Simple Idealized 1D NLSE Solver

A flexible and efficient solver for the 1D Nonlinear Schrödinger Equation (NLSE) using pseudo-spectral methods

Overview

This package provides a high-accuracy numerical framework for solving the 1D focusing NLSE:

$$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2 } + |\psi|^2 \psi = 0,$$

where $\psi(x,t)$ is the complex wave function. This equation models various physical phenomena including Bose-Einstein condensates, nonlinear optics, and water waves.

Numerical Method

The solver employs a Fourier pseudo-spectral method combined with high-order time integration:

Spatial Discretization

Using the Fourier transform $\hat{\psi}(k,t) = \mathcal{F}[\psi(x,t)]$, the NLSE becomes:

$$\frac{\partial \hat{\psi}}{\partial t} = -\frac{i}{2}k^2\hat{\psi} + i\mathcal{F}[|\psi|^2\psi],$$

where spatial derivatives are computed spectrally:

• $\mathcal{F}[\partial^2_x \psi] = -k^2 \hat{\psi}$
• Wavenumbers: $k_j = 2\pi j/L$ for $j \in [-N/2, N/2)$

Anti-Aliasing Filter

To prevent aliasing from the nonlinear term, we apply an exponential filter (2/3 rule):

$$\sigma(k) = \exp\left[-36\left(\frac{|k|}{k_{max}}\right)^{36}\right], \quad k_{max} = \frac{2\pi N}{3L}$$

Time Integration

The filtered equation in Fourier space:

$$\frac{d\hat{\psi}_f}{dt} = -\frac{i}{2}k^2\hat{\psi}_f + i\sigma(k)\mathcal{F}[|\mathcal{F}^{-1}[\hat{\psi}_f]|^2 \mathcal{F}^{-1}[\hat{\psi}_f]]$$

is solved using the 8th-order Dormand-Prince method (DOP853) with adaptive time-stepping, achieving relative tolerances down to $10^{-9}$.

Conservation Laws

The solver monitors three conserved quantities to ensure numerical stability:

• Mass (L² norm): $M = \int_{-\infty}^{\infty} |\psi|^2 dx$

• Momentum: $P = \int_{-\infty}^{\infty} \text{Im}(\psi^* \partial_x \psi) dx$

• Energy (Hamiltonian): $E = \int_{-\infty}^{\infty} \left[\frac{1}{2}|\partial_x \psi|^2 - \frac{1}{2}|\psi|^4\right] dx$

Key Features

• Spectral accuracy: Exponential convergence for smooth solutions
• Adaptive time-stepping: Automatic step size control based on error estimates
• JIT compilation: Optional Numba acceleration for performance-critical sections
• Stability monitoring: Real-time tracking of conservation laws
• Multiple initial conditions: Solitons, breathers, and modulation instability scenarios

Installation

From PyPI

pip install simple-idealized-1d-nlse

From source

git clone https://github.com/samuderasains/simple-idealized-1d-nlse.git
cd simple-idealized-1d-nlse
pip install -e .

Quick Start

# Run single scenario with YAML config
nlse-simulate single_soliton

# Run with TXT configuration file
nlse-simulate --config configs/txt/single_soliton.txt

# Run all predefined scenarios
nlse-simulate --all

# Run with verbose output
nlse-simulate single_soliton --verbose

Project Structure

simple_idealized_1d_nlse/
├── src/simple_idealized_1d_nlse/
│   ├── core/           # Core solver and numerical methods
│   ├── utils/          # Utility functions and helpers
│   ├── visualization/  # FiveThirtyEight-style plotting
│   └── io/            # Input/output handlers (YAML/TXT)
├── configs/           
│   ├── yaml/          # YAML configuration files
│   └── txt/           # TXT configuration files

../outputs/            # Simulation results (outside package)
../logs/              # Simulation logs (outside package)

Authors

• Sandy H. S. Herho - sandy.herho@email.ucr.edu
• Iwan P. Anwar
• Faruq Khadami
• Rusmawan Suwarman
• Dasapta E. Irawan

License

This project is licensed under the WTFPL License - see the LICENSE file for details.

Citation

If you use this software in your research, please cite:

@software{nlse_solver_2025,
  title = {Simple Idealized 1D NLSE Solver},
  author = {Herho, Sandy H. S. and Anwar, Iwan P. and Khadami, Faruq and 
           Suwarman, Rusmawan and Irawan, Dasapta E.},
  year = {2025},
  version = {0.0.4},
  url = {https://github.com/sandyherho/simple_idealized_1d_nlse}
}