Compactness of the set of multisoliton solutions of the nonlinear Schrödinger equation

Multisoliton solutions $\psi(x,t)$ of the nonlinear Schrödinger equation are considered which satisfy the condition of finite density:
$$ \lim_{x\to\pm\infty}\psi(x,t)=\frac12\omega e^{i\psi_\pm}. $$
It is proved that all these solutions satisfy the inequalities
$$ \sup_{\substack{-\infty<x<\infty\\-\infty<t<\infty}}\biggl|\frac{\partial^m} {\partial t^m}\frac{\partial^n}{\partial x^n}\psi(x,\,t)\biggr|\leqslant\frac14 (2\omega)^{1+n+2m}(n+2m)! $$
($m,n=0,1,2,\dots$), which implies solvability of the Cauchy problem for the nonlinear Schrödinger equation with an initial function $\psi(x,0)$ belonging to the closure of the set of nonreflecting potentials.