Numerical Methods for the Derivative Nonlinear Schrödinger Equation
-
Shu-Cun Li
and
Fang-Yuan Shi
Abstract
In this work, a second-order accuracy in both space and time Crank–Nicolson (C-N)-type scheme, a fourth-order accuracy in space and second-order accuracy in time compact scheme and a sixth-order accuracy in space and second-order accuracy in time compact scheme are proposed for the derivative nonlinear Schrödinger equation. The C-N-type scheme is tested to satisfy the conservation of discrete mass. For the two compact schemes, the iterative algorithm and the Thomas algorithm in block matrix form are adopted to enhance the computational efficiency. Numerical experiment is given to test the mass conservation for the C-N-type scheme as well as the accuracy order of the three schemes. In addition, the numerical simulation of binary collision and the influence on the solitary solution by adding a small random perturbation to the initial condition are also discussed.
Funding statement: This work is supported by National Natural Science Foundation of China (No. 11671044), the Science Challenge Project (No. TZ2016001) and Beijing Municipal Commission of Education (No. PXM2017\_014224\_000020).
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Articles in the same Issue
- Frontmatter
- A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay
- Numerical Methods for the Derivative Nonlinear Schrödinger Equation
- Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy
- Modeling of Supersonic/Hypersonic Boundary Layer Transition Using a Single-Point Approach
- A Novel Macromodel based on Krylov Subspace Projection Method for Micromixers with Serpentine Channels
- Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities
- Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions
- Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems
- Dynamics of Almost Periodic Solution for a Delayed Facultative Mutualism Model Involving Negative Feedback Terms
- Controllability of Fractional Evolution Inclusions with Noninstantaneous Impulses
- Analysis of a Delayed Predator–Prey System with Harvesting
- Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness
- Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline
- Numerical Simulation for Shale Gas Flow in Complex Fracture System of Fractured Horizontal Well
- Real-Time Control of a Rotary Inverted Pendulum using Robust LQR-based ANFIS Controller
- A Study of an Extended Generalized (2+1)-dimensional Jaulent–Miodek Equation
- RBFPUM with QR Factorization for Solving Water Flow Problem in Multilayered Soil
- Classical Magnetism and an Integral Formula Involving Modified Bessel Functions
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Articles in the same Issue
- Frontmatter
- A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay
- Numerical Methods for the Derivative Nonlinear Schrödinger Equation
- Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy
- Modeling of Supersonic/Hypersonic Boundary Layer Transition Using a Single-Point Approach
- A Novel Macromodel based on Krylov Subspace Projection Method for Micromixers with Serpentine Channels
- Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities
- Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions
- Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems
- Dynamics of Almost Periodic Solution for a Delayed Facultative Mutualism Model Involving Negative Feedback Terms
- Controllability of Fractional Evolution Inclusions with Noninstantaneous Impulses
- Analysis of a Delayed Predator–Prey System with Harvesting
- Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness
- Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline
- Numerical Simulation for Shale Gas Flow in Complex Fracture System of Fractured Horizontal Well
- Real-Time Control of a Rotary Inverted Pendulum using Robust LQR-based ANFIS Controller
- A Study of an Extended Generalized (2+1)-dimensional Jaulent–Miodek Equation
- RBFPUM with QR Factorization for Solving Water Flow Problem in Multilayered Soil
- Classical Magnetism and an Integral Formula Involving Modified Bessel Functions
- Lie Symmetry Analysis of Boundary Layer Stagnation-Point Flow and Heat Transfer of Non-Newtonian Power-Law Fluids Over a Nonlinearly Shrinking/Stretching Sheet with Thermal Radiation
