Analytical study of rational and double-soliton rational solutions governed by the KdV–Sawada–Kotera–Ramani equation with variable coefficients

In this paper, we investigate a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (3D-BMLP). By using bilinear forms under certain conditions, we obtain different wave structures for the 3D-BMLP. Among these waves, lump waves, breather waves, mixed waves, and multi-soliton wave solutions are constructed. The propagation and the dynamical behavior of the obtained solutions are discussed for different values of the free parameters. KEYWORDS (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, different wave structures, general bilinear form MSC CLASSIFICATION 35Q51; 35G99; 33F10 INTRODUCTIONDuring the last decades, solitons have been put forward in the study of nonlinear phenomena. Many researchers have extensively studied the dynamical behaviors for different nonlinear evolution equation models that are closely related to the concept of soliton by using different approaches. [1][2][3] The different types of solitons include the lump waves (usually called rogue waves in the presence of certain conditions), breather waves, and mixed waves that describe the interaction between lump wave and another type of soliton waves. 1,2,4,5 It is well known that solitons, of a diversity of physical structures, have attracted much attention in scientific fields, such as optical fiber communications, fluid dynamics, propagation of waves, marine engineering, fluid dynamics, plasma physics, incompressible fluid, ocean and rogue waves, photonics, and many other applications.As stated earlier, the Boiti-Leon-Manna-Pempinelli (BLMP) equation describes the fluid propagation and can be considered as a model for incompressible fluid. The BLMP equation describes the evolution of the horizontal velocity component of the water waves propagating in the xy-plane in an infinite narrow channel of constant depth. 6,7 It is worth noting that incompressible means that the effects of pressure on the fluid density are zero, where the density and the specific volume of the fluid do not change during the flow. Water and many liquids, at constant temperature, can be considered incompressible in most cases.The main objective of this work is to seek for lump waves, breather waves, multi-soliton waves, and mixed waves for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (3D-BLMP). 3,6 These solutions are obtained by using different choices for the arbitrary functions in bilinear forms. Lump waves, as special nonlinear wave phenomena, have been observed in many fields. 1,2 Lump waves are theoretically regarded as a limit form of soliton in some ways and propagate with higher propagating energy than general solitons. Recently, research works on the lump solutions have been growing more and more that led to promising findings. Therefore, theoretically researches on lump waves are helpful to better understand and predict possible extremes for nonlinear evolution systems.

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“…Here, we introduce the multi‐soliton solutions (double soliton solutions) for Equation . To this end, we assume that

f = A_{0} + A_{1} e^{ξ_{1}} + A_{2} e^{ξ_{2}} + A_{12} e^{ξ_{1} + ξ_{2}}, g = B_{0} + B_{1} e^{η_{1}} + B_{2} e^{η_{2}} + B_{12} e^{η_{1} + η_{2}},

where

\begin{matrix} right ξ_{1} & left = a_{1} x + a_{2} y + a_{3} z + a_{4} t, ξ_{2} = a_{5} x + a_{6} y + a_{7} z + a_{8} t, & right \\ right η_{1} & left = b_{1} x + b_{2} y + b_{3} z + b_{4} t, η_{2} = b_{5} x + b_{6} y + b_{7} z + b_{8} t, \end{matrix}

and a i , b i , A j , B j , A 12 , B 12 , i =1,2,…,8, j =0,1,2 are real constants.…”

Section: A Study Of Different Wave Structures Of Solitons For a 3d‐blmpmentioning

confidence: 99%

“…The different types of solitons include the lump waves (usually called rogue waves in the presence of certain conditions), breather waves, and mixed waves that describe the interaction between lump wave and another type of soliton waves …”

Section: Introductionmentioning

confidence: 99%

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

Osman

Wazwaz

2019

Math Methods in App Sciences

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127

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“…Furthermore, an efficient algorithm was applied to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients by Osman and Wazwaz in Osman and Wazwaz (2018). In addition, in , a variety of new types of nonautonomous combined multi-wave solutions of the (2+1)-dimensional variable coefficients KdV equation were derived by means of the generalized unified method (Wazwaz and Osman 2018;Osman 2016Osman , 2017.…”

Section: Introductionmentioning

confidence: 99%

General lump solutions, lumpoff solutions, and rogue wave solutions with predictability for the (2+1)-dimensional Korteweg-de Vries equation

et al. 2019

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In this work, the (2+1)-dimensional Korteweg-de Vries equation is investigated, which can be used to represent the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. By employing the properties of Bell's polynomial, we obtain bilinear representation of the equation with the aid of an appropriate transformation. Based on the obtained Hirota bilinear form, its lump solutions with localized characteristics are constructed in detail. We then derive the lumpoff solutions of the equation by studying a soliton solution generated by lump solutions. Furthermore, special rogue wave solutions with predictability are well presented, and the time and place of appearance are also derived. Finally, some graphic analysis is represented to better understand the propagation characteristics of the obtained solutions. It is hoped that our results provided in this work can be used to enrich the dynamic behaviors of the equation.

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“…Under using symbolic computation systems, the studying of the exact solutions of the NLEEs has attracted the attention of research community. A variety of approaches have been investigated and applied to the NLEEs, including the unified method (UM) [1][2][3][4][5] and its generalized form [6][7][8][9][10], the extended Jacobi elliptic function expansion method [11,12], the Bernoulli sub-equation function method [13,14], the sine-Gordon expansion method [15,16] and the Ricatti equation expansion [17,18].…”

Section: Introductionmentioning

confidence: 99%

New analytical internal long wave solutions described by the Benjamin Ono equation in deep water

Almutawa

2018

Journal of Scientific Research in Science

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Analytical study of rational and double-soliton rational solutions governed by the KdV–Sawada–Kotera–Ramani equation with variable coefficients

Cited by 75 publications

References 26 publications

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

General lump solutions, lumpoff solutions, and rogue wave solutions with predictability for the (2+1)-dimensional Korteweg-de Vries equation

New analytical internal long wave solutions described by the Benjamin Ono equation in deep water

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