In this seminar, I will discuss the local well-posedness (LWP) of the initial value problem for the Hirota-Satsuma system - a coupled system of two KdV-type equations. Physically, this system models nonlinear waves interactions of long waves with different dispersion relations, with applications in shallow water and stratified fluids. Since there are two equations in this system, we can suppose the initial data have different regularities, i.e., $(u_0, v_0) \in H^k (\mathbb{R}) \times H^s(\mathbb{R})$. Previous LWP results in the literature only suppose $k= s$ or $s = k + 1$. Using frequency-restricted estimates and the concept of integrated-by-parts strong solution, we were able to prove LWP in the general case $H^k (\mathbb{R}) \times H^s(\mathbb{R})$. This LWP is sharp except for only one obstruction which we are still investigating.

This is a joint work with Simão Correia and Jorge Drumond.