非线性薛定谔(NLS)方程的暗孤子从数学的角度是指具有非零边界条件的孤立波解。因为非零边界条件的特点，相对于亮孤子(即经典零边界条件的孤立波解)，NLS方程暗孤子的结构和动力学行为都要更为复杂。本项目将主要应用临界点理论和变分方法，研究NLS方程暗孤子理论中的两类具有非零边界条件的Gross–Pitaevskii(GP)方程，一类是含有Dirac delta相互作用势的local Gross–Pitaevskii(LGP)方程，另一类是含有Macdonald相互作用势的non-local Gross–Pitaevskii(NGP)方程，具体内容包括：二维LGP方程在整个亚音速区间内有限能量行波解的存在性；LGP方程最小能量行波解的唯一性；NGP方程的动量约束变分问题。我们将通过这些问题的研究着重发展变分泛函的scaling技巧和正则化方法，为临界点理论和变分法的应用提供新的思路。

The dark soliton of nonlinear Schrodinger (NLS) equation, from the point of view of mathematics, corresponds to the solitary wave solution of NLS equation with nonzero boundary condition. Because of the character of nonzero boundary condition, compared with bright soliton ,that is, the solitary wave solution of classic NLS equation with zero boundary condition, the structure and dynamics of dark solitons are much more complicated. In this project, we will apply the critical point theory and variational methods to study two classes of Gross–Pitaevskii (GP) equations with nonzero boundary conditions, arising in the dark soliton theory of NLS equations. One is the local Gross–Pitaevskii(LGP) equation with Dirac delta interaction potential, the other is the nonlocal Gross–Pitaevskii (NGP) equation with Macdonald interaction potential. Precisely, we will focus on the following problems: existence of finite energy traveling waves with any speed less than sound velocity for two dimensional LGP equation; uniqueness of least energy traveling waves for LGP equation; the variational problem with constrained momentum for NGP equation. Through the study on these problems, we will mainly develope the scaling technique and regularization approach for variational functional, which may provide some new ideas for the applications of critical point theory and variational methods.