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Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann–Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave equations with Z2xZ2 reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.
V. Gerdjikov, R. Ivanov and A. Kyuldjiev, On the N-wave equations and soliton interactions in two and three dimensions, Wave Motion, 48 (2011) 791-804. Available from the publisher here http://www.sciencedirect.com/science/journal/01652125/48 doi: 10.21427/pbt8-5p79