Eigenvector-dependent nonlinear eigenvalue problems are considered which arise from the finite difference discretizations of the Gross-Pitaevskii equation. Existence and uniqueness of positive eigenvector for both one and two dimensional cases and existence of antisymmetric eigenvector for one dimensional case are proved. In order to compute eigenpairs corresponding to excited states as well as ground state, homotopies for both one and two dimensional problems are constructed respectively and the homotopy paths are proved to be regular and bounded. Numerical results are presented to verify the theories derived for both one and two dimensional problems.
翻译:由于Gross-Pitaevskii等式的有限差异分解,人们会考虑到依赖雌性生物的非线性亚性价值问题。一维和二维均呈阳性乙型生物的存在和独特性得到证明,一维和二维都存在抗对等性乙型生物。为了计算与兴奋状态和地面状态相对应的乙型生物,分别构建了一维和二维问题的同质体,并证明同质性途径是固定和约束的。提供了数字结果,以核实一维和二维问题的理论。