非零边界条件下非线性薛定谔方程的Dbar Dressing方法
Dbar dressing method to nonlinear Schrödinger equation with nonzero boundary conditions
摘要：
将Dbar Dressing方法推广到研究具有非零边界条件的聚焦/散焦非线性薛定谔（NLS）方程.考虑了一类特殊的复变函数。该函数在中心为0的环外是亚纯的，在环内满足局部Dbar问题。将该函数的理论推广到构造非零边界条件的NLS方程的拉克斯对。在这个过程中，NLS势和Dbar问题的解决方案之间的关系建立。对Dbar问题引入一个确定的分布，得到聚焦/散焦NLS方程和守恒律。从特殊分布出发，给出了非零边界条件下聚焦/散焦NLS方程的显式解。
英文摘要：
The Dbar dressing method is extended to study the focusing/defocusing nonlinear Schrödinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered. The function is meromorphic outside an annulus with center 0 and satisfies a local Dbar problem inside the annulus. The theory of such function is extended to construct the Lax pair of the NLS equation with nonzero boundary condition. In this procedure, the relation between the NLS potential and the solution of the Dbar problem is established. A certain distribution for the Dbar problem is introduced to obtain the focusing/defocusing NLS equation and the conservation laws. The explicit solutions of the focusing/defocusing NLS equation with nonzero boundary condition are given from special distributions.
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