In this paper, we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation, same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov–Rubenchik systems. Convergence is reached in the topology \(L^2({\mathbb R})\times L^2({\mathbb R})\) and with an approximation in the energy space \(H^1({\mathbb R})\times L^2({\mathbb R})\). In the case of the Zakharov system, this is achieved without the condition \(\partial _t n(x,0) \in \dot{H}^{-1}({\mathbb R})\) for the wave component, improving previous results.

在本文中，我们表明三次非线性Schrödinger方程的解是Benney系统解的渐近极限。由于一维输运方程的特殊性，对于一维Zakharov和1d-Zakharov-Rubenchik系统的解可得到相同的结果。在拓扑\（L ^ 2（{\ mathbb R}）\ L ^ 2（{\ mathbb R}）\）中达到收敛，并且在能量空间中近似为\（H ^ 1（{\ mathbb R }）\次L ^ 2（{\ mathbb R}）\）。在Zakharov系统的情况下，这不需要波分量的条件\（\ partial _t n（x，0）\ in \ dot {H} ^ {-1}（{\ mathbb R}）\），改善以前的结果。