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.
The (ˆx, y) intervals are−70≤ˆx≤50,−30≤y≤30 for the upper row, and−60≤ˆx≤60,−30≤y≤30 for the lower row, where ˆx≡x−12tis the movingx-coordinate
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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.