Rogue waves for the Hirota equation on the Jacobi elliptic cn-function background
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ORIGINAL PAPER
Rogue waves for the Hirota equation on the Jacobi elliptic cn-function background Xia Gao · Hai-Qiang Zhang
Received: 1 April 2020 / Accepted: 21 July 2020 © Springer Nature B.V. 2020
Abstract In this paper, the exact rogue wave solutions for the Hirota equation are investigated on the Jacobi elliptic cn-function background. Under the Bargmann constraint, the differential constraints on the potential are obtained by the nonlinearization of spectral problem. Further, the eigenvalues are determined based on the differential constraint equations. The non-periodic solution of spectral problem is constructed, which can produce rogue periodic wave solutions of the Hirota equation. Moreover, the exact rogue periodic wave solutions are presented via the one- and two-fold Darboux transformation formulas. Finally, the generation mechanism and characteristics of rogue periodic waves are analyzed from the viewpoint of two- and threedimensional structures. Keywords Rogue periodic waves · Jacobi elliptic function · Hirota equation · Darboux transformation 1 Introduction Rogue waves (also called freak waves, giant waves and extreme waves) usually appear from nowhere and disappear without a trace [1–3]. The study of rogue waves has become an important topic in many fields such as in the thunderstorms, earthquakes and hurricanes [4]. X. Gao · H.-Q. Zhang (B) College of Science, University of Shanghai for Science and Technology, P. O. Box 253, Shanghai 200093, China e-mail: [email protected]
Rogue waves are often supposed to a signal of catastrophic phenomenon, and they possess huge amplitudes which are much higher than those of the surrounding waves. It has been demonstrated that the formation of rogue waves is related to the modulation instability of the plane waves and periodic waves [5]. The first mathematical description of rogue waves is the nonlinear Schrödinger (NLS) equation [6]. Later on, rogue waves have been found in other nonlinear evolution equations including the modified Korteweg–de Vries (mKdV) equation [7] and the Hirota equation [8]. In soliton theory, integrable nonlinear evolution equations on the nonzero background possess rich localized solutions, such as the rogue waves, breathers, lump waves, dark solitons and anti-dark solitons [3,8– 16]. In recent years, rogue waves on the periodic background, which are called rogue periodic waves, have become a hot research in nonlinear science. An algebraic method was proposed to construct the rogue wave solutions on the elliptic function background in Ref. [17]. This method combines the method of nonlinearization of spectral problem with the Darboux transformation (DT) approach. In Refs. [17,18], rogue wave solutions of the NLS equation and the mKdV equation have been obtained on the cn- and dn-function backgrounds. Very lately, the multi-breather and high-order rogue waves for the NLS equation on the elliptic function background have been presented with the algebrogeometric method and the DT method [19]. Subsequent work has revealed that rogue peri
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