This simple but very fundamental observation (in sharp contrast with the better behaved parabolic evolution) has necessitated radically different approach to the standard questions of local and global well-posedness, persistence of smoothness, stability of localized structures, etc. Indeed, tackling these questions took some time, and in fact the first rigorous mathematical results did not appear until the late 1970s.
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Stefanov, A. (2009). Decay and Strichartz Estimates for DNLS. 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_22
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