scholarly journals Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 1-10
Author(s):  
Aly R. Seadawy ◽  
Syed Tahir Raza Rizvi ◽  
Sarfraz Ahmad ◽  
Muhammad Younis ◽  
Dumitru Baleanu
Keyword(s):  
Liquid Crystal ◽  
Bilinear Form ◽  
Three Dimensional ◽  
Quadratic Function ◽  
Trigonometric Function ◽  
Director Field ◽  
Wave Dynamics ◽  
Hirota Bilinear Form ◽  
Wave Phenomena ◽  
Hirota Bilinear

Abstract The aim of this article was to address the lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation with the aid of Hirota bilinear technique. This model concerns in a massive nematic liquid crystal director field. By choosing the function f in Hirota bilinear form, as the general quadratic function, trigonometric function and exponential function along with appropriate set of parameters, we find the lump, lump-one stripe, multiwave and breather solutions successfully. We also interpreted some three-dimensional and contour profiles to anticipate the wave dynamics. These newly obtained solutions have some arbitrary constants and so can be applicable to explain diversity in qualitative features of wave phenomena.

Interaction phenomenon to dimensionally reducedp-gBKP equation

2018 ◽  
Vol 32 (06) ◽  
pp. 1850074 ◽  
Author(s):  
Runfa Zhang ◽  
Sudao Bilige ◽  
Yuexing Bai ◽  
Jianqing Lü ◽  
Xiaoqing Gao
Keyword(s):  
Three Dimensional ◽  
Quadratic Function ◽  
Cosine Function ◽  
Interaction Solutions ◽  
Hyperbolic Cosine ◽  
Hirota Bilinear Form ◽  
Dynamical Features ◽  
Hyperbolic Cosine Function ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

Based on searching the combining of quadratic function and exponential (or hyperbolic cosine) function from the Hirota bilinear form of the dimensionally reduced p-gBKP equation, eight class of interaction solutions are derived via symbolic computation with Mathematica. The submergence phenomenon, presented to illustrate the dynamical features concerning these obtained solutions, is observed by three-dimensional plots and density plots with particular choices of the involved parameters between the exponential (or hyperbolic cosine) function and the quadratic function. It is proved that the interference between the two solitary waves is inelastic.

scholarly journals Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-5 ◽  
Author(s):  
Yanni Zhang ◽  
Jing Pang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Three Dimensional ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Contour Plots ◽  
Dimensional Variable ◽  
Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

Lump, rogue wave, multi-waves and Homoclinic breather solutions for (2+1)-Modified Veronese Web equation

2021 ◽  
Vol 35 (04) ◽  
pp. 2150055
Author(s):  
S. T. R. Rizvi ◽  
Aly R. Seadawy ◽  
S. Ahmed ◽  
M. Younis ◽  
K. Ali
Keyword(s):  
Rogue Wave ◽  
Nonlinear Partial Differential Equation ◽  
Quadratic Function ◽  
Wave Transformation ◽  
Physical Parameters ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
Wave Phenomena ◽  
Hirota Bilinear ◽  
Linearly Degenerate

This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function [Formula: see text] in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

Dynamical analysis of solitary waves, lumps and interaction phenomena of a (2+1)-dimensional high-order nonlinear evolution equation

2019 ◽  
Vol 33 (16) ◽  
pp. 1950181 ◽  
Author(s):  
Bo Ren ◽  
Zhi-Mei Lou ◽  
Yong-Li Sun ◽  
Zhi-Wei He
Keyword(s):  
Solitary Waves ◽  
Bilinear Form ◽  
Nonlinear Evolution ◽  
Quadratic Function ◽  
High Order ◽  
Dynamical Analysis ◽  
Interaction Solutions ◽  
Variable Function ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A (2[Formula: see text]+[Formula: see text]1)-dimensional high-order nonlinear evolution (HNE) equation is considered in this paper. A Hirota bilinear form of the HNE equation is constructed by the dependent variable function. Solitary waves are derived by solving the Hirota bilinear form of the HNE equation. Lump waves of the HNE equation are obtained by introducing a positive quadratic function. By mixing an exponential function or two exponential functions with a quadratic function, interaction solutions between a lump and a one-soliton, and between a lump and a two-soliton are presented. For the interaction solution between a lump and a two-soliton, this kind of solution can be considered as a special rogue wave. The propagation phenomena of these explicit solutions are illustrated by some graphs.

