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

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 $s> -\frac34$ with initial data $(ϕ,ψ)$ 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.

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