, The Toda lattice. Ⅱ. Existence of integrals, Phys. Rev. B, 9 (1974), 1924-1925.  doi: 10.1103/PhysRevB.9.1924.

[12]

J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., 2002 (2002), 277-290.  doi: 10.1155/S1073792802105022.

[13]

M. KowalczykY. Liu and J. Wei, Singly periodic solutions of the Allen-Cahn equation and the Toda lattice, Comm. Partial Differential Equations, 40 (2015), 329-356.  doi: 10.1080/03605302.2014.947379.

[14]

C. S. LinJ. Wei and D. Ye, Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207.  doi: 10.1007/s00222-012-0378-3.

[15]

A. Malchiodi, Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.  doi: 10.1007/s11401-017-1082-9.

[16]

Y. Rébai, Solutions of semilinear elliptic equations with one isolated singularity, Differential Integral Equations, 12 (1999), 563-581. 

[17]

G. Teschl, Almost everything you always wanted to know about the Toda equation, Jahresber. Deutsch. Math.-Verein., 103 (2001), 149-162. 

[18]

M. Toda, Studies of a non-linear lattice, Phys. Rep., 18 (1975), 1-123.  doi: 10.1016/0370-1573(75)90018-6.

[19]

K. Wang, Stable and finite morse index solutions of Toda system, J. Differential Equations, 268 (2019), 60-79.  doi: 10.1016/j.jde.2019.08.006.

[20]

K. Wang and J. Wei, Finite morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044-1119.  doi: 10.1002/cpa.21812.

[21]

K. Wang and J. Wei, Second order estimate on transition layers, Adv. Math., 358 (2019), 106856, 85 pp. doi: 10.1016/j.aim.2019.106856.

show all references

References:
[1]

O. AgudeloM. del Pino and J. Wei, Solutions with multiple catenoidal ends to the Allen-Cahn equation in $\mathbb{R}^3$, J. Math. Pures Appl., 103 (2015), 142-218.  doi: 10.1016/j.matpur.2014.03.010.

[2]

O. AgudeloM. del Pino and J. Wei, Higher-dimensional catenoid, Liouville equation, and Allen-Cahn equation, Int. Math. Res. Not., 2016 (2016), 7051-7102.  doi: 10.1093/imrn/rnv350.

[3]

S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.  doi: 10.1007/s005260050080.

[4]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre D'une VariétéRiemannienne, Springer-Verlag, Berlin-New York, 1971.

[5]

L. CaffarelliR. Hardt and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18.  doi: 10.1007/BF01168999.

[6]

J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.

[7]

M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.

[8]

M. del PinoM. 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.

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.

[10]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.

[11]

H. Flaschka, The Toda lattice. Ⅱ. Existence of integrals, Phys. Rev. B, 9 (1974), 1924-1925.  doi: 10.1103/PhysRevB.9.1924.

[12]

J. Jost and G. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., 2002 (2002), 277-290.  doi: 10.1155/S1073792802105022.

[13]

M. KowalczykY. Liu and J. Wei, Singly periodic solutions of the Allen-Cahn equation and the Toda lattice, Comm. Partial Differential Equations, 40 (2015), 329-356.  doi: 10.1080/03605302.2014.947379.

[14]

C. S. LinJ. Wei and D. Ye, Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207.  doi: 10.1007/s00222-012-0378-3.

[15]

A. Malchiodi, Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.  doi: 10.1007/s11401-017-1082-9.

[16]

Y. Rébai, Solutions of semilinear elliptic equations with one isolated singularity, Differential Integral Equations, 12 (1999), 563-581. 

[17]

G. Teschl, Almost everything you always wanted to know about the Toda equation, Jahresber. Deutsch. Math.-Verein., 103 (2001), 149-162. 

[18]

M. Toda, Studies of a non-linear lattice, Phys. Rep., 18 (1975), 1-123.  doi: 10.1016/0370-1573(75)90018-6.

[19]

K. Wang, Stable and finite morse index solutions of Toda system, J. Differential Equations, 268 (2019), 60-79.  doi: 10.1016/j.jde.2019.08.006.

[20]

K. Wang and J. Wei, Finite morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044-1119.  doi: 10.1002/cpa.21812.

[21]

K. Wang and J. Wei, Second order estimate on transition layers, Adv. Math., 358 (2019), 106856, 85 pp. doi: 10.1016/j.aim.2019.106856.

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