scholarly journals Evolution Behaviour of Kink Breathers and Lump-M-Solitons (M → ∞) for the (3+1)-Dimensional Hirota-Satsuma-Ito-Like Equation

2021 ◽  
Author(s):  
Long-Xing Li
Keyword(s):  
Exponential Type ◽  
Bilinear Form ◽  
Nonlinear Dynamic ◽  
Wave Solution ◽  
Three Dimensional ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
The Given ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

Abstract In this paper, some novel lump solutions and interaction phenomenon between lump and kink M-soliton are investigated. Firstly, we study the evolution and degeneration behaviour of kink breather wave solution with difffferent forms for the (3+1)-dimensional Hirota-Satsuma-Ito-like equation by symbolic computation and Hirota bilinear form. In the process of degeneration of breather waves, some novel lump solutions are derived by the limit method. In addition, M-fifissionable soliton and the interaction phenomenon between lump solutions and kink M-solitons (lump-M-solitons) are investigated, the theorem and corollary about the conditions for the existence of the interaction phenomenon are given and proved further. The lump-M-solitons with difffferent types is studied to illustrate the correctness and availability of the given theorem and corollary, such as lump-cos type, lump-cosh-exponential type, lump cosh-cos-cosh type. Several three-dimensional fifigures are drawn to better depict the nonlinear dynamic behaviours including the oscillation of breather wave, the emergence of lump, the evolution behaviour of fission and fusion of lump-M-solitons and so on.

Lump solutions to a (2+1)-dimensional fourth-order nonlinear PDE possessing a Hirota bilinear form

2021 ◽  
pp. 2150160
Author(s):  
Wen-Xiu Ma ◽  
Solomon Manukure ◽  
Hui Wang ◽  
Sumayah Batwa
Keyword(s):  
Nonlinear Equation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Three Dimensional ◽  
Nonlinear Pde ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
Bilinear Formulation ◽  
Order Nonlinear Equation ◽  
Hirota Bilinear

Through the Hirota bilinear formulation, a (2+1)-dimensional combined fourth-order nonlinear equation is proposed, which possesses lump solutions. Two classes of lump solutions are presented explicitly in terms of the coefficients in the combined nonlinear equation. A set of examples of equations is provided to show the diversity of the considered combined nonlinear equation, together with three-dimensional plots, [Formula: see text]-curves and [Formula: see text]-curves of two specific lump solutions in two cases of the combined equation.

scholarly journals Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Xian Li ◽  
Yao Wang ◽  
Meidan Chen ◽  
Biao Li
Keyword(s):  
Symbolic Computation ◽  
Exponential Function ◽  
Bilinear Form ◽  
Quadratic Function ◽  
Hyperbolic Function ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hirota Bilinear Form ◽  
Dynamical Properties ◽  
Hirota Bilinear

Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters.

Characteristics of the breather waves, lump waves and semi-rational solutions in a generalized (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

2019 ◽  
Vol 33 (28) ◽  
pp. 1950350 ◽  
Author(s):  
Wei-Qi Peng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Rational Solutions ◽  
Bell Polynomial ◽  
Long Wave ◽  
Hirota Bilinear Form ◽  
Wave Limit ◽  
Parameter Constraints ◽  
Hirota Bilinear

In this work, we study a generalized (2[Formula: see text]+[Formula: see text]1)-dimensional asymmetrical Nizhnik–Novikov–Veselov (NNV) equation. Its Hirota bilinear form is constructed via the Bell polynomial. Based on the obtained bilinear form, the Nth-order breather waves are derived explicitly under certain parameter constraints. Moreover, we generate the nonsingular Nth-order lump waves through applying the long wave limit method. Additionally, we successfully present the semi-rational waves containing the combination of lump waves and single-soliton waves, the combination of lump waves and breather waves.

Lump-type solutions for the (4+1)-dimensional Fokas equation via symbolic computations

2017 ◽  
Vol 31 (25) ◽  
pp. 1750224 ◽  
Author(s):  
Li Cheng ◽  
Yi Zhang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Symbolic Computations ◽  
Hirota Bilinear Form ◽  
Free Parameters ◽  
Almost All ◽  
Hirota Bilinear

Based on the Hirota bilinear form, two classes of lump-type solutions of the (4[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Fokas equation, rationally localized in almost all directions in the space are obtained through a direct symbolic computation with Maple. The resulting lump-type solutions contain free parameters. To guarantee the analyticity and rational localization of the solutions, the involved parameters need to satisfy certain constraints. A few particular lump-type solutions with special choices of the involved parameters are given.

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