## Nagoya Mathematical Journal

### Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

#### Abstract

In this paper, coupled systems $$u_{t}+u_{xxx}+P(u,v)_{x}=0,$$ $$v_{t}+v_{xxx}+Q(u,v)_{x}=0$$ of Korteweg–de Vries type are considered, where $u=u(x,t)$, $v=v(x,t)$ are real-valued functions and where $x,t\in\mathbb{R}$. Here, subscripts connote partial differentiation and $$P(u,v)=Au^{2}+Buv+Cv^{2}\quad\mbox{and}\quad Q(u,v)=Du^{2}+Euv+Fv^{2}$$ are quadratic polynomials in the variables $u$ and $v$. Attention is given to the pure initial-value problem in which $u(x,t)$ and $v(x,t)$ are both specified at $t=0$, namely, $$u(x,0)=u_{0}(x)\quad\text{and}\quad v(x,0)=v_{0}(x),$$ for $x\in\mathbb{R}$. Under suitable conditions on $P$ and $Q$, global well-posedness of this problem is established for initial data in the $L^{2}$-based Sobolev spaces $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\gt -{3}/{4}$.

#### Article information

Source
Nagoya Math. J., Volume 215 (2014), 67-149.

Dates
First available in Project Euclid: 9 June 2014

https://projecteuclid.org/euclid.nmj/1402319947

Digital Object Identifier
doi:10.1215/00277630-2691901

Mathematical Reviews number (MathSciNet)
MR3263526

Zentralblatt MATH identifier
1372.35271

#### Citation

Bona, Jerry L.; Cohen, Jonathan; Wang, Gang. Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities. Nagoya Math. J. 215 (2014), 67--149. doi:10.1215/00277630-2691901. https://projecteuclid.org/euclid.nmj/1402319947

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