So, as my professor pointed out, there is a solitonic solution for the BNLS. It is not numerically stable and not Explicit, but it still serves as a good check. A solitonic solution for the BNLS is the solution of the following radial ODE:

$ - \Delta _r ^2 R(r) - R(r) + |R(r)|^{2\sigma} R(r) = 0 $

$ R'(0) = R'''(0) = 0, $ $R(\infty) = 0 $

In the case of $d = 1 $ , things get even simpler and you get the following ODE:

$R'''(r) R'(r) + \frac{1}{2}R'' ^2 (r) - \frac{1}{2} R^2 (r) + \frac {1}{2\sigma +2} R^{2 \sigma +2} (r) = 0 $

$ R''(0) = \sqrt{ R ^2(0) - \frac{1}{\sigma +1 } R ^{2\sigma +2} (0) } $ plus the previous initial condition.

This leaves us with a simple one parameter search of $ R(0)$.

Again, not a perfect sanity check but it should work. for more details, look at section 4 in this paper: http://www.math.tau.ac.il/~fibich/Manuscripts/dispersion.pdf

So, as my professor pointed out, there is a solitonic solution for the BNLS. It is not numerically stable and not Explicit, but it still serves as a good check. A solitonic solution for the BNLS is the solution of the following radial ODE:

$ - \Delta _r ^2 R(r) - R(r) + |R(r)|^{2\sigma} R(r) = 0 $

$ R'(0) = R'''(0) = 0, $ $R(\infty) = 0 $

In the case of $d = 1 $ , $\sigma = 4$ , things get even simpler and you get the following ODE:

$ -R^{(4)} -R + R^{9} = 0 $

This solution has an asymptotic sanity check for $r >> 1$, which states : $R(r) \sim ce^{-r/ \sqrt{2} } cos(r/ \sqrt{2} ) $

Again, not a perfect sanity check but it should work. for more details, look at section 4 in this paper: http://www.math.tau.ac.il/~fibich/Manuscripts/dispersion.pdf