2023
DOI: 10.1088/1674-1056/ac935b
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Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Abstract: 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… Show more

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Cited by 6 publications
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“…[1][2][3][4][5][6][7][8][9] In the shallow water, internal solitary waves can be described by the Korteweg-de Vries (KdV) equation to the first order. [10][11][12][13][14] A useful variant of the KdV equation is the extended KdV (eKdV) [15] η t + (c 0 + α 1 η + α 2 η 2 )η x + β 1 η xxx = 0, (1) where η(x,t) is the wave amplitude related to the isopycnic vertical displacement, t is the time variable, x is the spatial variable in the direction of wave propagation, the coefficients α 1 , α 2 , and β 1 are functions of the steady background stratification and shear through the linear eigenmode of interest, the linear phase speed c 0 is the eigenvalue of the Sturm-Louiville problem for the eigenmode, while the subscripts denote the partial derivatives. [16] For a two-layer fluid, the coefficients of Eq.…”
Section: Introductionmentioning
confidence: 99%
“…[1][2][3][4][5][6][7][8][9] In the shallow water, internal solitary waves can be described by the Korteweg-de Vries (KdV) equation to the first order. [10][11][12][13][14] A useful variant of the KdV equation is the extended KdV (eKdV) [15] η t + (c 0 + α 1 η + α 2 η 2 )η x + β 1 η xxx = 0, (1) where η(x,t) is the wave amplitude related to the isopycnic vertical displacement, t is the time variable, x is the spatial variable in the direction of wave propagation, the coefficients α 1 , α 2 , and β 1 are functions of the steady background stratification and shear through the linear eigenmode of interest, the linear phase speed c 0 is the eigenvalue of the Sturm-Louiville problem for the eigenmode, while the subscripts denote the partial derivatives. [16] For a two-layer fluid, the coefficients of Eq.…”
Section: Introductionmentioning
confidence: 99%
“…Numerous natural scenarios, such as nonlinear internal ocean waves that are stratified by density and current [2] and dispersion of small amplitude solutions [3], can be described by using a KdV equation. Subsequently, the KdV equation was further developed through the Miura transformation, leading to the famous modified Korteweg–de Vries (mKdV) equation [4]. Additionally, extensions of the mKdV equation to complex fields is well implemented in fluid mechanics, nonlinear optic, and dynamical systems [5].…”
Section: Introductionmentioning
confidence: 99%