  We study stable blow-up dynamics in the $L^2$-supercritical nonlinear
Schr\"{o}dinger equation in various dimensions. We first investigate the
profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4]
on the existence and local uniqueness of solutions of the cubic profile
equation to other $L^2$-supercritical nonlinearities and dimensions $d \geq 2$.
We then numerically observe the multi-bump structure of such solutions, and in
particular, exhibit the $Q_{1,0}$ solution, a candidate for the stable blow-up
profile. Next, using the dynamic rescaling method, we investigate stable
blow-up solutions in the $L^2$-supercritical NLS and confirm the square root
rate of the blow-up as well as the convergence of blow-up profiles to the
$Q_{1,0}$ profile.
