Algebraic and Differential Nonlinear Superposition Formulas

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• 作者：P. R. Gordoa
• 关键词：nonlinear superposition formula ; Bä ; cklund transformations
• 刊名：Theoretical and Mathematical Physics
• 出版年：2003
• 出版时间：2003
• 年：2003
• 卷：137
• 期：1
• 页码：1430-1438
• 全文大小：137 KB

文摘

Knowledge of the Lax pair and the Darboux transformation for a completely integrable system provides an iterative approach for generating exact solutions. This approach involves solving for the eigenfunction of the Lax pair at each step. But this process can be considerably simplified using the Bäcklund transformation and Bianchi''s permutability theorem. This allows constructing the so-called nonlinear superposition formula, which provides a new solution of the system in terms of three previous solutions. The advantage of this approach is that the differential order of the nonlinear superposition formulas is lower than that of the Lax pairs, and in some cases, these formulas reduce to algebraic equations. We consider the construction of new nonlinear superposition formulas in the form of both differential equations and algebraic equations.