Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system

Chen, Su-Su; Tian, Bo; Liu, Lei; Yuan, Yu‐Qiang; Zhang, Chen-Rong

doi:10.1016/j.chaos.2018.11.010

Cited by 75 publications

(16 citation statements)

References 38 publications

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“…Nonlinear partial differential equations containing time variables are generally referred to as nonlinear evolution equations (NLEEs) which can describe the state or process changing along with times in physics, dynamics, and other nature sciences. For the past decades, a variety of methods have sprung up to obtain exact solutions of NLEEs such as the homogeneous balance method [1], the generalized auxiliary equation technique [2], the inverse scattering method [3], the homotopy perturbation method [4], the optimal Galerkinhomotopy asymptotic method [5,6], the tan ðϕ/2Þ-expansion method [7], new Kudryashov's method [8], the function expansion method [9], the standard truncated Painlevé expan-sion method [10], Hirota's bilinear method [11][12][13][14][15], He's variational principle [16,17], binary Darboux transformation [18], Lie group analysis [19,20], Bäcklund transformation method [21], and the multiple Exp-function method [22]. Based on the above methods, plenty of exact solutions including soliton solution [23], lump solution [24], interaction solution [25], and rational solution [26] have been derived.…”

Section: Introductionmentioning

confidence: 99%

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

Yang

Manafian

Zia

et al. 2021

Advances in Mathematical Physics

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In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.

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Section: Introductionmentioning

confidence: 99%

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

Yang

Manafian

Zia

et al. 2021

Advances in Mathematical Physics

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“…As is well known, the model of many natural phenomena and the differential equations in the sciences and engineering are nonlinear and it is very important to obtain analytically or numerically accurate solutions. In order to achieve this goal, various methods have been developed for linear and nonlin-ear equations, such as the Exp-function method [5], the homotopy analysis method [6], the homotopy perturbation method [7], the (G'/G)-expansion method [8], the improved tan(φ/2)-expansion method [9,10], the Hirota bilinear method [11,12,[59][60][61][62][63][64][65][66][67], the He variational principle [13,14], the binary Darboux transformation [15], the Lie group analysis [16,17], the Bäcklund transformation method [18], and the multiple Exp-function method [19,20]. Moreover, many powerful methods have been used to investigate the new properties of mathematical models which are symbolizing serious real world problems [21,22,68,69].…”

Section: Introductionmentioning

confidence: 99%

Localized waves and interaction solutions to the fractional generalized CBS-BK equation arising in fluid mechanics

et al. 2021

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The Hirota bilinear method is employed for searching the localized waves, lump–solitons, and solutions between lumps and rogue waves for the fractional generalized Calogero–Bogoyavlensky–Schiff–Bogoyavlensky–Konopelchenko (CBS-BK) equation. We probe three cases including lump (combination of two positive functions as polynomial), lump–kink (combination of two positive functions as polynomial and exponential function) called the interaction between a lump and one line soliton, and lump–soliton (combination of two positive functions as polynomial and hyperbolic cos function) called the interaction between a lump and two-line solitons. At the critical point, the second-order derivative and the Hessian matrix for only one point will be investigated and the lump solution has one maximum value. The moving path of the lump solution and also the moving velocity and the maximum amplitude will be obtained. The graphs for various fractional orders α are plotted to obtain 3D plot, contour plot, density plot, and 2D plot. The physical phenomena of this obtained lump and its interaction soliton solutions are analyzed and presented in figures by selecting the suitable values. That will be extensively used to report many attractive physical phenomena in the fields of fluid dynamics, classical mechanics, physics, and so on.

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“…Moreover, plenty of nonlinear models have been widely studied. The NLPDEs perform a great role in plasma physics, ocean engineering, optical fibers, physics, biology, quantum physics, fluid mechanics, geochemistry, and many other scientific areas to explain the dynamical and physical processes 1‐27 . In this advanced era of science and technology, the study of nonlinear phenomena has become attractive field for scientists and engineers.…”

Section: Introductionmentioning

confidence: 99%

Multiple rogue wave, lump‐periodic, lump‐soliton, and interaction betweenk‐lump andk‐stripe soliton solutions for the generalized KP equation

Zhao

Manafian

Zaya

et al. 2020

Math Methods in App Sciences

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The multiple rogue wave solutions technique is engaged to seek the multifold soliton solutions for the generalized (2 + 1)‐dimensional Kadomtsev–Petviashvili (gKP) equation, which contains one wave, two waves, and triple waves solutions. The second‐order derivative will be perused to get the minimum or maximum amount of lump solution. For one case, the lump solution will be shown the bright‐dark lump structure, and for another case, the dark lump structure two small peaks and one deep hole can be present. Also, the interaction of lump with periodic waves and the interaction between the lump and two stripe solitons can be catched by introducing the Hirota forms. Simultaneously, the interaction between k‐lump and k‐stripe soliton wave solutions can be gained by the Hirota operator. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in diagrams by choosing proper amounts.

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Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system

Cited by 75 publications

References 38 publications

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

Localized waves and interaction solutions to the fractional generalized CBS-BK equation arising in fluid mechanics

Multiple rogue wave, lump‐periodic, lump‐soliton, and interaction betweenk‐lump andk‐stripe soliton solutions for the generalized KP equation

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