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 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.

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.

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.

Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed K. Elboree
Keyword(s):  
Symbolic Computation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Rogue Waves ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

Abstract Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

scholarly journals A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Jie Li ◽  
Chaudry Masood Khalique
Keyword(s):  
Three Dimensional ◽  
Lump Solution ◽  
Lump Solutions ◽  
Solution Structures ◽  
New Class ◽  
Contour Plots ◽  
Bilinear Formulation ◽  
Hirota Bilinear ◽  
Interesting Solution ◽  
Ito Equation

The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.

scholarly journals Abundant Symmetry-Breaking Solutions of the Nonlocal Alice–Bob Benjamin–Ono System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wang Shen ◽  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Quan-Yong Zhu ◽  
Jun-Chao Chen
Keyword(s):  
Symmetry Breaking ◽  
Gravity Waves ◽  
Bilinear Form ◽  
Time Reversal ◽  
Internal Gravity Waves ◽  
Explicit Solutions ◽  
Time Reversal Symmetry ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

The Benjamin–Ono equation is a useful model to describe the long internal gravity waves in deep stratified fluids. In this paper, the nonlocal Alice–Bob Benjamin–Ono system is induced via the parity and time-reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry-breaking soliton, breather, and lump solutions for this system are obtained through the derived Hirota bilinear form. By taking suitable constants in the involved ansatz functions, abundant fascinating symmetry-breaking structures of the related explicit solutions are shown.

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.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

2021 ◽  
pp. 2150437
Author(s):  
Liyuan Ding ◽  
Wen-Xiu Ma ◽  
Yehui Huang
Keyword(s):  
Symbolic Computation ◽  
Three Dimensional ◽  
Order Derivative ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Behaviors ◽  
Contour Plots ◽  
Hirota Bilinear ◽  
Ito Equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

scholarly journals Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

2021 ◽  
Author(s):  
Hongcai Ma ◽  
Shupan Yue ◽  
Yidan Gao ◽  
Aiping Deng
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bilinear Method ◽  
Test Function ◽  
Lump Solutions ◽  
Hirota Bilinear ◽  
Test Function Method

Abstract Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

scholarly journals Multiple Lump Solutions of the (4+1)-Dimensional Fokas Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Hongcai Ma ◽  
Yunxiang Bai ◽  
Aiping Deng
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Three Dimensional ◽  
Structural Features ◽  
Computation Method ◽  
Lump Solutions ◽  
Wave Solutions ◽  
Lump Wave

In this paper, we investigate multiple lump wave solutions of the new (4+1)-dimensional Fokas equation by adopting a symbolic computation method. We get its 1-lump solutions, 3-lump solutions, and 6-lump solutions by using its bilinear form. Moreover, some basic characters and structural features of multiple lump waves are explained by depicting the three-dimensional plots.

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