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: $$\{\begin{array}{l}{u}_t+u_{xxx}=-v{v}_x,v_t+\alpha v_{xxx}+\beta v_x=-(uv{)}_x,(u,v){\mathrm{\mid }}_{t=0}=(u_0,v_0)\in H^s(\mathbb{R})\times H^s(\mathbb{R}),\end{array}x\in \mathbb{R},t\in \mathbb{R},$$ where $\alpha \in \mathbb{R}\setminus \{0\}$ and $\beta \in \mathbb{R}$. Let $s^{\ast }(\alpha ,\beta )$ be the smallest value for which the IVP is locally analytically well-posed in $H^s(\mathbb{R})\times H^s(\mathbb{R})$ when $s>s^{}(\alpha ,\beta )$. Two interesting facts have already been known in literature: $s^{\ast }(\alpha ,0)=0$ for $\alpha \in (0,4)\setminus \{1\}$ and $s^{\ast }(4,0)=\frac{3}{4}$. Our key findings include the following: For $s^{\ast }(4,\beta )$, a significant reduction is observed, reaching $\frac{1}{2}$ for $\beta >0$ and $\frac{1}{4}$ for $\beta <0$. Conversely, when $\alpha \mathrm{\ne }4$, we demonstrate that the value of $\beta$ exerts no influence on $s^{\ast }(\alpha ,\beta )$. 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.