In this paper, the Cauchy problem for a class of coupled system of the four-dimensional cubic focusing nonlinear Schrödinger equations was investigated. By exploiting the double Duhamel method and the long-time Strichartz estimate, the global well-posedness and scattering were proven for the system below the ground state. In our proof, we first established the variational characterization of the ground state, and obtained the threshold of the global well-posedness and scattering. Second, we showed that the non-scattering is equivalent to the existence of an almost periodic solution by following the concentration-compactness/rigidity arguments of Kenig and Merle ^[17] (Invent. Math., 166 (2006), 645–675). Then, we obtained the global well-posedness and scattering below the threshold by excluding the almost periodic solution.

In this paper, the Cauchy problem for the nonlinear Schrödinger system

$$\{\begin{array}{l}i{\mathrm{\partial }}_t{u}_1(x,t)=\mathrm{\Delta }{u}_1(x,t)-|u_1(x,t){|}^{p-1}{u}_1(x,t)-|u_2(x,t){|}^{p-1}{u}_1(x,t),\\ i{\mathrm{\partial }}_t{u}_2(x,t)=\mathrm{\Delta }{u}_2(x,t)-|u_2(x,t){|}^{p-1}{u}_2(x,t)-|u_1(x,t){|}^{p-1}{u}_2(x,t),\end{array}$$

was investigated in $d$ space dimensions. For $1<p\le 1+2/d$, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when $d=1,p=3$.