A modified numerical scheme for the cubic Schrödinger equation

Abstract

A numerical scheme based on the use of rational approximants in a two-time level recurrence relation is applied to the cubic Schrödinger equation. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved using an already known modified predictor-corrector (MPC) scheme. The efficiency of the proposed method is tested to the single and the double-soliton waves and the results arising from the experiments are compared with the relevant ones known in the available bibliography.

An Improved Second-Order Numerical Method for the Generalized Burgers–fisher Equation

“…BgF equation: comparisons with known MPC methods The MPC method has already been used in the numerical solution of the nonlinear sine-Gordon equations in one (Sg1) [7] and two (Sg2) [3,5,6] dimensions, Boussinesq (Bq) [4], Schrödinger (NLS) [9] and modified Burgers (MBg) [8] ρ(10) = 1.000627ρ(10) = 1.000122 10 −4 10 ρ(10) = 1.000006ρ(10) = 1.000001 [19] δ = 2 10 −3 1 MPC 0.6272687 × 10 −2 0.4428246 × 10 −2 α = 0.1 PC 0.6276611 × 10 −2 0.4429159 × 10 −2 β = −0.0025 10 MPC 0.9905620 × 10 −2 0.6528471 × 10 −2 PC 0.9911409 × 10 −2 0.6533182 × 10 −2 10 −3 10 ρ(10) = 1.000605ρ(10) = 1.000464 10 −4 10 ρ(10) = 1.000007ρ(10) = 1.000005 [21] δ = 1 10 −3 1 MPC 0.6303023 × 10 −3 0.4600117 × 10 −3 α = 0.01 PC 0.6306929 × 10 −3 0.4600706 × 10 −3 β = 0.01 10 MPC 0.6848886 × 10 −3 0.4712857 × 10 −3 PC 0.6853077 × 10 −3 0.4714072 × 10 −3 10 −3 10 ρ(10) = 1.000544ρ(10) = 1.000193 10 −4 10 ρ(10) = 1.000006ρ(10) = 1.000002 T 7. Known results for the MPC and the PC methods (E(t) is the energy at time level t and c is the velocity).…”

“…Known results for the MPC and the PC methods (E(t) is the energy at time level t and c is the velocity). MPC and PC methods from the above papers are given in Table 7, in which [9] e = |E(t) − E(0)| PC − |E(t) − E(0)| MPC denotes the difference of the errors of the methods. From Table 7, the following conclusions are obtained:…”

“…This process, which is known as a modified predictor-corrector (MPC) method, in contrast to the iterative classical predictor-corrector (PC) method, is always explicit and is applied once. It has also been applied successfully with second-, third-and fourth-order in time approximations in other nonlinear partial differential equations (PDEs) [3][4][5][6][7][8][9], giving an improvement in the accuracy over the PC method. Owing to the need to minimize the calculations mentioned above, the order of the approximation used for this process will always be a subject of research.…”

A Schrödinger-type algorithm for solving the Schrödinger equations via Phragmén–Lindelöf inequalities

“…The nonlinear system (1.1) has been proved to possess wide application fields in many real world problems such as anomalous diffusion [2,4,15], disease models [6,9,21], ecological models [26], synchronization of chaotic systems [1,27], etc.…”

Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

“…The Schrödinger equation may describe many physical phenomena in optics, mechanics, and plasma physics, and it plays a very important role in various areas of mathematical physics. Numerical methods for this problem have been investigated extensively, e.g., see [1][2][3] for finite difference methods, [4][5][6][7][8] for finite element methods (FEMs), and [9][10][11] for others. Especially, the superconvergence analysis of FEMs for the Schrödinger equation have been studied successfully.…”