eirikfagerbakke/nlse1d

1D Cubic Nonlinear Schrödinger Equation (NLSE)

Overview

This dataset contains numerical solutions to the 1D cubic nonlinear Schrödinger equation (NLS).

The governing equation is: $$i{u}_t=u_{xx}+u\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{.}i\; u\_t\; =\; u\_\{xx\}\; +\; u\; |u|^2.$$

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Initial Conditions

The initial conditions are generated as random periodic functions composed of a finite number of Fourier modes.

Each sample is defined as: $$u_0(x)=\frac{1}{\sqrt{n_{\text{modes}}}}\sum _{k=1}^{n_{\text{modes}}}{A}_k{e}^{i(kx+{\varphi }_k)},u\_0(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{n\_\{\backslash text\{modes\}\}\}\}\; \backslash sum\_\{k=1\}^\{n\_\{\backslash text\{modes\}\}\}\; A\_k\; \backslash ,\; e^\{i\; (k\; x\; +\; \backslash phi\_k)\},$$ where:

• $$A_k\sim \mathcal{U}(a_{\min },a_{\max })A\_k\; \backslash sim\; \backslash mathcal\{U\}(a\_\{\backslash min\},\; a\_\{\backslash max\})$$ are random amplitudes,
• $${\varphi }_k\sim \mathcal{U}(0,2\pi )\backslash phi\_k\; \backslash sim\; \backslash mathcal\{U\}(0,\; 2\backslash pi)$$ are random phases.

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Numerical Solver

• Integrator: scipy.integrate.solve_ivp
• Method: DOP853 (explicit Runge–Kutta of order 8(5,3))
• Spatial discretization: 6th-order central finite differences
• Boundary conditions: Periodic

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Dataset details

The equations were solved on $$x\in [0,2\pi )x\; \backslash in\; [0,\; 2\backslash pi)$$ with 256 points and $$t\in [0,2)t\; \backslash in\; [0,\; 2)$$ with 256 points.

The solution is stored in the fields "data_real" and "data_imag".

from datasets import load_dataset
train_dataset = load_dataset("eirikfagerbakke/nls", split="train").with_format("numpy")
u_sample = train_dataset[0]["data_real"] + 1j * train_dataset[0]["data_imag"]