(2 + 1)维 Konopelchenko-Dubrovsky 方程的 Wronskian 解、Bäcklund 变换和 Painlevé 分析

Di Gao, Wen-Xiu Ma, Xing Lü
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摘要

本文的主要工作是构建 Wronskian 解,并研究 (2 + 1) 维 Konopelchenko-Dubrovsky 方程的可整性特征。首先,利用沃伦斯基技术获得沃伦斯基解的充分条件。根据 Wronskian 形式,在行列式中选取满足线性偏微分方程的元素,即可得到孤子解。其次,分别通过 Hirota 双线性方法和贝尔多项式理论直接导出了双线性贝克隆变换和贝尔多项式型贝克隆变换。最后,Painlevé 分析证明了该方程具有 Painlevé 特性,并构建了 Painlevé 型 Bäcklund 变换,通过选择适当的种子解求解精确解族。研究表明,Wronskian 技术、Bäcklund 变换、贝尔多项式和 Painlevé 分析可用于获得非线性演化方程的精确解,如孤子解、单波解和双波解。
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Wronskian solution, Bäcklund transformation and Painlevé analysis to a (2 + 1)-dimensional Konopelchenko–Dubrovsky equation
The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution.
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