The (n+1)-dimensional VNLS system

The multi-component (n+1)-dimensional vector nonlinear Scrodinger equation is as follows:

i$$\bf\phi_{t}$$ = $$\nabla^{2}_{}$$$$\bf\phi$$ + F$$\bf\phi$$

where $\bf\phi$ is an m-vector of complex valued functions, and F is a real valued function of $\bf\phi$, t,$\bf x$. Before generating the system, we substituted $\bf\phi$ = $\bf\phi_a$ + i$\bf\phi_b$ and seperated the original PDE into real and imaginary parts. This allowed us to work with a real system of equations, but in twice the number of depenent functions.

The determining system consists of the dependent variables $\Phi$01a, $\Phi$01b, ..., $\Phi$ma, $\Phi$mb, T and X01, X02,..., Xn which depend on the independent variables $\phi$01a, $\phi$01b, ..., $\phi$ma, $\phi$mb, t and x01, x02,..., xn. There is also the arbitrary function F($\phi$01a,$\phi$01b,...,$\phi$ma,$\phi$mb, t, x01, x02,..., xn).

This class is extremely broad, but the determining systems for this fairly general class are available here (1.68 MB) for m = 1..4, n = 1..10 (the files in the archive are named vnlsfull_n_m).

The case for which we have provided benchmarks is for the following form of F:

F = $$\bf\phi$$ ^. $$\bf\phi^{*}_{}$$ = $$\phi$$01a² + $$\phi$$01b² + ... + $$\phi$$ma² + $$\phi$$mb²

The systems with this form of the potential can be downloaded here (1.99 MB) for m = 1..4, n = 1..10 (the files in the archive are named vnlssqr_n_m).

		Input		PureElim
Components	Dimension	Equations	Length	Time (sec)	Mem (MB)
1	1	46	6097	0.03	0.78
1	2	120	13001	0.04	0.84
1	3	222	23093	0.06	0.86
1	4	352	36805	0.09	0.90
1	5	510	54569	0.17	0.98
1	6	696	76817	0.25	1.04
1	7	910	103981	0.37	1.15
1	8	1152	136493	0.52	1.21
1	9	1422	174785	0.80	1.36
1	10	1720	219289	1.17	1.64

		Input		PureElim
Components	Dimension	Equations	Length	Time (sec)	Mem (MB)
2	1	248	43955	0.34	1.13
2	2	624	84639	0.56	1.52
2	3	1144	142243	0.69	1.45
2	4	1808	218687	0.59	1.25
2	5	2616	315891	0.74	1.48
2	6	3568	435775	1.31	1.69
2	7	4664	580259	2.93	2.57
2	8	5904	751263	2.05	2.23
2	9	7288	950707	4.42	3.11
2	10	8816	1180511	3.94	2.76

		Input		PureElim
Components	Dimension	Equations	Length	Time (sec)	Mem (MB)
3	1	678	154665	1.67	1.90
3	2	1704	285333	1.84	1.97
3	3	3126	467037	2.03	2.12
3	4	4944	704817	2.70	2.16
3	5	7158	1003713	3.62	2.48
3	6	9768	1368765	5.19	3.27
3	7	12774	1805013	6.66	3.50
3	8	16176	2317497	9.43	3.74
3	9	19974	2911257	9.92	4.05
3	10	24168	3591333	27.30	6.28

		Input		PureElim
Components	Dimension	Equations	Length	Time (sec)	Mem (MB)
4	1	1408	394099	5.44	2.93
4	2	3552	709691	5.45	3.92
4	3	6528	1143347	14.45	7.54
4	4	10336	1705435	8.07	3.34
4	5	14976	2406323	20.77	5.57
4	6	20448	3256379	15.87	5.40
4	7	26752	4265971	27.77	5.90
4	8	33888	5445467	28.39	8.79
4	9	41856	6805235	67.25	10.03
4	10	50656	8355643	80.90	10.24