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

Ma, Wen-Xiu; You, Yuncheng

doi:10.1090/S0002-9947-04-03726-2

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
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by Wen-Xiu Ma and Yuncheng You PDF

Trans. Amer. Math. Soc. 357 (2005), 1753-1778 Request permission

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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