In our considerations up to now, we have focused on localized solutions of the DNLS equation. In this chapter, we consider a different, yet important, class of solutions of the DNLS, namely the plane waves. Plane waves are spatially uniform (in the modulus) solutions, characterized by a wave number of the spatial modulation of their real and imaginary part, and an associated frequency (of temporal oscillation). They exist both in the continuum and in the discrete form of the NLS equation and one of the fundamental elements of their importance in this dispersive wave setting is that they are unstable, under appropriate conditions, to modulations, through a mechanism known as the modulational instability (MI).
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Kevrekidis, P.G. (2009). Extended Solutions and Modulational Instability. In: The Discrete Nonlinear Schrödinger Equation. Springer Tracts in Modern Physics, vol 232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89199-4_6
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