It is shown that the problem of dispersive self-modulation of waves including one transverse dimension and the three-dimensional self-focusing and self-defocusing theories has exact nontrivial analytic solutions. These are solutions compatible with cylindrical geometry and self-similar solutions that describe intermediate asymptotic, or long-time, behavior. These solutions are all obtained from symmetry considerations only.
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L. Gagnon (1990) studied this question.
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