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Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions


Authors: Wen-Xiu Ma and Yuncheng You
Journal: Trans. Amer. Math. Soc. 357 (2005), 1753-1778
MSC (2000): Primary 35Q53, 37K10; Secondary 35Q51, 37K40
DOI: https://doi.org/10.1090/S0002-9947-04-03726-2
Published electronically: December 22, 2004
MathSciNet review: 2115075
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Abstract: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.


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Additional Information

Wen-Xiu Ma
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: mawx@math.usf.edu

Yuncheng You
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: you@math.usf.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03726-2
Keywords: Integrable equation, soliton theory
Received by editor(s): June 2, 2003
Published electronically: December 22, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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