Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy

Geng, Xianguo; He, Guoliang; Wu, Lihua

doi:10.1016/j.geomphys.2019.01.005

Cited by 28 publications

(9 citation statements)

References 39 publications

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“…The CDGSKE was widely applied in the area of fluid dynamics [4][5][6][7]. In the past decades, many different solutions to CDGSKE were developed by the analytical or numerical methods, including the dressing method [7], Darboux transformation [4], Backlund transformation in bilinear forms [8], Hirota's bilinear method [9], the exp-function method [10], the exp[-φ(z)]-expansion method [11], the Riemann theta function method [12], the variational approach [13,14], the Fourier spectral method [15], and the reproducing kernel method [16], and so on. There are also some research results about the non-linear (2+1)-D CDGKSE, which can be seen as the general form of CDGSKE [17].…”

Section: Introductionmentioning

confidence: 99%

Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Chen

2022

THERM SCI

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This paper focuses on solving the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation (FCDGSKE). We propose two analytical methods based on the fractional complex transform, the variational iteration method and the homotopy perturbation method. The approximated solutions to the initial value problems associated with FCDGSKE are provided without linearization and complicated calculation. Numerical results show the main merits of the analytical approaches.

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Section: Introductionmentioning

confidence: 99%

Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Chen

2022

THERM SCI

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“…The interaction phenomenon between the lump waves and stripe solitons in the 2D-CDGKSE, by making use of the HBM, was investigated [7]. The CDGKS hierarchy associated with a matrix spectral problem was suggested, based on Lenard recursion equations [8]. In [9], Bernoulli sub-equation function method was applied to obtain some new exact oscillating solutions.…”

Section: Introductionmentioning

confidence: 99%

Towering and internal rogue waves induced by two-layer interaction in non-uniform fluid. A 2D non-autonomous gCDGKSE

Abdel‐Gawad

2022

Nonlinear Dyn

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A Generalized (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation (2D- gCDGKSE) is an integro-differential equation that describes tow-layer fluid interaction. The non-autonomous (2+1)-dimensional gCDGKSE (NAUT-gCDGKSE) was rarely considered in the literature. In the previous works, the concepts of two-layer fluid interaction and non-uniform fluid were not explored. This motivated us to focus the attention on these themes. Our objective is to inspecting waves structures in non-uniform fluid which describes fluid flows near a solid boundary. Thus, the present work is completely new. Our objective, here, is to inspect waves which are similar to those created in waterfall, water waves behind dams, boat sailing, in the network of canals during water release, and internal waves in submarine. In a uniform fluid, rogue waves occur in open oceans and seas, while in the present case of non-uniform fluid, towering and internal rogue waves occur near barriers (islands) and near submarine, respectively. This was consolidated experimentally, as it was shown that rogue wave is produced in a water tank (which is with solid boundary). The exact solutions of NAUT-gCDGKSE are derived here, by implementing the extended unified method (EUM). In applications, it is found that the EUM is of lower time cost in symbolic computation, than when using Lie symmetry, Darboux and AutoBucklund transformations. The results obtained here are evaluated numerically, and they are displayed in graphs. They reveal multiple waves structures with relevance to waves created near a solid boundary. Among them are towering and internal rogue waves, internal (hollowed) and bulge-U-shape wave and S-shape wave, water fall, saddle wave, and dromoions.

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“…Levi and Ragnisco constructed the Darboux transformation for SK equation [25] (see also [34,3]) and a nonlinear superposition formula was found by Hu and Li [22]. Most recently, Geng, He and Wu constructed the algebro-geometric solutions for the SK hierarchy [15]. For more results and properties of the SK equation, one is referred to [2,11,13,19,20,35,30,37,44,33] and the references there.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

Mao¹,

Liu²,

Xue³

2021

JNMP

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In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the supersymmetric Sawada-Kotera (SSK) equation.The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.

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Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy

Cited by 28 publications

References 39 publications

Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Towering and internal rogue waves induced by two-layer interaction in non-uniform fluid. A 2D non-autonomous gCDGKSE

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

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