The Majda-Biello system on the half-line

The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter $\alpha$ as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for $0<\alpha<1$ or $1<\alpha<4$ all these problems are well-posed for initial data in Sobolev spaces $H^s$, $s\ge 0$. For $\alpha=1$ or $\alpha>4$ well-posedness holds for Dirichlet data if $s>-3/4$, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For $\alpha=4$ well-posedness of all problems holds for $s\ge 3/4$. The Robin and Neumann boundary data are in $H^{s/3}$ while the Dirichlet boundary data are in $H^{(s+1)/3}$. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained here are optimal.