The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self‐Consistent Source

Yali Shen,Fengqin Zhang,Xiaomei Feng

doi:10.1155/2012/872385

Yali Shen, Fengqin Zhang + Show 1 more

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https://doi.org/10.1155/2012/872385

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Journal: Discrete Dynamics in Nature and Society	Publication Date: Jan 1, 2012
Citations: 1	License type: CC BY 3.0

Affiliation: Yuncheng University

Abstract

The Painlevé property and Bäcklund transformation for the KdV equation with a self‐consistent source are presented. By testing the equation, it is shown that the equation has the Painlevé property. In order to further prove its integrality, we give its bilinear form and construct its bilinear Bäcklund transformation by the Hirota′s bilinear operator. And then the soliton solution of the equation is obtained, based on the proposed bilinear form.

Highlights

It is well known that some nonlinear partial differential equations such as the soliton equations with self-consistent sources have important physical applications
The soliton solutions of some equations such as the KdV, AKNS, and nonlinear schrodinger equation with self-consistent sources are obtained through the inverse scattering method 1, 2
The Hirota bilinear method has been successfully used in the search for exact solutions of continuous and discrete systems, and in the search for new integrable equations by testing for multisoliton solutions or Backlund transformation 5, 6

Summary

Introduction

It is well known that some nonlinear partial differential equations such as the soliton equations with self-consistent sources have important physical applications. The soliton solutions of some equations such as the KdV, AKNS, and nonlinear schrodinger equation with self-consistent sources are obtained through the inverse scattering method 1, 2. In 3 a Darboux transformation, positon and negaton solutions to a Schrodinger self-consistent source equation are further constructed. Two classes of exponential and rational traveling wave solutions with arbitrary wave numbers are Discrete Dynamics in Nature and Society computed by applying the proposed bilinear Backlund transformation see 7 for details. It is a good reference for solving many high-dimensional soliton equations.

PainlEvE Test

Conclusion

Bilinear Form

Next we come to the second part of the proof: