Lump soliton, mixed lump stripe and periodic lump solutions of a (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Zhao, Zhonglong; Chen, Yong; Han, Bo

doi:10.1142/s0217984917501573

Cited by 109 publications

(31 citation statements)

References 33 publications

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“…For instance, the Hirota's bilinear transformation, the generalized bilinear transformation, the inverse‐scattering transformation, the Painlevé analysis approach, and the Darboux transformation . Recently, the lump‐type and mixed‐lump‐type solutions, analytical and rationally localized, of NPDEs have been extensively studied by many researchers . However, there is no systematic study on the lump solutions of any differential‐difference equations, such as the Toda equation.…”

Section: Introductionmentioning

confidence: 99%

Lump solutions of the 2D Toda equation

Sun

2020

Math Methods in App Sciences

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In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method.

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Section: Introductionmentioning

confidence: 99%

Lump solutions of the 2D Toda equation

Sun

2020

Math Methods in App Sciences

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“…In 2015, Ma investigated the lump solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation by means of the bilinear forms and positive quadratic functions [13]. Inspired by this work, many researchers studied the lump solutions of the integrable systems, such as the (2+1)-dimensional Sawada-Kotera equation [14], the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [15], the (3+1)dimensional B-type KP equation [16], and the generalized Calogero-Bogoyavlenskii-Schiff equation [17]. Furthermore, the generalized bilinear forms can also be used to derive the lump solutions [18].…”

Section: Introductionmentioning

confidence: 99%

Rogue Wave and Multiple Lump Solutions of the (2+1)‐Dimensional Benjamin‐Ono Equation in Fluid Mechanics

Zhao

Gao

2019

Complexity

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In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8lump solutions are presented, respectively. The 3-lump wave has a "triangular" structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.

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“…(1) based on a multi-dimensional Riemann theta function and Hirota's bilinear method. Zhao et al [47] have presented the lump stripe solution of Eq. (1) by using bilinear form.…”

Section: Introductionmentioning

confidence: 99%

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Liu

Wen

2019

Adv Differ Equ

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In this work, the (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation is investigated. Hirota's bilinear method is used to determine the N-soliton solutions for this equation, from which the M-lump solutions are obtained by using long wave limit when N is even (i.e., N = 2M). Then, taking N = 5 as an example, we discuss some novel mixed lump-soliton and lump-soliton-breather solutions by using long wave limit and choosing special conjugate complex parameters from the five-soliton solution. Figures are plotted to reveal the dynamical features of such obtained lump and mixed interaction solutions. These results may be useful for understanding the propagation phenomena of nonlinear localized waves.

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Lump soliton, mixed lump stripe and periodic lump solutions of a (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Cited by 109 publications

References 33 publications

Lump solutions of the 2D Toda equation

Lump solutions of the 2D Toda equation

Rogue Wave and Multiple Lump Solutions of the (2+1)‐Dimensional Benjamin‐Ono Equation in Fluid Mechanics

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

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