Figure 1.
A simple case of the two particles interactions
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Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations
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 |
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.
Keywords:
Even solutions,
Toda system,
degree argument approach,
perturbation theory,
non-integrable system.
Mathematics Subject Classification:
Primary: 34B30, 34D10, 34E99.
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
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References:
| [1] |
M. A. Agrotis, P. 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 Pino, M. 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 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.
doi: 10.1016/j.aim.2010.01.003. |
| [4] |
M. Del Pino, M. 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. Lee, C. S. Lin, J. 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. Lee, C. S. Lin, W. 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. Chai, C. 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. Agrotis, P. 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 Pino, M. 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 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.
doi: 10.1016/j.aim.2010.01.003. |
| [4] |
M. Del Pino, M. 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. Lee, C. S. Lin, J. 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. Lee, C. S. Lin, W. 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. Chai, C. 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 2.
A simple case of the three particles interactions
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