We consider a damped/driven nonlinear Schrödinger equation in \(\mathbb {R}^n\), where n is arbitrary, \({\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,\) under odd periodic boundary conditions. Here \(\eta (t,x)\) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy \( \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, \) uniformly in \(t\ge 0\) and \(\nu >0\). In this work we prove that for small \(\nu >0\) and any initial data, with large probability the Sobolev norms \(\Vert u(t,\cdot )\Vert _m\) with \(m>2\) become large at least to the order of \(\nu ^{-\kappa _{n,m}}\) with \(\kappa _{n,m}>0\), on time intervals of order \(\mathcal {O}(\frac{1}{\nu })\). It proves that solutions of the equation develop short space-scale of order \(\nu \) to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.

我们考虑\（\ mathbb {R} ^ n \）中的阻尼/驱动非线性Schrödinger方程，其中n是任意的，\（{\ mathbb {E}} u_t- \ nu \ Delta u + i | u | ^ 2u = \ sqrt {\ nu} \ eta（t，x），\ quad \ nu> 0，\）在奇周期边界条件下。这里\（\ eta（t，x）\）是一个随机力，时间为白色，空间为平滑。已知解的Sobolev范数在\（t \ ge 0 \）和\（\中均满足\（\ Vert u（t）\ Vert _m ^ 2 \ le C \ nu ^ {-m}，\）nu> 0 \）。在这项工作中，我们证明了对于较小的\（\ nu> 0 \）和任何初始数据，Sobolev范数\（\ Vert u（t，\ cdot）\ Vert _m \）具有\（m> 2 \ ）至少成为大到的顺序\（\ NU ^ { - \卡帕_ {N，M}} \）与\（\卡帕_ {N，M}> 0 \） ，上的顺序的时间间隔\（\ mathcal {O}（\ frac {1} {\ nu}）\）。证明了该方程的解在正数上发展了\（\ nu \）阶的短空间尺度，并严格建立了方程的（直接）能量级联。