Non-homogeneous boundary value problems for coupled KdV-KdV systems posed on the half line

Abstract

In this article, we study an initial-boundary-value problem of a coupled KdV-KdV system on the half line ${\mathbb{R}}^{+}$ with non-homogeneous boundary conditions: \begin{equation*} \left\{ \begin{array}{l} u_t+v_x+u u_x+v_{xxx}=0, \quad v_t+u_x+(vu)_x+u_{xxx}=0, \quad u(x,0)=ϕ(x),\quad v(x,0)=ψ(x), \quad u(0,t)=h_1(t),\quad v(0,t)=h_2(t),\quad v_x(0,t)=h_3(t), \end{array} \right. \qquad x,\,t>0. \end{equation*} It is shown that the problem is locally unconditionally well-posed in $H^s({\mathbb{R}}^{+})\times H^s({\mathbb{R}}^{+})$ for $s>-\frac{3}{4}$ with initial data $(\ddot{I}•,\ddot{I}ˆ)$ in $H^s({\mathbb{R}}^{+})\times H^s({\mathbb{R}}^{+})$ and boundary data $(h_1,h_2,h_3)$ in $H^{\frac{s+1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s+1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s}{3}}({\mathbb{R}}^{+})$. The approach developed in this paper can also be applied to study more general KdV-KdV systems posed on the half line.

The title is slightly modified. The organization of the paper is adjusted. Some results are rephrased and several proofs are substantially shortened. We also add and update some references