Research article

Well-posedness of initial value problem of Hirota-Satsuma system in low regularity Sobolev space

  • Received: 21 November 2021 Revised: 04 January 2022 Accepted: 13 January 2022 Published: 25 January 2022
  • MSC : 35E15, 35Q53

  • In this paper, we study the initial value problem of Hirota-Satsuma system:

    $ \begin{equation} \notag \left\{ \begin{array}{ll} u_t-\alpha(u_{xxx}+6uu_x) = 2\beta vv_x, & \ x\in {\mathbb{R}}, \ t\ge 0, \\ v_t+v_{xxx}+3uv_x = 0, & x\in {\mathbb{R}}, \ t\ge 0, \\ u(0, x) = \phi(x), \; \; v(0, x) = \psi(x), & x\in {\mathbb{R}}, \end{array} \right. \end{equation} $

    where $ \alpha\in {\mathbb{R}} $, $ \beta\in {\mathbb{R}} $; $ u = u(x, t) $, $ v = v(x, t) $ are real functions. Aided by Fourier restrict norm method, we show that $ \forall s > -\frac 18 $ initial value problem (0.1) is locally well-posed in $ H^s({\mathbb{R}})\times H^{s+1}({\mathbb{R}}) $ which improved the results of [7].

    Citation: Xiangqing Zhao, Zhiwei Lv. Well-posedness of initial value problem of Hirota-Satsuma system in low regularity Sobolev space[J]. AIMS Mathematics, 2022, 7(4): 6702-6710. doi: 10.3934/math.2022374

    Related Papers:

  • In this paper, we study the initial value problem of Hirota-Satsuma system:

    $ \begin{equation} \notag \left\{ \begin{array}{ll} u_t-\alpha(u_{xxx}+6uu_x) = 2\beta vv_x, & \ x\in {\mathbb{R}}, \ t\ge 0, \\ v_t+v_{xxx}+3uv_x = 0, & x\in {\mathbb{R}}, \ t\ge 0, \\ u(0, x) = \phi(x), \; \; v(0, x) = \psi(x), & x\in {\mathbb{R}}, \end{array} \right. \end{equation} $

    where $ \alpha\in {\mathbb{R}} $, $ \beta\in {\mathbb{R}} $; $ u = u(x, t) $, $ v = v(x, t) $ are real functions. Aided by Fourier restrict norm method, we show that $ \forall s > -\frac 18 $ initial value problem (0.1) is locally well-posed in $ H^s({\mathbb{R}})\times H^{s+1}({\mathbb{R}}) $ which improved the results of [7].



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    [10] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1–21. https://doi.org/10.1215/S0012-7094-93-07101-3 doi: 10.1215/S0012-7094-93-07101-3
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