# Failure of the $N$-wave interaction approximation without imposing periodic boundary conditions

Haas, Tobias; Schneider, Guido

##### Abstract:
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 k_0 in the original 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.

 Zugehörige Institution(en) am KIT Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-882858 KITopen-ID: 1000088285 Verlag KIT, Karlsruhe Umfang 25 S. Serie CRC 1173 ; 2018/41 Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) Schlagwörter counter example, no approximation, TWI, new proof
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