Rational solutions and their interactions with kink and periodic waves for a nonlinear dynamical phenomenon

2021 ◽  
pp. 2150236
Author(s):  
Aly R. Seadawy ◽  
Syed T. R. Rizvi ◽  
M. Aamir Ashraf ◽  
Muhammad Younis ◽  
Maria Hanif
Keyword(s):  
Analytical Solutions ◽  
Soliton Solutions ◽  
Periodic Waves ◽  
Trigonometric Functions ◽  
Rational Solutions ◽  
Lump Solutions ◽  
Nonlinear Dynamical ◽  
Interaction Phenomenon

Lump (rational) waves and their interactions with kink and periodic waves, periodic cross-lump solutions will be discussed for (2+1)-dimensional Maccari-system in this paper. With the combination of rational, exponential, and trigonometric functions, we will study various lump soliton solutions. We will find out analytical solutions with interaction phenomenon and also describe them in graphical ways.

General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

2020 ◽  
pp. 2150057
Author(s):  
Xin-Mei Zhou ◽  
Shou-Fu Tian ◽  
Ling-Di Zhang ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Soliton Solutions ◽  
High Order ◽  
Rational Solutions ◽  
Lump Solutions ◽  
Long Wave ◽  
Wave Limit

In this work, we investigate the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation. Based on its bilinear form, the [Formula: see text]th-order breather solutions of the gBK equation are successful given by taking appropriate parameters. Furthermore, the [Formula: see text]th-order lump solutions of the gBK equation are obtained via the long-wave limit method. In addition, the semi-rational solutions are generated to reveal the interaction between lump solutions, soliton solutions, and breather solutions.

Lump solitons and interaction phenomenon to a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

2019 ◽  
Vol 33 (32) ◽  
pp. 1950395 ◽  
Author(s):  
Na Liu ◽  
Yansheng Liu
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Nonlinear Waves ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hirota’S Bilinear Form ◽  
Interaction Phenomenon

This paper studies lump solutions and interaction solutions for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. With the help of symbolic computation and Hirota’s bilinear form, we obtain bright–dark lump solutions, lump-soliton solutions, and lump-kink solutions. Meanwhile, the dynamics of the obtained three classes of solutions are analyzed and exhibited mathematically and graphically. These results provide us with useful information to grasp the propagation processes of nonlinear waves.

The discrete tanh method for solving the nonlinear differential-difference equations

2020 ◽  
Vol 34 (19) ◽  
pp. 2050177 ◽  
Author(s):  
A. Houwe ◽  
M. Inc ◽  
S. Y. Doka ◽  
B. Acay ◽  
L. V. C. Hoan
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Order Derivative ◽  
Trigonometric Functions ◽  
Rational Solutions ◽  
Fractional Order Derivative ◽  
Tanh Method

We investigated analytical solutions for the nonlinear differential-difference equations (DDEs) having fractional-order derivative. We employed the discrete tanh method in computations. Performance of trigonometric functions, dark one solitons and rational solutions are discussed in detail. The results with reliable parameters are illustrated via 2-D and 3-D graphs.

Lump, mixed lump-soliton, and periodic lump solutions of a (2+1)-dimensional extended higher-order Broer–Kaup System

2020 ◽  
Vol 34 (33) ◽  
pp. 2050384
Author(s):  
Fan Guo ◽  
Ji Lin
Keyword(s):  
Rogue Wave ◽  
Soliton Solutions ◽  
Elastic Collision ◽  
Auxiliary Function ◽  
Higher Order ◽  
Trigonometric Functions ◽  
Resonant Line ◽  
Lump Solutions ◽  
Truncated Painlevé Expansion ◽  
Line Soliton

In this paper, a (2+1)-dimensional extended higher-order Broer–Kaup system is introduced and its bilinear form is presented from the truncated Painlevé expansion. By taking the auxiliary function as the ansatzs including quadratic, exponential, and trigonometric functions, lump, mixed lump-soliton, and periodic lump solutions are derived. The mixed lump-soliton solutions are classified into two cases: the first one describes the non-elastic collision between one lump and one line soliton, which exhibits fission and fusion phenomena. The second one depicts the interaction consisting of one lump and two line soliton, which generates a rogue wave excited from two resonant line solitons.

Chirped and chirp-free optical solitons for Heisenberg ferromagnetic spin chains model

2021 ◽  
pp. 2150139
Author(s):  
Syed Tahir Raza Rizvi ◽  
Aly R. Seadawy ◽  
Ishrat Bibi ◽  
Muhammad Younis
Keyword(s):  
Graphical Representation ◽  
Spin Dynamics ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Optical Solitons ◽  
Spin Chains ◽  
Solitary Wave Solutions ◽  
Rational Solutions ◽  
Wave Solutions ◽  
Ode Method

In this paper, we study (2+1)-dimensional non-linear spin dynamics of Heisenberg ferromagnetic spin chains equation (HFSCE) for various soliton solutions. We obtain two types of optical solitons i.e. chirp free and chirped solitons. We obtain bright and bright-like soliton, singular-like solitons, periodic and rational solutions, Weierstrass elliptic functions solutions and other solitary wave solutions for HFSCE with the aid of sub-ODE method. At the end, we present graphical representation of our solutions.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

2019 ◽  
Vol 20 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jianqing Lü ◽  
Sudao Bilige ◽  
Xiaoqing Gao
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Special Case ◽  
Interaction Phenomenon

AbstractIn this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

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.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

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