June  2022, 21(6): 1895-1916. doi: 10.3934/cpaa.2021075

On the even solutions of the Toda system: A degree argument approach

1. 

Department of Mathematics, University of Science and Technology of China, Hefei 230027, China

2. 

Department of Mathematics, Towson University, Towson, MD 21252, USA

3. 

School of Mathematical Sciences and Physics, North China Electric Power University, Beijing 102206, China

* Corresponding author

Dedicated to Professor Goong Chen on the occasion of his seventieth birthday

Received  November 2020 Revised  March 2021 Published  June 2022 Early access  April 2021

Fund Project: Y. Liu is partially supported by "The Fundamental Research Funds for the Central Universities WK3470000014, " and NSFC no. 11971026. X. Yong is partially supported by the Beijing Natural Science Foundation no. Z200001

In this paper, we show the existence of even solutions with prescribed asymptotic behavior at infinity. Instead of using the integrability of the Toda system, the novel idea here is a degree argument approach. Perturbation theory has also been used in our study. Our method can be generalized to explore non-integrable systems with exponential type nonlinearities.

Citation: Yong Liu, Jing Tian, Xuelin Yong. On the even solutions of the Toda system: A degree argument approach. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1895-1916. doi: 10.3934/cpaa.2021075
References:
[1]

M. A. AgrotisP. A. Damianou and C. Sophocleous, The Toda lattice is super-integrable, Phys. A, 365 (2006), 235-243.  doi: 10.1016/j.physa.2006.01.001.

[2]

M. Del PinoM. Kowalczyk and F. Pacard, 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.

[3]

M. Del PinoM. KowalczykF. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.

[4]

M. Del PinoM. Kowalczyk and 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.

[5]

F. Gesztesy, H. Holden, J. Michor and G. Teschl, Soliton equations and their algebro-geometric solutions: Vol. II. 1 + 1-dimensional discrete models, in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2008. doi: 10.1017/CBO9780511543203.

[6]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Springer-Verlag, 1976.

[7]

R. Hirota, The direct method in soliton theory, in Cambridge Tracts in Mathematics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511543043.

[8]

V. I. Inozemtsev, The finite Toda lattices, Commun. Math. Phys., 121 (1989), 629-638. 

[9]

J. Jost and G. Wang, Analytic aspects of the Toda system: I. A Moser-Trudinger inequality, Commun. Pure Appl. Anal., 51 (2001), 1289-1319.  doi: 10.1002/cpa.10004.

[10]

Y. Kodama and B. Shipman, Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint, J. Phys. A, 51 (2018), 353001, 39pp. doi: 10.1088/1751-8121/aacecf.

[11]

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.

[12]

Y. LeeC. S. LinJ. Wei and W. Yang, Degree counting and Shadow system for Toda system of rank two: One bubbling, J. Differ. Equ., 7 (2018), 4343-4401.  doi: 10.1016/j.jde.2017.12.018.

[13]

Y. LeeC. S. LinW. Yang and L. Zhang, Degree counting for Toda system with simple singularity: one point blow up, J. Differ. Equ., 5 (2020), 2163-2209.  doi: 10.1016/j.jde.2019.09.016.

[14]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equations on tori, Ann. Math., (2010), 911–954. doi: 10.4007/annals.2010.172.911.

[15]

C. L. ChaiC. S. Lin and C. L. Wang, Mean field equations, hyperelliptic curves, and modular forms: I, Cambridge J. Math., 12 (2015), 127-274.  doi: 10.4310/CJM.2015.v3.n1.a3.

[16]

Y. Liu, Even solutions of the Toda system with prescribed asymptotic behavior, Commun. Pure Appl. Anal., 10 (2011), 1779-1790.  doi: 10.3934/cpaa.2011.10.1779.

[17]

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., (1975), 467–497.

[18]

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

[19]

M. Toda, Theory of Nonlinear Lattices, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83219-2.

show all references

References:
[1]

M. A. AgrotisP. A. Damianou and C. Sophocleous, The Toda lattice is super-integrable, Phys. A, 365 (2006), 235-243.  doi: 10.1016/j.physa.2006.01.001.

[2]

M. Del PinoM. Kowalczyk and F. Pacard, 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.

[3]

M. Del PinoM. KowalczykF. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.

[4]

M. Del PinoM. Kowalczyk and 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.

[5]

F. Gesztesy, H. Holden, J. Michor and G. Teschl, Soliton equations and their algebro-geometric solutions: Vol. II. 1 + 1-dimensional discrete models, in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2008. doi: 10.1017/CBO9780511543203.

[6]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Springer-Verlag, 1976.

[7]

R. Hirota, The direct method in soliton theory, in Cambridge Tracts in Mathematics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511543043.

[8]

V. I. Inozemtsev, The finite Toda lattices, Commun. Math. Phys., 121 (1989), 629-638. 

[9]

J. Jost and G. Wang, Analytic aspects of the Toda system: I. A Moser-Trudinger inequality, Commun. Pure Appl. Anal., 51 (2001), 1289-1319.  doi: 10.1002/cpa.10004.

[10]

Y. Kodama and B. Shipman, Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint, J. Phys. A, 51 (2018), 353001, 39pp. doi: 10.1088/1751-8121/aacecf.

[11]

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.

[12]

Y. LeeC. S. LinJ. Wei and W. Yang, Degree counting and Shadow system for Toda system of rank two: One bubbling, J. Differ. Equ., 7 (2018), 4343-4401.  doi: 10.1016/j.jde.2017.12.018.

[13]

Y. LeeC. S. LinW. Yang and L. Zhang, Degree counting for Toda system with simple singularity: one point blow up, J. Differ. Equ., 5 (2020), 2163-2209.  doi: 10.1016/j.jde.2019.09.016.

[14]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equations on tori, Ann. Math., (2010), 911–954. doi: 10.4007/annals.2010.172.911.

[15]

C. L. ChaiC. S. Lin and C. L. Wang, Mean field equations, hyperelliptic curves, and modular forms: I, Cambridge J. Math., 12 (2015), 127-274.  doi: 10.4310/CJM.2015.v3.n1.a3.

[16]

Y. Liu, Even solutions of the Toda system with prescribed asymptotic behavior, Commun. Pure Appl. Anal., 10 (2011), 1779-1790.  doi: 10.3934/cpaa.2011.10.1779.

[17]

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., (1975), 467–497.

[18]

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

[19]

M. Toda, Theory of Nonlinear Lattices, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83219-2.

Figure 1.  A simple case of the two particles interactions
Figure 2.  A simple case of the three particles interactions
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