The Peregrine soliton $$Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$$ Q ( x , t ) = e it ( 1 - 4 ( 1 + 2 i t ) 1 + 4 x 2 + 4 t 2 ) is an exact solution of the 1d focusing nonlinear Schrödinger equation (NLS) $$iB_t+B_{xx}=-2|B|^2B$$ i B t + B xx = - 2 | B | 2 B , having the feature that it decays to $$e^{it}$$ e it at the spatial and time infinities, and with peaks and troughs in a local region. It is considered as a prototype of the rogue wave by the ocean waves community. The 1D NLS is related to the full water wave system in the sense that asymptotically it is the envelope equation for full water waves. In this paper, working in the framework of water waves which decay non-tangentially, we give a rigorous justification of the NLS from the full water waves equation on long time scale in a regime that allows for the partial justification of the Peregrine soliton. As a byproduct, we prove the long time existence of solutions for the full water waves equation with small initial data in space of the form $$H^s(\mathbb {R})+H^{s'}(\mathbb {T})$$ H s ( R ) + H s ′ ( T ) , where $$s\geqq 4, s'>s+\frac{3}{2}$$ s ≧ 4 , s ′ > s + 3 2 .

中文翻译：

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来自二维全水波的游隼孤子的部分证明

游隼孤子 $$Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$$Q ( x , t ) = e it ( 1 - 4 ( 1 + 2 it ) 1 + 4 x 2 + 4 t 2 ) 是一维聚焦非线性薛定谔方程 (NLS) $$iB_t+B_{xx}=-2|B 的精确解|^2B$$ i B t + B xx = - 2 | 乙 | 2 B ，具有在空间和时间无穷远处衰减到 $$e^{it}$$ e 的特征，并且在局部区域具有波峰和波谷。它被海浪界认为是流氓海浪的原型。一维 NLS 与全水波系统有关，因为它渐近地是全水波的包络方程。在本文中，在非切向衰减的水波框架下工作，我们从长时间尺度上的全水波方程给出了 NLS 的严格证明，该制度允许游隼孤子的部分证明。作为副产品，我们证明了在空间中具有小初始数据的全水波方程解的长期存在性，形式为 $$H^s(\mathbb {R})+H^{s'}(\mathbb { T})$$ H s ( R ) + H s ′ ( T ) ，其中 $$s\geqq 4, ​​s'>s+\frac{3}{2}$$ s ≧ 4 , s ′ > s + 3 2 .