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: $$\left\{ \begin{array}{l} u_{t} + u_{xxx} = - v v_x, v_{t} + \alpha v_{xxx} + \beta v_x = - (uv)_{x}, (u,v)|_{t=0} = (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R},$$ where $\alpha \in \mathbb{R}\setminus \{0\}$ and $\beta \in \mathbb{R}$. Let $s^{*}(\alpha, \beta)$ be the smallest value for which the IVP is locally analytically well-posed in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ when $s > s^{}(\alpha, \beta)$. Two interesting facts have already been known in literature: $s^{*}(\alpha, 0) = 0$ for $\alpha \in (0,4)\setminus\{1\}$ and $s^*(4,0) = \frac34$. Our key findings include the following: For $s^{*}(4,\beta)$, a significant reduction is observed, reaching $\frac12$ for $\beta > 0$ and $\frac14$ for $\beta < 0$. Conversely, when $\alpha \neq 4$, we demonstrate that the value of $\beta$ exerts no influence on $s^*(\alpha, \beta)$. These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of $\alpha$ and $\beta$ in the well-posedness of the system.