Darboux transformation and soliton solution for the (2+1)-dimensional complex modified Korteweg-de Vries equations

Yesmakhanova, Kuralay; Shaikhova, Gaukhar; Bekova, Guldana; Myrzakulov, Ratbay

doi:10.1088/1742-6596/936/1/012045

Cited by 26 publications

(9 citation statements)

References 10 publications

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“…Then, by formulas (21), (22), we finally obtain solutions of the mVN equations (16,17) for the inverted catenoid → , ,…”

Section: Proof Mkdv Solutionmentioning

confidence: 99%

“…The obtained results can also be applied in the physical sciences by considered as [12], [13], [16], [17]. [6], [7] жұмыстарында табылған геометриялық мағынасы минималды беттiң инверсиясына анықтама бередi, ары қарай мВН теңдеуiнiң нақты шешiмдерiн табу арқылы инверсияланған беттердi сипаттай аламыз.…”

Section: Proof Mkdv Solutionunclassified

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Soliton Deformation of Inverted Catenoid

Kurmanbayev

Yesmakhanova

2021

News of NAN RK. SGTS

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The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

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“…Then, by formulas (21), (22), we finally obtain solutions of the mVN equations (16,17) for the inverted catenoid → , ,…”

Section: Proof Mkdv Solutionmentioning

confidence: 99%

Section: Proof Mkdv Solutionunclassified

Soliton Deformation of Inverted Catenoid

Kurmanbayev

Yesmakhanova

2021

News of NAN RK. SGTS

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“…Starting with the zero seed, the soliton and deform soliton solutions have been obtained in Refs. [26,27]. On the other hand, the periodic solutions are studied in Refs.…”

Section: Introductionmentioning

confidence: 99%

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Yuan

Ghanbari

2023

Chinese Phys. B

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Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method has been used to derive several positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg-de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied which shows that multi-positons of (2+1)-dimensional equations are decomposed into multi-solitons as well as the (1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

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“…Nonlinear partial differential equations are broadly used to model nonlinear processes in many areas of mathematical biology, physics, chemistry [18,1]. As a result of interest in those problems, different analytical solution methods as Hirota's bilinear method [6,7], Darboux transformation method [10,21], sine-cosine method [19,17], hyperbolic tangent method [11,12] and so on were developed.…”

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confidence: 99%

Exact Solutions of the Lakshmanan – Porsezian – Daniel Equation

Shaikhova¹,

Syzdykova²,

Кудайбергенов³

2022

ПМ&Ф

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In this paper, the Lakshmanan-Porsezian-Daniel (LPD) equation is considered. This equation is integrable and admits Lax pair. The LPD equation is the generalization of the nonlinear Schrodinger (NLS) equation and described by Ablowitz-Kaup-Newell-Segur (AKNS) system. Using the sine-cosine method and the hyperbolic tangent method a variety of new exact solutions are obtained. These methods are effective tools for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions are found to be important for the explanation of some practical physical problems.

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Darboux transformation and soliton solution for the (2+1)-dimensional complex modified Korteweg-de Vries equations

Cited by 26 publications

References 10 publications

Soliton Deformation of Inverted Catenoid

Soliton Deformation of Inverted Catenoid

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Exact Solutions of the Lakshmanan – Porsezian – Daniel Equation

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