Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose–Einstein condensates (BEC), which is governed by the Gross–Pitaevskii equation (GPE).  choose the chemical potential λ as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrodinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as ll as excited-states of the SEP in the variational form.  also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(h t ) in the discrete H 1 norm is achieved, where t = 2a ndt = 1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential. © 2008 Elsevier B.V. All rights reserved.
['S.-L. Chang', 'C.-S. Chien', 'Z. C. Li']
I am writing a research paper on a idea A finite difference continuation method for computing energy levels of Bose-Einstein condensates
[{"from": "human", "value": "I am writing a research paper on a idea A finite difference continuation method for computing energy levels of Bose-Einstein condensates"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose\u2013Einstein condensates (BEC), which is governed by the Gross\u2013Pitaevskii equation (GPE).  choose the chemical potential \u03bb as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrodinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as ll as excited-states of the SEP in the variational form.  also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(h t ) in the discrete H 1 norm is achieved, where t = 2a ndt = 1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential. \u00a9 2008 Elsevier B.V. All rights reserved."}]