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Characteristics of rogue waves on a periodic background for the Hirota equation. (English) Zbl 1524.35596

Summary: We consider the rogue dn-periodic waves (the rogue wave solutions on the dn-periodic waves background) for the Hirota equation by using Darboux transformation. We take Jacobian elliptic function dn as a seed solution, which is modulationally unstable as regards long wave perturbations. Through nonlinearization of the Lax pair for Hirota equation, the corresponding periodic eigenfunctions are successfully obtained. Based on these periodic eigenfunctions, we further construct the solutions of the Lax pair equations with dn-periodic wave seed solutions. In addition, numerical simulations are presented to reveal the phenomena of these solutions under different parameters choices.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
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