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

Xiangqing Zhao; Zhiwei Lv; Xiangqing Zhao; Zhiwei Lv

doi:10.3934/math.2022374

AIMS Mathematics

2022, Volume 7, Issue 4: 6702-6710. doi: 10.3934/math.2022374

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Research article

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

Xiangqing Zhao ^,,
Zhiwei Lv

Department of Mathematics, Suqian University, Suqian 223800, China

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].
- Hirota-Satsuma system,
- initial value problem,
- Fourier restrict norm method,
- local well-posed
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

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Abstract

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].

References

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