A universal way to determine Hirota's bilinear equation of KdV type

2013 ◽  
Vol 54 (8) ◽  
pp. 081506
Author(s):  
Yi-Chao Ye ◽  
Zi-Xiang Zhou
Keyword(s):  
Bilinear Equation

scholarly journals A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

2018 ◽  
Vol 07 (04) ◽  
pp. 1840001
Author(s):  
A. N. W. Hone ◽  
F. Zullo
Keyword(s):  
Taylor Series ◽  
Taylor Expansion ◽  
Tau Function ◽  
Tau Functions ◽  
Sigma Function ◽  
Special Cases ◽  
Hirota Bilinear Equation ◽  
Order Four ◽  
Hirota Bilinear ◽  
Bilinear Equation

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

A Study on Rational Solutions to a KP-like Equation

2015 ◽  
Vol 70 (4) ◽  
pp. 263-268 ◽  
Author(s):  
Yufeng Zhang ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equation ◽  
Prime Number ◽  
Nonlinear Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Differential Operators ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Bilinear Equation

AbstractA KP-like nonlinear differential equation is introduced through a generalised bilinear equation which possesses the same bilinear form as the standard KP bilinear equation. By symbolic computation, nine classes of rational solutions to the resulting KP-like equation are generated from a search for polynomial solutions to the corresponding generalised bilinear equation. Three generalised bilinear differential operators adopted are associated with the prime number p=3.

scholarly journals The homogeneous balance of undetermined coefficients method and its application

2016 ◽  
Vol 14 (1) ◽  
pp. 816-826 ◽  
Author(s):  
Yi Wei ◽  
Xin-Dang He ◽  
Xiao-Feng Yang
Keyword(s):  
Partial Differential Equations ◽  
Boussinesq Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pdes ◽  
The Kdv Equation ◽  
Generalized Boussinesq Equation ◽  
Positive Integers ◽  
Bilinear Equation ◽  
Homogeneous Balance

AbstractThe homogeneous balance of undetermined coefficients method is firstly proposed to solve such nonlinear partial differential equations (PDEs), the balance numbers of which are not positive integers. The proposed method can also be used to derive more general bilinear equation of nonlinear PDEs. The Eckhaus equation, the KdV equation and the generalized Boussinesq equation are chosen to illustrate the validity of our method. The proposed method is also a standard and computable method, which can be generalized to deal with some types of nonlinear PDEs.

General description on extended homoclinic orbit solutions of the KdV-type bilinear equations

2020 ◽  
pp. 2150092
Author(s):  
Shu-Zhi Liu ◽  
Da-Jun Zhang
Keyword(s):  
Soliton Solution ◽  
Homoclinic Orbit ◽  
Soliton Solutions ◽  
General Description ◽  
De Vries ◽  
Bilinear Derivatives ◽  
Bilinear Equations ◽  
Special Case ◽  
Bilinear Equation

The Korteweg–de Vries (KdV)-type bilinear equations always allow 2-soliton solutions. In this paper, for a general KdV-type bilinear equation, we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution. Two properties of bilinear derivatives are developed to deal with bilinear equation deformations. A non-integrable (3+1)-dimensional bilinear equation is employed as an example.

Lump solutions and interaction behaviors to the (2+1)-dimensional extended shallow water wave equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850387 ◽  
Author(s):  
Wenguang Cheng ◽  
Tianzhou Xu
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Dynamic Properties ◽  
Quadratic Function ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Shallow Water Wave ◽  
Double Exponential ◽  
Combination Function ◽  
Bilinear Equation

In this paper, the exact solutions to the (2[Formula: see text]+[Formula: see text]1)-dimensional extended shallow water wave (SWW) equation are investigated by using its bilinear form and ansatz techniques. Following the method given by Ma [Phys. Lett. A 379 (2015) 1975–1978], two classes of lump solutions are constructed by searching for positive quadratic function solutions to the associated bilinear equation. Furthermore, two kinds of interaction solutions between a lump and solitary waves are presented by taking the solution of the associated bilinear equation as a linear combination function of a quadratic function and the double exponential function, one of which is the interaction solution between a lump and an exponentially decayed soliton, and the other one is the interaction solution between a lump and an exponentially decayed twin soliton. Finally, some figures are given to illustrate the dynamic properties of these obtained solutions.

Construction of rational solutions for the (2+1)-dimensional Broer–Kaup system

2019 ◽  
Vol 33 (30) ◽  
pp. 1950377 ◽  
Author(s):  
Yun-Hu Wang
Keyword(s):  
Linear Equation ◽  
Symbolic Computation ◽  
Linear Form ◽  
Rational Solutions ◽  
Dynamical Features ◽  
Truncated Painlevé Expansion ◽  
Bilinear Equation

Based on the quartic–linear form of the (2[Formula: see text]+[Formula: see text]1)-dimensional Broer–Kaup system which is derived from its truncated Painlevé expansion, three kinds of rational solutions are obtained through ansatz and symbolic computation with Maple. In general, these kinds of solutions obtained from quartic–linear equation are different from the ones which are generated via bilinear equation. Figures are presented to show the dynamical features of these solutions.

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