Rogue wave and lump patterns in several integrable equations are asymptotically predicted by the root distributions of Umemura polynomials associated with the third Painlevé equation.
±(N−1), thenU N(z;µ)has a zero root of multiplicityN 0(N0 + 1)/2, where N0 =N− |µ|
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Rogue-wave and lump patterns associated with the third Painlev\'{e} equation
Rogue wave and lump patterns in several integrable equations are asymptotically predicted by the root distributions of Umemura polynomials associated with the third Painlevé equation.