The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the H1-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.
Citation: Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation[J]. Electronic Research Archive, 2020, 28(4): 1545-1562. doi: 10.3934/era.2020081
[1] | Hamed Karami, Pejman Sanaei, Alexandra Smirnova . Balancing mitigation strategies for viral outbreaks. Mathematical Biosciences and Engineering, 2024, 21(12): 7650-7687. doi: 10.3934/mbe.2024337 |
[2] | Pannathon Kreabkhontho, Watchara Teparos, Thitiya Theparod . Potential for eliminating COVID-19 in Thailand through third-dose vaccination: A modeling approach. Mathematical Biosciences and Engineering, 2024, 21(8): 6807-6828. doi: 10.3934/mbe.2024298 |
[3] | Holly Gaff, Elsa Schaefer . Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering, 2009, 6(3): 469-492. doi: 10.3934/mbe.2009.6.469 |
[4] | Stefanie Fuderer, Christina Kuttler, Michael Hoelscher, Ludwig Christian Hinske, Noemi Castelletti . Data suggested hospitalization as critical indicator of the severity of the COVID-19 pandemic, even at its early stages. Mathematical Biosciences and Engineering, 2023, 20(6): 10304-10338. doi: 10.3934/mbe.2023452 |
[5] | Vinicius Piccirillo . COVID-19 pandemic control using restrictions and vaccination. Mathematical Biosciences and Engineering, 2022, 19(2): 1355-1372. doi: 10.3934/mbe.2022062 |
[6] | ZongWang, Qimin Zhang, Xining Li . Markovian switching for near-optimal control of a stochastic SIV epidemic model. Mathematical Biosciences and Engineering, 2019, 16(3): 1348-1375. doi: 10.3934/mbe.2019066 |
[7] | Zi Sang, Zhipeng Qiu, Xiefei Yan, Yun Zou . Assessing the effect of non-pharmaceutical interventions on containing an emerging disease. Mathematical Biosciences and Engineering, 2012, 9(1): 147-164. doi: 10.3934/mbe.2012.9.147 |
[8] | Majid Jaberi-Douraki, Seyed M. Moghadas . Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences and Engineering, 2014, 11(5): 1045-1063. doi: 10.3934/mbe.2014.11.1045 |
[9] | Lili Liu, Xi Wang, Yazhi Li . Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity. Mathematical Biosciences and Engineering, 2023, 20(12): 20914-20938. doi: 10.3934/mbe.2023925 |
[10] | Eunha Shim . Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1615-1634. doi: 10.3934/mbe.2013.10.1615 |
The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the H1-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.
[1] |
An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. (1993) 71: 1661-1664. ![]() |
[2] |
A new integrable shallow water equation. Adv. Appl. Mech. (1994) 31: 1-33. ![]() |
[3] |
Analysis on the blow-up of solutions to a class of interable peakon equations. J. Funct. Anal. (2016) 270: 2343-2374. ![]() |
[4] |
Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. (1998) 181: 229-243. ![]() |
[5] | Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (1998) 26: 303-328. |
[6] |
Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. (2011) 173: 559-568. ![]() |
[7] |
Global weak solutions for a shallow water equation. Commun. Math. Phys. (2000) 211: 45-61. ![]() |
[8] |
Stability of peakons. Comm. Pure Appl. Math. (2000) 53: 603-610. ![]() |
[9] |
A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser, Boston, MA, 1997, 93–101. doi: 10.1007/978-1-4612-2434-1_5
![]() |
[10] |
Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D (1981/82) 4: 47-66. ![]() |
[11] |
An extension of integrable peakon equations with cubic nonlinearity. Nonlinearity (2009) 22: 1847-1856. ![]() |
[12] | C. He, X. C. Liu and C. Qu, Orbital stability of peakons and the trains of peakons for an integrable two-component Novikov system, work in progress. |
[13] |
A. Himonas and D. Mantzavinos, The initial value problem for a Novikov system, J. Math. Phys. 57 (2016), 21pp. doi: 10.1063/1.4959774
![]() |
[14] |
A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 10pp. doi: 10.1088/1751-8113/41/37/372002
![]() |
[15] |
Blow-up phenomenon for the integrable Novikov equation. J. Math. Anal. Appl. (2012) 385: 551-558. ![]() |
[16] |
J. Kang, X. Liu, P. J. Olver and C. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017), 26pp. doi: 10.3842/SIGMA.2017.035
![]() |
[17] |
Global weak solutions to the Novikov equation. J. Funct. Anal. (2013) 265: 520-544. ![]() |
[18] |
Two-component generalizations of the Novikov equation. J. Nonlinear Math. Phys. (2019) 26: 390-403. ![]() |
[19] |
On bi-Hamiltonian structure of two-component Novikov equation. Phys. Lett. A (2013) 377: 257-261. ![]() |
[20] |
Stability of peakons for the Novikov equation. J. Math. Pures Appl. (2014) 101: 172-187. ![]() |
[21] |
H. Lundmark and J. Szmigielski, An inverse spectral problem related to the Geng-Xue two-component peakon equation, Mem. Amer. Math. Soc., 244 (2016), 87pp. doi: 10.1090/memo/1155
![]() |
[22] |
J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1
![]() |
[23] |
Perturbative symmetry approach. J. Phys. A (2002) 35: 4775-4790. ![]() |
[24] |
V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 14pp. doi: 10.1088/1751-8113/42/34/342002
![]() |
[25] |
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E (1996) 53: 1900-1906. ![]() |
[26] |
The periodic Cauchy problem for Novikov's equation. Int. Math. Res. Not. (2011) 2011: 4633-4648. ![]() |
[27] |
X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A., 44 (2011), 17pp. doi: 10.1088/1751-8113/44/5/055202
![]() |
[28] |
On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. (2000) 53: 1411-1433. ![]() |
1. | Inkyung Ahn, Seongman Heo, Seunghyun Ji, Kyung Hyun Kim, Taehwan Kim, Eun Joo Lee, Jooyoung Park, Keehoon Sung, Investigation of nonlinear epidemiological models for analyzing and controlling the MERS outbreak in Korea, 2018, 437, 00225193, 17, 10.1016/j.jtbi.2017.10.004 | |
2. | Carl-Etienne Juneau, Tomas Pueyo, Matt Bell, Genevieve Gee, Pablo Collazzo, Louise Potvin, Lessons from past pandemics: a systematic review of evidence-based, cost-effective interventions to suppress COVID-19, 2022, 11, 2046-4053, 10.1186/s13643-022-01958-9 |