scholarly journals Binary Bell polynomials, Hirota bilinear approach to Levi equation

2017 ◽  
Vol 293 ◽  
pp. 565-574 ◽  
Author(s):  
Yaning Tang ◽  
Weijian Zai ◽  
Siqiao Tao ◽  
Qing Guan
Keyword(s):  
Bell Polynomials ◽  
Binary Bell Polynomials ◽  
Hirota Bilinear ◽  
Bilinear Approach

scholarly journals Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-guang Cheng ◽  
Biao Li ◽  
Yong Chen
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Dimensional Variable ◽  
Binary Bell Polynomials ◽  
Hirota Bilinear

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.

Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations

2017 ◽  
Vol 72 (10) ◽  
pp. 963-972
Author(s):  
Xiao-Yu Wu ◽  
Bo Tian ◽  
Lei Liu ◽  
Yan Sun
Keyword(s):  
Lattice Spacing ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Bound State ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bright Solitons ◽  
Dark Solitons ◽  
Spacing Parameter ◽  
Binary Bell Polynomials

AbstractUnder investigation in this paper are the coupled Ablowitz–Ladik equations, which are linked to the optical fibres, waveguide arrays, and optical lattices. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation. Bright/dark one- and two-soliton solutions are also obtained. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright solitons increase as the value of the lattice spacing increases. Increasing value of the lattice spacing can lead to the increase of both the bright solitons’ amplitudes and velocities, and the decrease of the velocities of the dark solitons. The lattice spacing parameter has no effect on the amplitudes of the dark solitons. Overtaking interaction between the unidirectional bright two solitons and a bound state of the two equal-velocity solitons is presented. Overtaking interaction between the unidirectional dark two solitons and the two parallel dark solitons is also plotted.

scholarly journals Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Xuejun Zhou ◽  
Onur Alp Ilhan ◽  
Fangyuan Zhou ◽  
Sutarto Sutarto ◽  
Jalil Manafian ◽  
...  
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Regularity Of Solutions ◽  
Bell Polynomials ◽  
Soliton Theory ◽  
Dimensional Variable ◽  
Binary Bell Polynomials

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

Solitons and integrability for a (2+1)-dimensional generalized variable-coefficient shallow water wave equation

2017 ◽  
Vol 31 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Ya-Le Wang ◽  
Yi-Tian Gao ◽  
Shu-Liang Jia ◽  
Zhong-Zhou Lan ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Shallow Water Wave ◽  
Binary Bell Polynomials

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg–de Vries (KdV) equation and the Calogero–Bogoyavlenskii–Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Abundant interaction between lump and k-kink, periodic and other analytical solutions for the (3+1)-D Burger system by bilinear analysis

2021 ◽  
Vol 35 (13) ◽  
pp. 2150173
Author(s):  
Na Zhao ◽  
Jalil Manafian ◽  
Onur Alp Ilhan ◽  
Gurpreet Singh ◽  
Rana Zulfugarova
Keyword(s):  
Analytical Solutions ◽  
Periodic Wave ◽  
Bell Polynomials ◽  
Solitary Wave Solutions ◽  
Soliton Theory ◽  
Periodic Wave Solutions ◽  
Bilinear Operators ◽  
Wave Solutions ◽  
Kink Wave ◽  
Hirota Bilinear

In this paper, we study the (3+1)-dimensional Burger system which is considered in soliton theory and generated by considering the Hirota bilinear operators. The bilinear frame to the Burger system by using the multi-dimensional Bell polynomials is constructed. Also, based on the binary Bäcklund transformations, the generalized Bell polynomials are written. We retrieve some novel exact analytical solutions, containing interaction between lump and two kink wave solutions, interaction between lump and periodic wave solutions, interaction between stripe and periodic solutions, breather wave solutions, cross-kink wave solutions, interaction between kink and periodic wave solutions, multi-wave solutions, and finally solitary wave solutions for the (3+1)-dimensional Burger system by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences.

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