## The (n+1)-dimensional VNLS system

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

i = + F

where is an m-vector of complex valued functions, and F is a real valued function of , t,. Before generating the system, we substituted = + i 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 01a, 01b, ..., ma, mb, T and X01, X02,..., Xn which depend on the independent variables 01a, 01b, ..., ma, mb, t and x01, x02,..., xn. There is also the arbitrary function F(01a,01b,...,ma,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 = . = 01a2 + 01b2 + ... + ma2 + 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

Allan Wittkopf 2000-06-15