AKNS and NLS hierarchies, MRW solutions, $P_n$ breathers, and beyond

Abstract : We describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and their mixed and deformed versions. The definition of the AKNS hierarchy and its deformed versions is given in the Sec. II. We also consider the continuous symmetries of the related equations and the related spectral curves. This work continues and summarises some of our previous studies dedicated to the rogue wave-like solutions associated with AKNS, nonlinear Schrödinger, and KP hierarchies. The general scheme is illustrated by the examples of small rank n, n ⩽ 7, rational or quasi-rational solutions. In particular, we consider rank-2 and rank-3 quasi-rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics, hydrodynamics, and many other branches of science.
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Submitted on : Thursday, December 20, 2018 - 10:16:58 AM
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Vladimir Matveev, Aleksandr Smirnov. AKNS and NLS hierarchies, MRW solutions, $P_n$ breathers, and beyond. Journal of Mathematical Physics, American Institute of Physics (AIP), 2018, 59 (9), pp.091419. ⟨10.1063/1.5049949⟩. ⟨hal-01961697⟩



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