Abstract
In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.
Highlights
Darboux transformation (DT) is an important technique to construct exact solutions of nonlinear partial differential equations [1]
The main significance of the DT is that infinite sequences of solutions to nonlinear equations can be generated by algebraic procedures
The nonlinear Schrödinger (NLS) equation is widely used in physics [8,9,10,11,12,13,14,15], nonlinear optics [16,17], and soft condensed matter physics [18] and there has been a vast amount of literature involving the NLS equation over the years
Summary
Darboux transformation (DT) is an important technique to construct exact solutions of nonlinear partial differential equations [1]. 2. Darboux Transformation and Exact Solutions for the NLS Equations 2.1. From Equations (2), (3) and (6), we obtain the following spectral problem. We assume that the new matrix U has the same type of with U. It means that pj(x, t) of U and pj(x, t) of U have the same structures. We assume that the new matrix V has the same type with V, which means that they have the same structures only pj(x, t) of V transformed into pj(x, t) of V.
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