ANALYTICAL INVESTIGATION OF THE CAUDREY–DODD–GIBBON–KOTERA–SAWADA EQUATION USING SYMBOLIC COMPUTATION

2013 ◽  
Vol 27 (06) ◽  
pp. 1250124 ◽  
Author(s):  
XIAO-GE XU ◽  
XIANG-HUA MENG ◽  
CHUN-YI ZHANG ◽  
YI-TIAN GAO

In this paper, the Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation is analytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its N-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Bäcklund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation.

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  

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.


2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.


2019 ◽  
Vol 33 (05) ◽  
pp. 1950019 ◽  
Author(s):  
Iftikhar Ahmed ◽  
Aly R. Seadawy ◽  
Dianchen Lu

In this study, based on the Hirota bilinear method, mixed lump-solitons, periodic lump and breather soliton solutions are derived for (2 + 1)-dimensional extended KP equation with the aid of symbolic computation. Furthermore, dynamics of these solutions are explained with 3d plots and 2d contour plots by taking special choices of the involved parameters. Through the mixed lump-soliton solutions, we observe two fusion phenomena, first from interaction of lump and single soliton and other from interaction of lump with two solitons. In both cases, lump moves gradually towards soliton and transfers energy until it completely merges with the solitons. We also observe new characteristics of periodic lump solutions and kinky breather solitons.


2012 ◽  
Vol 424-425 ◽  
pp. 564-567
Author(s):  
Bang Qing Li ◽  
Cong Wang

Applying a symbolic computation algorithm, namely, the improved Hirota bilinear method, a new type of the N-soliton solutions is obtained for the (2+1)-dimensional nonlinear dissipative Zabolotskaya-Khokhlov system. The solutions can be expressed explicitly. Furthermore, the evolution process is investigated for the N-soliton solutions


2021 ◽  
pp. 2150437
Author(s):  
Liyuan Ding ◽  
Wen-Xiu Ma ◽  
Yehui Huang

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.


Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.


2009 ◽  
Vol 87 (12) ◽  
pp. 1227-1232 ◽  
Author(s):  
Abdul-Majid Wazwaz

In this work we study two systems of coupled KdV and coupled KP equations. The Hirota bilinear method is applied to show that these two systems are completely integrable. Multiple-soliton solutions and multiple singular-soliton solutions are derived for each system. The resonance phenomenon is examined as well.


2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-guang Cheng ◽  
Biao Li ◽  
Yong Chen

The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, theN-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.


2021 ◽  
Vol 6 (10) ◽  
pp. 11046-11075
Author(s):  
Wen-Xin Zhang ◽  
◽  
Yaqing Liu

<abstract><p>In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.</p></abstract>


2020 ◽  
pp. 2150060
Author(s):  
Wen-Tao Huang ◽  
Cheng-Cheng Zhou ◽  
Xing Lü ◽  
Jian-Ping Wang

Under investigation in this paper is the dynamics of dispersive optical solitons modeled via the Schrödinger–Hirota equation. The modulation instability of solutions is firstly studied in the presence of a small perturbation. With symbolic computation, the one-, two-, and three-soliton solutions are obtained through the Hirota bilinear method. The propagation and interaction of the solitons are simulated, and it is found the collision is elastic and the solitons enjoy the particle-like interaction properties. In the end, the asymptotic behavior is analyzed for the three-soliton solutions.


Sign in / Sign up

Export Citation Format

Share Document