Various kinds of high-order solitons to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

Abstract

In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation which is used to describe the wave phenomenon in fluid mechanics is investigated. Based on the bilinear representation, perturbation method and Taylor expansion approach, we derive various kinds of high-order solitons including the N-kink soliton, n-order lump-type soliton and mixture solution of kink soliton and lump-type soliton. First, N-kink soliton solution is obtained by the bilinear representation and perturbation method. Second, by using the Taylor expansion approach for the 2n-kink soliton solution, n-order lump-type soliton is obtained. Third, by mean of the Taylor expansion approach for 2n-kink soliton solution in the N-kink soliton solution (1 < 2n < N), we construct the mixture solution consisting of (N − 2n)-kink soliton and n-order lump-type soliton. Interestingly, the collision between kink soliton and lump-type soliton can give rise to a high-order lump-type soliton.

On integrability of an extended Bogoyavlenskii‐Kadomtsev‐Petviashvili equation: Multiple soliton solutions

“…where u k (x, y, t) 0 s (k = 0,1,2, …) are the functions of x, y, and t. Substituting (7) in Equation 3, and following the Painlevé analysis, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] we obtain the characteristic equation with five resonances at k = −1, 1, 4, 5, and 6. The resonance at k = −1 is related to singular manifold ψ(x, y, t) = 0.…”

“…For α ≠ 0, the equation is a modification of the CBS equation, also understood as a modification of the Kadomtsev-Petviashvili equation (KP). [1][2][3][4][5][6][7][8] This equation was derived in Reference 1 by a reduction of the well-known (3 + 1)dimensional KP equation. The BKP equation (1) describes the propagation of nonlinear waves in a variety of scientific fields such as plasmas, fluid dynamics, shallow waters waves, tsunami, and many others.…”

“…Many systematic powerful methods are used to study the nonlinear equations, integrable and nonintegrable. The generalized symmetry method, Painlevé analysis, the inverse scattering method, the Bäcklund transformation method, the conservation law method, the Cole-Hopf transformation, Bell polynomials method, tanh method, and the Hirota bilinear method [3][4][5][6][7][8][9][10][11][12][13][14][15][16] are mostly used in addition to other techniques. The Hirota's bilinear method is mostly used for the determination of multiple soliton solutions for a vast number of nonlinear equations.…”

Lump-Type Wave and Interaction Solutions of the Bogoyavlenskii–Kadomtsev–Petviashvili Equation

“…In this paper, we will devote to investigating the Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation [25][26][27][28][29][30][31][32]:…”

“…e bilinear structures and multiple wave solutions have been constructed by means of the binary Bell polynomials method [31]. In [32], Wang and Fang have also investigated various kinds of high-order solitons by employing the perturbation method and Taylor expansion approach. In this paper, we will mainly investigate the lumptype wave solution of the BKP equation 1by the Hermitian quadratic form.…”