Under some conditions on the function $F$ the nonlinear Schrödinger equation $i\psi _t =-\psi _{xx} + F(|\psi |^2)\psi ,\quad \psi = \psi (x, t) \in \mathbb {C},$ admits a of bounded solutions $w(x|\sigma (t))$, which parameters $\sigma = \sigma (t) \in \mathbb {R}^4$ depend explicitely on time $t$. The Cauchy problem for the Schrödinger equation with the initial data $\psi (x, 0) = w(x|\sigma _0(0)) + \chi _0(x)$ is considered where $\chi _0$ is assumed to have the sufficiently small norm $N = \|(1 + x^2)\chi _0\|_2 + \|\chi _0'\|_2.$ If the spectrum of the linearization of the Schrödinger equation on the soliton $w(\cdot |\sigma _0(0))$ has the simplest structure in some natural sense, the asymptotic behavior of $\psi $ as $t\to +\infty $ is given by the formula (in $\mathbf {L}_2$-norm) : $\psi = w(\cdot |\sigma _+(t)) + exp(-il_0t)f_+ + o(1),$ here $\sigma _+(0)$ is close to $\sigma _0(0)$, $l_0 =-\partial ^2_x$, $f_+ \in \mathbf {L}_2(\mathbb {R})$ and is sufficiently small.