After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states: $$ i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} \psi (x,t) = 0 $$ $$ \psi(x,0) = \psi_0 (x) \in H^2, x \in\ \mathbb{R}^d $$

Since my main purpose is to run a sanity check to my simulation, other more "soft" checks will be welcomed. I work with $ d = 1 $ and various integer values of $\sigma$.