Dynamic patterns of high-order rogue waves for Sasa–Satsuma equation

Mu, Gui; Qin, Zhenyun

doi:10.1016/j.nonrwa.2016.01.001

Cited by 65 publications

(51 citation statements)

References 51 publications

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“…Note added: When this paper was accepted to be published, I noticed that Mu and Qin presented the high order rogue waves for the focusing Sasa-Satsuma model by the generalized Darboux transformation (or dressing transformation) [23], which were overlapped with a part of subsection 4.2 with a different form.…”

Section: Liming Lingmentioning

confidence: 99%

The algebraic representation for high order solution of Sasa-Satsuma equation

Ling¹

2016

DCDS-S

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In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method. 1975 1976 LIMING LING then equation (2) could be rewritten asSSE (4) is one of a limited number of integrable models and has been a field of active research for the past two decades. Thanks to the integrability, the sophisticated soliton construct underlying this wave equation can, therefore, be achieved using an array of mathematical tools such as inverse scattering transform [22,14,17,33], Darboux transformation (DT) [27,30,1,4,32,25], Hirota bilinear method [6,5,26], and others. Although the DT for the SSE was given in the previous study, the reduction of DT is not complete. In this work, we reestablish the DT through the loop group method [29]. With this way, we can obtain the DT completely. On the other hand, since the spectral problem for SSE (4) is 3 × 3 and possesses deep reduction, the high order DT can not be obtained directly. Based on the elementary DT [21,7] and analysis of spectral parameters, one can obtain the generalized DT [2,9,10,19]. As direct applications, the single solitonic solutions, multi-solitonic solution and the high order solitonic solutions are given through this method.In the previous literature, most of works concentrate on the soliton solution for SSE on the zero background [27,28]. Recently, there are some works focus on the solutions of SSE with the non-vanishing background (NVBC). For instance: the breather solution [30, 1], W-shape soliton [32], rational W-shape soliton [34] and the twist rogue wave solution [4] are obtained for the focusing SSE in the recent literature. In this work, we construct the solitonic solution for both the focusing and defocusing SSE on the NVBC systematically. For the focusing case, when the spectral parameter is located in the image axis, we can obtain the soliton solution and resonant soliton solution, periodical solution, half periodical solution, rational W-shape soliton, resonant rational W-shape soliton, and their high order ones; when the spectral parameter is not located in the image and real axis, we can obtain the breather solution, resonant breather solution, rogue wave solution, composite rogue wave solution and their high order ones. For the defocusing case, when the spectral parameter is located on the segment of real axis, we can obtain the dark soliton, W-shape dark soliton; when the spectral parameter is not located on the real and image axis, we can obtain the breather solution and its high order ones.The structure of this paper is organized in the following: In section 2, we give the DT and generalized DT for SSE through the loop group method. The reduction for DT was analyzed by spectral...

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Section: Liming Lingmentioning

confidence: 99%

The algebraic representation for high order solution of Sasa-Satsuma equation

Ling¹

2016

DCDS-S

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“…Thus, they have attracted a lot of attention in the physics and nonlinear waves communities in recent years. Analytical expressions of rogue waves have been derived in a large number of integrable nonlinear wave equations, such as the nonlinear Schrödinger (NLS) equation [19][20][21][22][23][24][25][26][27][28], the derivative NLS equation [29,30], the three-wave interaction equation [31], the Davey-Stewartson equations [32,33], and many others [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Rogue waves have also been observed in water tanks [54,55] and optical fibers [56][57][58].…”

Section: Introductionmentioning

confidence: 99%

General Rogue Waves in the Boussinesq Equation

Yang

2020

J. Phys. Soc. Jpn.

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We derive general rogue wave solutions of arbitrary orders in the Boussinesq equation by the bilinear Kadomtsev-Petviashvili (KP) reduction method. These rogue solutions are given as Gram determinants with 2N − 2 free irreducible real parameters, where N is the order of the rogue wave. Tuning these free parameters, rogue waves of various patterns are obtained, many of which have not been seen before. Compared to rogue waves in other integrable equations, a new feature of rogue waves in the Boussinesq equation is that the rogue wave of maximum amplitude at each order is generally asymmetric in space. On the technical aspect, our contribution to the bilinear KP-reduction method for rogue waves is a new judicious choice of differential operators in the procedure, which drastically simplifies the dimension reduction calculation as well as the analytical expressions of rogue wave solutions.

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“…Besides, another popular method to derive a variety of solutions to soliton equation theoretically is the Darboux transformation [52,53,54,55,56,57,58,59], but these obtained solutions have demonstrated that the bilinear method is a feasible scheme in computing different types solutions. Examples of these solutions include solitons, breathers, rogue waves, and many other types of rational solutions.…”

Section: Soliton Breather Solutions Of the Nonlocal Mel'nikov Equationmentioning

confidence: 99%

Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

Liu

Zheng

2018

Nonlinear Dyn

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Inspired by the works of Ablowitz, Mussliman and Fokas, a partial reverse space-time nonlocal Mel'nikov equation is introduced. This equation provides two dimensional analogues of the nonlocal Schrödinger-Boussinesq equation. By employing the Hirota's bilinear method, soliton, breathers and mixed solutions consisting of breathers and periodic line waves are obtained. Further, taking a long wave limit of these obtained soliton solutions, rational and semi-rational solutions of the nonlocal Mel'nikov equation are derived. The rational solutions are lumps. The semi-rational solutions are mixed solutions consisting of lumps, breathers and periodic line waves. Under proper parameter constraints, fundamental rogue waves and a semi-rational solution of the nonlocal Schrödinger-Boussinesq equation are generated from solutions of the nonlocal Mel'nikov equation.

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Dynamic patterns of high-order rogue waves for Sasa–Satsuma equation

Cited by 65 publications

References 51 publications

The algebraic representation for high order solution of Sasa-Satsuma equation

The algebraic representation for high order solution of Sasa-Satsuma equation

General Rogue Waves in the Boussinesq Equation

Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

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