Consider the nonlinear Schrödinger equation $$ \left\{\begin{array}{l} i u_t + \Delta u = |u|^p u, \quad p > 0 \\ u(0, x) = u_0(x) \in L^2 \end{array}\right. $$ Given the initial data and the solution, I understand how to formally find scattering. Suppose $u$ is a forward solution to the initial data $u_0$, then it has the form: $$ u(t) = e^{i t \Delta} u_0 - i \int_0^t e^{i(t-s) \Delta}|u|^p u(s) \, ds $$ We have: $$ e^{-i t \Delta} u(t) = u_0 - i \int_0^t e^{-i s \Delta}|u|^p u(s) \, ds $$ Therefore, formally: $$ u_{+} = u_0 - i \int_0^{\infty} e^{-i s \Delta}|u|^p u(s) \, ds $$ which is the scattering.
My question is, given an $L^2$ function $\varphi(x)$, how can we find a solution $u$ that scatters to $e^{it \Delta}\varphi$? This means: $$ \left\|u(t,x) - e^{i t \Delta} \varphi\right\|_2 \rightarrow 0, \quad \text{as } t \rightarrow \infty $$