半直线上的KdV方程：Dirichlet到Neumann映射
The KdV equation on the half-line: The Dirichlet to Neumann map
摘要：
考虑半直线$x \geq 0$上KdV方程$u_t + u_x + 6uu_x + u_{xxx} = 0$的初边值问题.对于适定问题，可以指定初始数据$u(x,0)$以及三个边界值之一$\{u(0,t), u_x(0,t), u_{xx}(0,t)\}$;其他两个边界值仍然未知。我们提供的Dirichlet以及两个Neumann问题的未知边值的非线性积分方程组的一个特征。的特征是有效的，在这个意义上，积分方程可以解决扰动的所有订单在一个定义良好的递归计划。
英文摘要：
We consider initial-boundary value problems for the KdV equation $u_t + u_x + 6uu_x + u_{xxx} = 0$ on the half-line $x \geq 0$. For a well-posed problem, the initial data $u(x,0)$ as well as one of the three boundary values $\{u(0,t), u_x(0,t), u_{xx}(0,t)\}$ can be prescribed; the other two boundary values remain unknown. We provide a characterization of the unknown boundary values for the Dirichlet as well as the two Neumann problems in terms of a system of nonlinear integral equations. The characterizations are effective in the sense that the integral equations can be solved perturbatively to all orders in a well-defined recursive scheme.
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