Validation#

This page documents numerical results for the representative models used as baselines in the paper. It is not an exhaustive verification of every theory supported by soliton_solver; models not included here have not been individually validated in this section. The tables report grid-refinement and benchmark comparisons for the selected cases.

The example scripts define the model parameters and initial conditions for these baseline runs. Their numerical results are then compared across grid resolutions and, where available, with published benchmarks. These checks demonstrate the solver’s behavior on the selected paper cases, but should not be interpreted as validation of all model implementations or parameter regimes.

Paper baseline models#

The model parameters and initial states are those specified by the cited example scripts, with the grid varied for the refinement calculations. The listed \(N_x\) and \(N_y\) are the full grid dimensions, including halo points; the domain lengths are \(L_x\) and \(L_y\). Unless overridden in a model configuration, the shared solver settings are halo=2, courant=0.5, killkinen=True, and time_step=None. The lattice spacings are calculated as \(h_x=L_x/(N_x-1)\) and \(h_y=L_y/(N_y-1)\). The tables report the discrete energy and, for models with a topological invariant, the corresponding topological charge. Energies are given in the normalization of each model and compared with the cited benchmark where available.

Baby Skyrme model#

Source: soliton_solver/examples/baby_skyrme_gl.py.

Category	Baseline setting
Domain	`xsize=20.0`, `ysize=20.0`
Ansatz / constraint	`ansatz="neel"`, `unit_magnetization=True`
Initialization	`mode="ground"`
Run command	`python -m soliton_solver.examples.baby_skyrme_gl`

The standard and aloof calculations use the same domain, ansatz, constraint, and grid-refinement sequence, with the potential choice varied between runs. The broken-potential calculation uses N=3 and its separately listed model parameters. Within each potential’s refinement sequence, the physical domain and model parameters are held fixed.

Standard potential#

Category	Baseline setting
Model	`mpi=0.3162`, `kappa=1.0`, `potential="standard"`

Grid Size \(N_x\)	Grid Size \(N_y\)	Grid Spacing \(h_x\)	Grid Spacing \(h_y\)	Topological Charge \(\mathcal{Q}\)	Energy \(E\)
128	128	0.15748	0.15748	-1.0000	19.6781
256	256	0.07843	0.07843	-1.0000	19.6778
512	512	0.03914	0.03914	-1.0000	19.6778

Reference: The benchmark is taken from [D. Foster, Baby Skyrmion chains, Nonlinearity 23, 465 (2010)].
Benchmark: Foster reports a standard-potential energy of \(E=19.65\); the computed 512-by-512 energy is \(E=19.6778\), a difference of \(0.0278\) (about \(0.14\%\)).
Conclusion: The energy is unchanged to four decimal places between the 256-by-256 and 512-by-512 grids, and the topological charge remains \(\mathcal{Q}=-1\). This indicates grid convergence at these resolutions, while the finest-grid energy is about \(0.14\%\) above the cited value.

Aloof potential#

Category	Baseline setting
Model	`mpi=0.3162`, `kappa=1.0`, `potential="Aloof"`

Grid Size \(N_x\)	Grid Size \(N_y\)	Grid Spacing \(h_x\)	Grid Spacing \(h_y\)	Topological Charge \(\mathcal{Q}\)	Energy \(E\)
128	128	0.15748	0.15748	-0.9999	20.3163
256	256	0.07843	0.07843	-1.0000	20.3161
512	512	0.03914	0.03914	-1.0000	20.3161

Reference: The benchmark is taken from [P. Salmi and P. Sutcliffe, Aloof baby Skyrmions, J. Phys. A 48, 035401 (2015)].
Benchmark: Salmi and Sutcliffe report an energy of \(E=20.27\); the computed 512-by-512 energy is \(E=20.3161\), a difference of \(0.0461\) (about \(0.23\%\)).
Conclusion: The energy changes by only \(0.0002\) from the 128-by-128 grid to the 256-by-256 grid and is unchanged to four decimal places at 512-by-512; the topological charge converges to \(\mathcal{Q}=-1\). The finest-grid energy is about \(0.23\%\) above the cited value.

