• Previous Article
    Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies
  • CPAA Home
  • This Issue
  • Next Article
    Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments
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  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 & Applied Analysis, doi: 10.3934/cpaa.2021075
Figure 1.  A simple case of the two particles interactions
Figure 2.  A simple case of the three particles interactions
[1]

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

[2]

Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759

[3]

Manuel del Pino, Michal Kowalczyk, Juncheng Wei. The Jacobi-Toda system and foliated interfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 975-1006. doi: 10.3934/dcds.2010.28.975

[4]

Chunyan Ji, Daqing Jiang. Persistence and non-persistence of a mutualism system with stochastic perturbation. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 867-889. doi: 10.3934/dcds.2012.32.867

[5]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[6]

Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739

[7]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[8]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[9]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

[10]

Johan Matheus Tuwankotta, Eric Harjanto. Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation. Journal of Computational Dynamics, 2019, 6 (2) : 469-483. doi: 10.3934/jcd.2019024

[11]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[12]

Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial & Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73

[13]

Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3011-3042. doi: 10.3934/dcds.2013.33.3011

[14]

Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235

[15]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

[16]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120

[17]

Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233

[18]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[19]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015

[20]

Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653

2019 Impact Factor: 1.105

Article outline

Figures and Tables

[Back to Top]