论文标题
耦合的KDV-KDV系统的非均匀边界值问题在半线上构成
Non-homogeneous boundary value problems for coupled KdV-KdV systems posed on the half line
论文作者
论文摘要
在本文中，我们研究了半行$ \ mathbb {r}^+ $在非均匀边界条件上的耦合kdv-kdv系统的初始符合性问题：\ begin {equination*} \ left \ left \ left \ leet \ okit u_t+v_x+u_x+v_ {xxx} = 0，\ quad v_t+u_x+（vu）_x+u_ {xxx} = 0，\ quad u（x，0）= ϕ（x），\ quad v（x，0）=ψ（x），\ quad u（0，t）= h_1（t），\ quad v（0，t）= h_2（t），\ quad v_x（0，t）= h_3（t），\ end \ end {array} \ right。 \ qquad x，\，t> 0。 \ end {equation*}表明，该问题在$ h^s（\ Mathbb {r}^+）中置于无条件的无条件范围内，$ s> - \ frac34 $具有初始data $ in $ h^s in $ h^s n $ h^s（\ frac34 $） h^{s}（\ mathbb {r}^+）$和边界数据$（h_1，h_2，h_3）$ in $ h^{\ frac {\ frac {s+1} {3}}}}}}}}（\ mathbb {r}^+times h^{\ frac {s+1} {3}}}（\ mathbb {r}^+）\ times h^{\ frac {s} {3}}}}}（\ mathbb {r}^+）$。本文中开发的方法也可以应用于一半线上提出的更通用的KDV-KDV系统。
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.
