Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2005-03-01
Trans. Amer. Math. Soc., 357 (2005), No.5, 1753-1778
Nonlinear Sciences
Exactly Solvable and Integrable Systems
26 pages including 12 figures
Scientific paper
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.
Ma Wen-Xiu
You Yuncheng
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