In this paper, we investigate the initial value problem (IVP henceforth) associated with the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity. First, by using Fourier restriction norm method and Tao’s [k, Z]-multiplier method, we establish a trilinear estimate on the Bourgain space \(X_{s,b}.\) Then, combining the trilinear estimate with the contraction mapping principle, we prove that IVP is locally well-posed for the initial data \((u_0(x),v_0(x))\in H^s({\mathbb {R}})\times H^s({\mathbb {R}})\) with \(s\ge \frac{1}{4}\).

中文翻译：

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具有Kerr律非线性的摄动非线性Schrödinger方程的三线性估计。

在本文中，我们将研究与具有Kerr律非线性的摄动非线性Schrödinger方程有关的初值问题（此后称为IVP）。首先，通过使用傅立叶约束范数方法和Tao的[ k，  Z ]乘数方法，在布尔加因空间\（X_ {s，b}。\）上建立三线性估计，然后，将三线性估计与收缩映射原理相结合，我们证明IVP对于初始数据\（（u_0（x），v_0（x））\在H ^ s（{\ mathbb {R}}）\ times H ^ s（{\ mathbb {R}}）\）和\（s \ ge \ frac {1} {4} \）。