2011 # Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

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(14 citation statements)

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“…To apply this method in systematic way, we combine the splitting steps via the standard second order Strang splitting [21]. The flowchart of this method can be described by the following steps.…”

confidence: 99%

“…To apply this method in systematic way, we combine the splitting steps via the standard second order Strang splitting [21]. The flowchart of this method can be described by the following steps.…”

confidence: 99%

“…Substituting Equation (19), Equation (20) and Equation (21) into Equation (15) O k h + . The boundary conditions (2) and the system given in the Equation (22) consists of 2 N + equations in 2 N + unknown.…”

confidence: 99%

“…Its numerical solutions have been researched by many authors. For example, finite difference method [10] [11], quasi-interpolation scheme [12], quadratic B-spline finite element scheme [13], compact split-step finite difference method and pseudo-spectral collocation method [14] [15], exponential spline method [16], spline methods [17] [18], split-step orthogonal spline collocation method [19], a high-order and accurate method [20], linearly implicit conservative scheme [21].…”

confidence: 99%

“…In such algorithms, handling multidimensional problems is transformed into solving a series of one-dimensional problems by introducing intermediate variables. Moreover, many other numerical methods are also proposed to solve multidimensional Schrödinger equations, such as the collocation method [21,22,27,33], the Galerkin method [27,31], and the mesh-free methods [25,31,32]. The above methods are effective for solving Schrödinger equations under certain conditions.…”

confidence: 99%

“…However, many of them will encounter severe difficulties in uniformly solving the three-dimensional generalized nonlinear Schrödinger equation (1). For example, the time-splitting methods need to obtain the density by solving analytically a nonlinear differential equation [7,8,21,26], which is an extremely difficult task for general damping term ( ). And when the classical collocation method [25] and Galerkin type method [32] are employed to solve directly the nonlinear Schrödinger equation, the matrices generated in the spatial discretization of nonlinear terms will be dependent on the time-dependent unknown vector and must be recalculated at each time step [25,32,38], thereby consuming considerable computing resources.…”

confidence: 99%