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 $ H^1 $-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
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 $ H^1 $-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. ![]() |