Broken potential#

Category	Baseline setting
Model	`mpi=1.0`, `kappa=1.0` , `N=3`, `potential="Broken"`

Grid Size \(N_x\)	Grid Size \(N_y\)	Grid Spacing \(h_x\)	Grid Spacing \(h_y\)	Topological Charge \(\mathcal{Q}\)	Energy \(E\)
128	128	0.15748	0.15748	-0.9994	34.7516
256	256	0.07843	0.07843	-1.0000	34.7843
512	512	0.03914	0.03914	-1.0000	34.7864

Reference: The benchmark is taken from [J. Jäykkä, M. Speight, and P. Sutcliffe, Broken baby Skyrmions, Proc. R. Soc. A. 468, 1085 (2012)].
Benchmark: Jäykkä, Speight, and Sutcliffe report an energy of \(E=34.79\); the computed 512-by-512 energy is \(E=34.7864\), a difference of \(-0.0036\) (about \(0.010\%\) below the reference).
Conclusion: The topological charge converges to \(\mathcal{Q}=-1\), and the energy approaches the cited value as the grid is refined. Between 256-by-256 and 512-by-512, the energy changes by \(0.0021\); further refinement would help establish convergence of the energy.

Bose-Einstein condensate#

Source: soliton_solver/examples/bose_einstein_condensate_gl.py.

Category	Baseline setting
Physical parameters	`beta=1000.0`
Domain	`xsize=24.0`, `ysize=24.0`
Solver	`time_step=0.006`
Initialization	`mode="ground"`
Run command	`python -m soliton_solver.examples.bose_einstein_condensate_gl`

Slowly rotating BEC#

Category	Baseline setting
Physical parameters	`omega_rot=0.5`

Grid Size (\(N_x\))	Grid Size (\(N_y\))	Grid Spacing (\(h_x\))	Grid Spacing (\(h_y\))	Energy (\(E\))
128	128	0.18898	0.18898	11.1000
256	256	0.09412	0.09412	11.1052
512	512	0.04697	0.04697	11.1054

Reference: H. Chena, G. Dongb, W. Liuc, and Z. Xie, Second-order flows for computing the ground states of rotating Bose-Einstein condensates, J. Comput. Phys. 475, 111872 (2023).
Benchmark: For \(\beta=1000\) and \(\Omega=0.5\), the reference energy is \(E=11.0954\). The computed 512-by-512 energy is \(E=11.1054\), which is \(0.0100\) (about \(0.0901\%\)) above the reference.
Conclusion: The energy changes by \(0.0002\) between the 256-by-256 and 512-by-512 grids. The finest-grid energy is about \(0.0901\%\) above the cited benchmark.

Rapidly rotating BEC#

Category	Baseline setting
Physical parameters	`omega_rot=0.9`

Grid Size (\(N_x\))	Grid Size (\(N_y\))	Grid Spacing (\(h_x\))	Grid Spacing (\(h_y\))	Energy (\(E\))
128	128	0.18898	0.18898	6.4873
256	256	0.09412	0.09412	6.3649
512	512	0.04697	0.04697	6.3651

Reference: The benchmark is taken from [H. Chena, G. Dongb, W. Liuc, and Z. Xie, Second-order flows for computing the ground states of rotating Bose-Einstein condensates, J. Comput. Phys. 475, 111872 (2023)].
Benchmark: For \(\beta=1000\) and \(\Omega=0.9\), the reference energy is \(E=6.3601\); the computed 512-by-512 result is \(E=6.3651\), a difference of \(0.0050\) (about \(0.079\%\)).
Conclusion: The computed energy changes by only \(0.0002\) between the 256-by-256 and 512-by-512 grids, indicating grid convergence at these resolutions, although the finest-grid result remains about \(0.079\%\) above the reference value.

Validation

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