Failure of the N‐wave interaction approximation without imposing periodic boundary conditions,ZAMM - Journal of Applied Mathematics and Mechanics

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Failure of the N‐wave interaction approximation without imposing periodic boundary conditions
ZAMM - Journal of Applied Mathematics and Mechanics ( IF 2.3 ) Pub Date : 2020-04-07 , DOI: 10.1002/zamm.201900230
Tobias Haas ₁ , Guido Schneider ₁

Affiliation

The N‐wave interaction (NWI) system appears as an amplitude system in the description of N nonlinearly interacting and linearly transported wave packets in dispersive wave systems such as the water wave problem or problems in nonlinear optics. The purpose of this paper is twofold. First we give a new simplified proof for the failure of the NWI approximation in case of resonances which are located at integer multiples of a basic wave number k₀ in the original

2 π / k_{0}

‐spatially periodic dispersive wave system. Secondly, we give a first rigorous proof that an amplitude system fails in the description of an original system, without imposing periodic boundary conditions on the original system.

中文翻译：

没有施加周期性边界条件的N波相互作用近似的失败

在水波问题或非线性光学问题等色散波系统中，N个非线性相互作用和线性传输的波包的描述中，N波相互作用（NWI）系统以幅度系统的形式出现。本文的目的是双重的。首先，在共振位于原始波数k _0的整数倍的情况下，我们给出了NWI近似失效的新简化证明。