Wronskian, Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Huang, Qian-Min; Gao, Yi–Tian

doi:10.1007/s11071-017-3630-y

Cited by 49 publications

(15 citation statements)

References 28 publications

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“…In this paper, based on the Hirota's bilinear form and Mathematica software [25][26][27][28][29][30][31][32][33][34][35], the lump and interaction solutions between lump and solitary wave of a generalized (3 + 1)dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles are studied. Their physical structures are described in some 3D plots.…”

Section: Discussionmentioning

confidence: 99%

Interaction phenomena between lump and solitary wave of a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Liu¹,

Zhu²,

He³

et al. 2020

Commun. Theor. Phys.

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In this paper, a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation is studied in liquid with gas bubbles. Based on the Hirota’s bilinear form and symbolic computation, lump and interaction solutions between lump and solitary wave are obtained, which include a periodic-shape lump solution, a parabolic-shape lump solution, a cubic-shape lump solution, interaction solutions between lump and one solitary wave, and between lump and two solitary waves. The spatial structures called the bright lump wave and the bright-dark lump wave are discussed. Interaction behaviors of two bright-dark lump waves and a periodic-shape bright lump wave are also presented. Their interactions are shown in some 3D plots.

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Section: Discussionmentioning

confidence: 99%

Interaction phenomena between lump and solitary wave of a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Liu¹,

Zhu²,

He³

et al. 2020

Commun. Theor. Phys.

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“…Shallow water waves, known as one type of water waves with small depth relative to the water wavelength, have been studied for the applications in hydraulic engineering and environmental engineering [14][15][16][17][18][19]. In this paper, we will investigate the following (2+1)dimensional extended shallow water wave equation [20][21][22][23][24][25][26],…”

Section: Introductionmentioning

confidence: 99%

Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

Deng¹,

Gao²,

Yu³

et al. 2021

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Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2+1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H -soliton, M -breather and J -lump solutions have been constructed via the modified Pfaffian technique, where H , M and J are the positive integers. One-breather solutions with a real function ϕ ( y ) are derived, where y is the scaled space variable, we notice that ϕ ( y ) influences the shapes of the background planes. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ ( y ) are obtained with the condition, where k 1 R and k 1 I are the real constants, we notice that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ ( y ) is changed from a linear function to a periodic function.

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“…Periodic wave solutions were derived by Tang and Zai via the extended homoclinic test approach [31]. Wronskian, Pfaffian and periodic wave solutions were constructed by Huang and Gao [32]. Important contributions were also made on exact solutions of the (3+1)-dimensional BLMP equation in [33][34][35][36][37][38][39][40], especially on the lump-type and breather-type solutions.…”

Section: Introductionmentioning

confidence: 99%

Novel localized wave solutions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Sun¹,

Qi²,

An³

2020

Commun. Theor. Phys.

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Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.

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Wronskian, Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Cited by 49 publications

References 28 publications

Interaction phenomena between lump and solitary wave of a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Interaction phenomena between lump and solitary wave of a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

Novel localized wave solutions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

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