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On the Cauchy problem and peakons of a two-component Novikov system

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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 Cauchy problem in Besov spaces Bsp,r with 1 ⩽ p, r ⩽ + ∞, s > max{1 + 1/p, 3/2} and Sobolev spaces Hs(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.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11631007, 11471174 and 11471259).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

    Changzheng Qu

  2. School of Mathematics, Northwest University, Xi’an, 710127, China

    Ying Fu

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  1. Changzheng Qu
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  2. Ying Fu
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Correspondence to Changzheng Qu.

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Qu, C., Fu, Y. On the Cauchy problem and peakons of a two-component Novikov system. Sci. China Math. 63, 1965–1996 (2020). https://doi.org/10.1007/s11425-019-9557-6

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  • DOI: https://doi.org/10.1007/s11425-019-9557-6

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