Multi-kink solutions and soliton fission and fusion of Sharma–Tasso–Olver equation

Chen, Ai-Hua

doi:10.1016/j.physleta.2010.03.054

Cited by 55 publications

(25 citation statements)

References 9 publications

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“…Lots of approaches have been applied to study STO equation including first integral method [7], bi-Hamiltonian formulation method [8], the fractional sub-equation method [9], the sine-cosine method [10] and q-functions approach [11]. Furthermore, several transformations which involve Cole-Hopf transformation, fractional complex transform [12], Darboux transformation [13] and Bäcklund transformation have been used to discuss the STO equation [13,14]. 2 The fission and fusion in Sharma-Tasso-Olver-Burgers equation Firstly, we give a brief introduction of the Sharma-Tasso-Olver-Burgers (STOB) equation which will be useful in what follows.…”

Section: Introductionmentioning

confidence: 99%

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Soliton molecules in Sharma–Tasso–Olver–Burgers equation

Yan

Sen-Yue

2020

Applied Mathematics Letters

115

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Soliton molecules have been experimentally discovered in optics and theoretically investigated for coupled systems. This paper is concerned with the formation of soliton molecules by the resonant mechanism for a noncoupled system, the Sharma-Tasso-Olver-Burgers (STOB) equation. In terms of introducing velocity resonance conditions, we derive the soliton (kink) molecules, half periodic kink (HPK) molecules and breathing soliton molecule of STOB equation. Meanwhile, the fission and fusion phenomena among kinks, kink molecules, HPKs and HPK molecules have been revealed. Moreover, we also discuss the central periodic kink solutions from the multiple solitary wave solutions.Solitons have been experimentally found in plasma physics and optics, as well as in nonlinear science area including Bose-Einstein condensation and DNA mechanical waves [15,16]. It is wellknown that solitons have an important application in a variety of areas, such as atmospheric and ocean dynamics [17], optical fibers [18]-[22], photonic crystals [23] and plasmas [24]. Later on the soliton molecules which are the soliton bound states have attracted considerable attention. Soliton molecules have been observed in optical systems [25]-[29] and analyzed in Bose-Einstein condensates [30]. Various theoretical proposals to form soliton molecules have been established [31]-[33]. It is known that soliton molecules in coupled systems have been well discussed [34]. Recently, the breathing soliton molecules have been experimentally observed in a mode-locked fiber laser [28]. The resonant theory of solitons is applicable to a variety of integrable systems, such as the KP-(II) equation [35, 36] and Novikov-Veselov equation [37] in which Wronskian representation of the τ -function have been employed. By restricting different resonant conditions on the solitons, it may derive many types resonant excitations. The resonant solutions have been generated by using linear superposition principle and the Bell polynomials for the bilinear equations [38]. By virtue of Hirotas direct method and the Bäcklund transformation, the researchers study the fission and fusion of the solitary wave solution of the Burgers and STO equations [14]. Furthermore, the interaction between the lump soliton and the pair of resonance stripe solitons form a rogue wave solution [39]. The rational solutions to the modified Korteweg-de Vries equation have been analyzed by imposing the wave number resonance constraints [40]. Lately, by means of introducing velocity resonant mechanism, Lou investigated some types of soliton molecules in fluid systems [41] and nonlinear optical systems [42] involving the third-fifth order Korteweg de-Vries (KdV) equation, the KdV-Sawada-Kotera (KdVSK) equation, the KdV-Kaup-Kupershmidt (KdVKK) equation, the Hirota system and the potential modified KdV-sG system which derive kink-kink molecule, kink-antikink molecule, kink-breather molecule and breatherbreather molecule etc. Although the structure and properties of the integrable systems have been widely investigated,...

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Section: Introductionmentioning

confidence: 99%

“…The single soliton solutions of (13) and (14) with k 1 k 2 < 0 are the exact solutions of STOB equation. Then two solitary waves occur interaction which makes two solitary waves fuse to a resonant solution by (15).…”

Section: Introductionmentioning

confidence: 99%

Soliton molecules in Sharma–Tasso–Olver–Burgers equation

Yan

Sen-Yue

2020

Applied Mathematics Letters

115

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“…With relevance to multiwave solutions. [6][7][8][9][10][11][12][13] On the other hand, when the inhomogeneities of media are taken into account, that is the NLEE are considered with space variable coefficients. This case represents more realistic models than those which are considered with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

On multiple soliton similariton‐pair solutions, conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

Abdel‐Gawad

Tantawy

Yusuf

2019

Math Methods in App Sciences

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In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.

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“…the fusion behavior coexists with the elastic collision. Although elastic collision and solitary waves fusion have been investigated in [5][6][7][8][9][10] by many authors recently, to the authors' knowledge, the elastic-fusion-coupled interaction is a new phenomenon for the Boussinesq equation (1). The solutions with this property could be useful to describe the run-up of a tsunami wave on a vertical wall [11].…”

Section: Introductionmentioning

confidence: 99%