On the Cauchy problem of a new integrable two-component Novikov equation
- PDF / 368,652 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 13 Downloads / 199 Views
On the Cauchy problem of a new integrable two-component Novikov equation Yongsheng Mi1 · Daiwen Huang2 Received: 6 February 2020 / Accepted: 20 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong solutions maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. Keywords Besov spaces · Camassa–Holm type equation · Local well-posedness · Persistence properties Mathematics Subject Classification 35Q53 · 35A01 · 35B44 · 35B65
1 Introduction In this paper, we consider the following Cauchy problem of the new integrable twocomponent Novikov equation ⎧ m t + (u 2 + v 2 )m x + 3(uu x + vvx )m − n(uvx − u x v) = 0, t > 0, x ∈ R, ⎪ ⎪ ⎨ n t + (u 2 + v 2 )n x + 3(uu x + vvx )n − m(u x v − uvx ) = 0, t > 0, x ∈ R, (1.1) t > 0, x ∈ R, m = u − u x x , n = v − vx x , ⎪ ⎪ ⎩ u(0, x) = u 0 (x), v(0, t) = v0 (x), x ∈ R.
Communicated by Adrian Constantin.
B
Yongsheng Mi [email protected]
1
College of Mathematics and Statistics„ Yangtze Normal University, Chongqing 408100, People’s Republic of China
2
Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
123
Y. Mi, D. Huang
The new integrable two-component Camassa–Holm equation (1.1) was found by Li [53]. It was shown in [53] that the system (1.1) appears in the bi-Hamiltonian form δW δH u δm = (m, n) , = (m, n) δδm H δW v t δn δn
(1.2)
with Hamiltonian pair 3m∂ + 2m x (∂ 3 − 4∂)−1 (3m∂ + m x , 3n∂ + n x ) 3n∂ + 2n x n + ∂ −1 (−n, m) −m 1 3m∂ −1 m + n∂ −1 n m∂ −1 n + n∂ −1 m (∂ 2 − 1) (1.3) (m, n) = − (∂ 2 − 1) m∂ −1 n + n∂ −1 m m∂ −1 m + 3n∂ −1 n 2 (m, n) =
1 2
The associated Hamiltonian functional is H = mv + nud x, and W is nonlocal and looks very complicated, so we omit it. Recently, the local well-posedness, peakons, the blow-up, wave-breaking phenomena of the Cauchy problem (1.1) is studied by Qu et al. in [64]. In recent decades, the Camassa–Holm (CH) type equations raised a lot of interest because of their specific properties, one of which is that they possess peakon solutions (peaked soliton solutions with discontinuous derivatives at the peaks). The most celebrated member of them is the Camassa–Holm equation [18] u t − u t x x + 3uu x = 2u x u x x + uu x x x
(1.4)
modelling the unidirectional propagation of shallow water waves over a flat bottom, u(t, x) stands for the fluid velocity at time t in the spatial direction x. It is a wellknown integrable equation describing the velocity dynamics of shallow water waves. This equation spontaneously exhibits emergence of singular s
Data Loading...
