Effect of lower order terms on the well-posedness of Majda-Biello systems

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: \(\begin{aligned} \left\{ \begin{array}{l} u_{t} + u_{xxx} = - v v_x,\\ v_{t} + \alpha v_{xxx} + \beta v_x = - (uv)_{x},\\ (u,v)|_{t=0} = (u_0,v_0) \in H^{s}(\mathbb {R}) \times H^{s}(\mathbb {R}), \end{array} \right. \quad x \in \mathbb {R}, \, t \in \mathbb {R}, \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t+u_{\mathit{xxx}}=-v{v}_x,\hfill \\ v_t+\alpha v_{\mathit{xxx}}+\beta v_x=-{(uv)}_x,\hfill \\ {(u,v)|}_{t=0}=(u_0,v_0)\in H^s(\mathbb{R})\times H^s(\mathbb{R}),\hfill \end{array}\right)x\in \mathbb{R},t\in \mathbb{R},\end{array}$$ where \(\alpha \in \mathbb {R}\setminus \{0\}\) $\alpha \in \mathbb{R}\backslash \{0\}$ and \(\beta \in \mathbb {R}\) $\beta \in \mathbb{R}$ . Let \(s^{*}(\alpha , \beta )\) $s_{}^{\ast }(\alpha ,\beta )$ be the smallest value for which the IVP is locally analytically well-posed in \(H^{s}(\mathbb {R})\times H^{s}(\mathbb {R}) \) $H^s(\mathbb{R})\times H^s(\mathbb{R})$ when \(s > s^{*}(\alpha , \beta )\) $s>s_{}^{\ast }(\alpha ,\beta )$ . Two interesting facts have already been known in literature: \(s^{*}(\alpha , 0) = 0\) $s_{}^{\ast }(\alpha ,0)=0$ for \(\alpha \in (0,4){\setminus }\{1\}\) $\alpha \in (0,4)\backslash \{1\}$ and \(s^*(4,0) = \frac{3}{4}\) $s^{\ast }(4,0)=\frac{3}{4}$ . Our key findings include the following:

For \(s^{*}(4,\beta )\) $s_{}^{\ast }(4,\beta )$ , a significant reduction is observed, reaching \(\frac{1}{2}\) $\frac{1}{2}$ for \(\beta > 0\) $\beta >0$ and \(\frac{1}{4}\) $\frac{1}{4}$ for \(\beta < 0\) $\beta <0$ .