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.

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.

Painlevé Integrability and N-Soliton Solution for the Whitham- Broer-Kaup Shallow Water Model Using Symbolic Computation

2008 ◽  
Vol 63 (5-6) ◽  
pp. 253-260 ◽  
Author(s):  
Cheng Zhang ◽  
Bo Tian ◽  
Xiang-Hua Meng ◽  
Xing Lü ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Water Model ◽  
Shallow Water Model ◽  
Graphic Analysis ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Singular Manifolds

With the help of symbolic computation, the Whitham-Broer-Kaup shallow water model is analyzed for its integrability through the Painlev´e analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the Hirota bilinear form is obtained and the corresponding N-soliton solution with graphic analysis is also given. Furthermore, a bilinear auto-Bäcklund transformation is constructed for the Whitham-Broer-Kaup model, from which a one-soliton solution is presented.

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.

Lump-Type Solutions to the (3+1)-Dimensional Jimbo-Miwa Equation

2016 ◽  
Vol 17 (7-8) ◽  
pp. 355-359 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

Abstract:Taking advantage of the Hirota bilinear form, four classes of lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation are presented through symbolic computation with Maple. Special choices of the involved parameters guaranteeing analyticity of the fourth solution are given, together with two particular lump-type solutions.

Abundant mixed lump-kink solutions to a (2+1)-dimensional bSK equation

2018 ◽  
Vol 32 (26) ◽  
pp. 1850313 ◽  
Author(s):  
Zeguang Liu
Keyword(s):  
Bilinear Form ◽  
Logarithmic Transformation ◽  
Symbolic Computations ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear Equation ◽  
Simulation Results ◽  
Hirota Bilinear ◽  
Bilinear Equation

In this paper, we study lump-kink solutions of a (2+1)-dimensional bidirectional Sawada–Kotera equation and discuss their dynamics. A Hirota bilinear form of a (2+1)-dimensional bidirectional Sawada–Kotera equation is deduced via a dependent logarithmic transformation. Based on this Hirota bilinear equation, we obtain eight classes of lump-kink solutions which combine stripe soliton and lump soliton by using symbolic computations. Our simulation results with the appropriate choice of the arbitrary parameters that show the motion of lump soliton and the process of interaction between lump soliton and a stripe soliton.

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 solutions to the BKP equation by symbolic computation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640028 ◽  
Author(s):  
Jing-Yun Yang ◽  
Wen-Xiu Ma
Keyword(s):  
Symbolic Computation ◽  
Kp Equation ◽  
Lump Solutions ◽  
Positivity Condition ◽  
Independent Variables ◽  
Computation Process ◽  
Starting Point ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Bkp Equation

Lump solutions are rationally localized in all directions in the space. A general class of lump solutions to the (2+1)-dimensional B-Kadomtsev–Petviashvili (BKP) equation is presented through symbolic computation with Maple. The Hirota bilinear form of the equation is the starting point in the computation process. Like the KP equation, the resulting lump solutions contain six arbitrary parameters. Two of the parameters are due to the translation invariances of the BKP equation with the independent variables, and the other four need to satisfy a nonzero determinant condition and the positivity condition, which guarantee analyticity and rational localization of the solutions.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

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