New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

Abstract

We introduce a new form of the Painlevé integrable (3+1)-dimensional combined potential Kadomtsev--Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation (pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new (3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new (3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined pKP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

“…In according to Hirota's bilinear method, one-, two-, three-and four-soliton solutions are obtained. Equation (2) is evidently integrable with respect to N-soliton solutions, and the complete integrability of this combined integrable equation has been performed through Painlevé analysis [107]. In addition, by taking the complex conjugate conditions to the soliton solutions and imposing restrictions, one breath-wave and two breath-wave solutions can be gained.…”

Multiple solitons with fission and multi waves interaction solutions of a (3+1)-dimensional combined pKP-BKP integrable equation

“…In this section, we initiate our investigation by considering the soliton solution of order one for equation (5), as outlined in [3]. The constraint condition takes the following form:…”

“…So, we use the Hirota bilinear form (HBF) for the model under consideration. The HBF of equation ( 5) was determined by the authors in [3] through the utilization of…”

“…where c 5 ≠ 0 and c j are constants for j = 1, 2, L 6. Very recently, Wazwaz introduced a new combined version of pKP-BKP equation in (3+1)-dimension which is presented as[3] …”

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

“…Nonlinear evolution equations have been used to describe the nonlinear physical phenomena in such fields as fluid mechanics, plasma physics, optical fibres, and solid state physics. [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.…”