scholarly journals Nonlinear Evolution Equations Generated from the Backlund Transformation for the Boussinesq Equation

1977 ◽  
Vol 57 (3) ◽  
pp. 797-807 ◽  
Author(s):  
R. Hirota ◽  
J. Satsuma
Keyword(s):  
Evolution Equations ◽  
Boussinesq Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation

Abundant New Multiple Soliton-like Solutions and Rational Solutions of the (2+1)-Dimensional Broer-Kaup Equation

2001 ◽  
Vol 56 (12) ◽  
pp. 816-824 ◽  
Author(s):  
Zhenya Yan
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Heat Equations ◽  
Backlund Transformation ◽  
Rational Solutions ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

Abstract In this paper we firstly improve the homogeneous balance method due to Wang, which was only used to obtain single soliton solutions of nonlinear evolution equations, and apply it to (2 + 1)-dimensional Broer-Kaup (BK) equations such that a Backlund transformation is found again. Considering further the obtained Backlund transformation, the relations are deduced among BK equations, well-known Burgers equations and linear heat equations. Finally, abundant multiple soliton-like solutions and infinite rational solutions are obtained from the relations.

Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

2016 ◽  
Vol 30 (24) ◽  
pp. 1650309
Author(s):  
Lin Wang ◽  
Qixing Qu ◽  
Liangjuan Qin
Keyword(s):  
Bilinear Form ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation ◽  
Wronskian Determinant ◽  
Bilinear Bäcklund Transformation ◽  
Bilinear Equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

APPLICATIONS OF THE LIE ALGEBRA gl(2)

2009 ◽  
Vol 23 (14) ◽  
pp. 1763-1770 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Lie Algebra ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation

With the help of a subalgebra of the Lie algebra gl (2) (still denoted by gl (2)), a non-isospectral evolution-equation hierarchy is obtained, whose reduction is similar to that given by Li. Employing the Lie algebra gl (2) again, we work out nonlinear evolution equations with exponential terms and produce their Bäcklund Transformation as well as some exact solutions.

FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

2009 ◽  
Vol 23 (12) ◽  
pp. 1519-1524 ◽  
Author(s):  
FUCAI YOU ◽  
TIECHENG XIA ◽  
JIAO ZHANG
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Evolution Equations ◽  
Boussinesq Equation ◽  
Nonlinear Evolution ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Partial Differential ◽  
Kdv Equations

Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

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