On the Cauchy problem and peakons of a two-component Novikov system
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https://doi.org/10.1007/s11425-019-9557-6
On the Cauchy problem and peakons of a two-component Novikov system Changzheng Qu1,∗ & Ying Fu2 1School
of Mathematics and Statistics, Ningbo University, Ningbo 315211, China; of Mathematics, Northwest University, Xi’an 710127, China
2School
Email: [email protected], [email protected] Received May 16, 2019; accepted June 17, 2019
Abstract
We study a two-component Novikov system, which is integrable and can be viewed as a two-
component generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the s Cauchy problem in Besov spaces Bp,r with 1 6 p, r 6 +∞, s > max{1 + 1/p, 3/2} and Sobolev spaces H s (R)
with s > 3/2, and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system. Keywords
two-component Novikov system, Hamiltonian structure, Camassa-Holm type equation, well-
posedness, peaked soliton MSC(2010)
35B30, 35G25
Citation: Qu C Z, Fu Y. On the Cauchy problem and peakons of a two-component Novikov system. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-9557-6
1
Introduction
In this paper, we are mainly concerned with the Cauchy problem of the two-component Novikov system [41] m ⃗ t + 3(⃗u · ⃗ux )m ⃗ + |⃗u|2 m ⃗ x − (⃗u ∧ ⃗ux )m ⃗ = 0,
t > 0,
x ∈ R,
(1.1)
where ⃗u = (u, v)t , m ⃗ = (m, n) = ⃗u − ⃗uxx , ⃗u ∧ ⃗ux = ⃗u ⊗ ⃗ux − ⃗ux ⊗ ⃗u, subject to the initial condition ⃗u(0, x) = ⃗u0 (x),
x ∈ R,
and u(t, x) and v(t, x) are time-dependent functions on R. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019 ⃝
math.scichina.com
link.springer.com
2
Qu C Z et al.
Sci China Math
In the cases of v = 0 or u = v in (1.1), it is equivalent to the Novikov equation mt + 3uux m + u2 mx = 0,
m = u − uxx ,
(1.2)
up to a scale invariance. System (1.1) can be viewed as a two-component generalization of the Novikov equation (1.2). Indeed, the system (1.1) can be extended to systems involving any multi-component ⃗u = (u1 , u2 , . . . , uk ) (k > 3). The Novikov equation (1.2) was introduced by Novikov [49] in the classification of symmetries for equations of the form mt = F (u, ux , m, mx ),
m = u − uxx
(1.3)
while it possesses higher-order generalized symmetries. Equation (1.2) was proved to be integrable since it admits the Lax-pair and bi-Hamilt
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