The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation
The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation
复制标题
DOI：
10.1088/0951-7715/25/4/1011
复制
发表时间：
2012-04
期刊：
影响因子：
1.7
通讯作者：
A. Fokas;B. Pelloni
中科院分区：
文献类型：
--
作者：
A. Fokas;B. Pelloni
我们分析了椭圆型正弦-戈登方程在上半平面上的Dirichlet问题。我们根据Riemann-Hilbert问题表达解q(x, y)，该问题的跳跃矩阵由特定函数b（λ）唯一定义，显式地以给定的Dirichlet数据g0(x) = q（x, 0）和未知的Neumann边界值g1(x) = qy（x, 0）表示，其中g0(x)和g1(x)通过全局关系{b(λ) = 0， λ大于或等于0}相关联。此外，我们证明了后一种关系可以用来表征Dirichlet-to-Neumann映射，即用g0(x)表示g1(x)。这似乎提供了第一个这样的映射被明确表征为非线性可积椭圆PDE的情况，而不是演化PDE。
We analyse the Dirichlet problem for the elliptic sine-Gordon equation in the upper half plane. We express the solution q(x, y) in terms of a Riemann–Hilbert problem whose jump matrix is uniquely defined by a certain function b(λ), , explicitly expressed in terms of the given Dirichlet data g0(x) = q(x, 0) and the unknown Neumann boundary value g1(x) = qy(x, 0), where g0(x) and g1(x) are related via the global relation {b(λ) = 0, λ ⩾ 0}. Furthermore, we show that the latter relation can be used to characterize the Dirichlet-to-Neumann map, i.e. to express g1(x) in terms of g0(x). It appears that this provides the first case that such a map is explicitly characterized for a nonlinear integrable elliptic PDE, as opposed to an evolution PDE.