Darboux transformations for the nonlinear Schrödinger equations
Darboux transformations for the nonlinear Schrödinger equations
复制标题
DOI：
10.1088/0305-4470/29/23/029
复制
发表时间：
1996-12
期刊：
影响因子：
--
通讯作者：
Manuel Mañas
中科院分区：
文献类型：
--
作者：
Manuel Mañas
利用带有算子谱参数的Lax对的向量解的Grammian型行列式构造了AKNS/ZS系统的Darboux变换。研究了具有标准色散和反常色散的非线性薛定谔方程的达布变换约化问题。对于非线性Schrodinger方程的一个给定的种子解，给出了两个不同的新解族，其中一个新解族与它的一个新的向量Lax对有关.在第一个新解族中，我们给出了与对角矩阵有关的拓扑解，它们对振幅和非零背景具有不同的渐近变元.对于反常色散的情况下，它们代表连续变形的明亮的n-孤子的解决方案，这是恢复为零背景。特别是这些解决方案包含的组合的多个同宿轨道的聚焦非线性薛定谔方程。与约旦块，我们发现合理的变形刚刚描述的解决方案，以及纯理性的解决方案。第二族不仅包含上述解，而且包含更广泛的解。例如，在标准色散情况下，我们能够获得暗孤立子解。
Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schrodinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schrodinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schrodinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.