A fourth-order real-space algorithm for solving local Schrödinger equations

2001 ◽  
Vol 115 (15) ◽  
pp. 6841-6846 ◽  
Author(s):  
J. Auer ◽  
E. Krotscheck ◽  
Siu A. Chin
Keyword(s):  
Fourth Order ◽  
Real Space ◽  
Schrodinger Equations ◽  
Schrödinger Equations

The Conservative Splitting High-Order Compact Finite Difference Scheme for Two-Dimensional Schrödinger Equations

2017 ◽  
Vol 15 (01) ◽  
pp. 1750079
Author(s):  
Bo Wang ◽  
Dong Liang ◽  
Tongjun Sun
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Two Dimensions ◽  
High Order ◽  
Optimal Error Estimate ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Time Step ◽  
Order Accuracy

In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrödinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the scheme satisfies the charge and energy conservations and it is unconditionally stable. We also prove the optimal error estimate of fourth-order accuracy in spatial step and second-order accuracy in time step. The scheme can be easily implemented and extended to higher dimensional problems. Numerical examples are presented to confirm our theoretical results.

scholarly journals A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrödinger equations

2017 ◽  
Vol 14 (02) ◽  
pp. 249-300
Author(s):  
Tristan Roy
Keyword(s):  
Fundamental Solution ◽  
Fourth Order ◽  
Spatial Localization ◽  
Weak Form ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Free Solution ◽  
Present Contribution ◽  
Finite Sum ◽  
Uniformly Bounded

We prove a weak form of the soliton resolution conjecture for — [Formula: see text] uniformly bounded in time — solutions of semilinear fourth-order Schrödinger equations, in dimensions [Formula: see text], and with a mass supercritical-energy subcritical power type nonlinearity, by using a strategy developed by T. Tao in 2007. More precisely, we prove that the solutions are decomposed into a sum of two terms: a free solution and a nonradiative term that approaches asymptotically an object that has similar properties to those of a finite sum of solitons. The asymptotic behavior of the nonradiative term is derived from its asymptotic frequency localization and its asymptotic spatial localization. There are two main differences between the present contribution and T. Tao’s earlier work. The first one appears when we prove the asymptotic frequency localization: we fill a gap of regularity by using the better dispersive properties of the high frequency pieces of the free solution. The second one appears when we prove the asymptotic spatial localization. A key estimate depends on the fundamental solution that does not have an explicit form. We overcome the difficulty by introducing a modified fundamental solution and exploiting the symmetries of the characters of the phases.

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