【作者】 邓淑芳；

【导师】 陈登远； 张大军；

【作者基本信息】 上海大学 ， 计算数学， 2004， 博士

【摘要】 本文利用Hirota方法、Wronskian技巧和B(a|¨)cklund变换研究了一些等谱，非等谱与具自容源孤子方程的多孤子解。在第二章中通过新的双线性导数公式，利用Hirota方法得到了KP方程，非线性自偶网格方程，Toda链和非线性Schr(o|¨)dinger方程新的单孤子，双孤子解，并猜测出新N孤子解的表达式。特别由这些新解可导出原经典解。第三章叙述了Wronskian行列式的定义与性质，并以KP和Toda链方程为例，证明其具有Wronskian形式的新解。然后在第四章中由KP方程的谱问题与时间发展式导出具自容源KP方程，利用Hirota截断技术，可得单孤子解，双孤子解，三孤子解等等，并猜测出N孤子解的一般表达式。此外利用Wronskian行列式的性质和某些特殊的处理方法，证明了具自容源的KP方程具有Wronskian形式的解。通过直接计算证明了由Hirota方法猜测的N孤子解的表达式与Wronskian形式的N孤子解是一致的。类似于第二章的求解过程，利用Hirota方法得到了具自容源KP方程的新解。在第五章中我们分别给出了非等谱KP和KdV方程的双线性形式和双线性B(a|¨)cklund变换。利用Hirota方法得到了非等谱KP和KdV方程的多孤子解的表达式。但是与等谱情形不同的是由Hirota方法得到的f的表达式与Wronskian形式解的表达式f在恢复非等谱方程的N孤子解时是不一致的。由非等谱KP和KdV方程的双线性B(a|¨)cklund变换出发，利用Hirota方法和Wronskian技巧分别得到这些方程解的表达式并讨论了其解的一致性。需要指出的是在等谱方程B(a|¨)cklund变换的求解中，一般是由方程的已知解求出新解，再以所得的解作为已知解，求出更新解，周而复始。但是在非等谱KP方程双线性B(a|¨)cklund变换中这种规则是不成立的。把孤子方程的B(a|¨)cklund变换作一些修正，利用修正的B(a|¨)cklund变换，可以得到孤子方程的新解，在本文的最后一章中以KP方程为例说明了这一点。在附录中，我们给出论文中所求出孤子解的相应图形。 本文中利用Hirota方法，Wronskian技巧和双线性B(a|¨)cklund变换对孤子方程的求解技巧，可推广到其他孤子方程。更多还原

【Abstract】 In this paper, we consider the solution of some soliton equations by Hirota method, Wronskian technique and B(?)cklund transformation. The novel multisoliton solutions for the KP equation, the nonlinear lumped self-dual network equations, the Toda lattice and the nonlinear Schr(?)dinger equation are derived by using of Hirota direct method. The KP equation and the Toda lattice have also solutions in new Wronskian form . In addition, taking the KP equation as example we also show the novel solutions obtained by Backlund transformation are coincidence with the novel solution obtained through Hirota method. The above three methods are easily to be extended to some other soliton equations.The paper also proposes a KP equation with self-consistent sources from the linear problem of the KP system. One-, two- and even three-soliton solutions are successively constructed through the standard Hirota’s approach. On the basis of this, We conjecture further a general formula of N-soliton solution. We also use Wronskian technique to give Wronski determinant solutions. By virtue of some determinantal identities, solution is verified by direct substitution into the bilinear equations of the KP equation with self-consistent sources. The coincidence of the N-soliton solutions obtained by Hirota method and Wronskian technique is proved. The novel multisoliton solutions for the KP equation with self-consistent sources are also obtained by Hirota method.Apart from that, the bilinear equation and bilinear B(?)cklund transformation for the nonisospectral KP equation and the nonisospectral KdV equation are obtained. Exact solutions are constructed in terms of Wronskian and are verified by direct substitution to the satisfy the bilinear equation and the assiciated B(?)cklund transofrmation respectively. These two nonisospectral equations are also solved through the Hirota method. Finally, some figures are presented to show the shape and motion of the soliton solutions for some equations.更多还原