A 3×3 Matrix Spectral Problem And Its Darboux Transformation

Nonlinear differntial equation(s)(NDE) is an important mathematical model for describing physical phenomenon and an important field in the contemporary study of nonlinear science. That exploring and developing new methods to solve NDE and showing the interactive laws among the soliton solutions is one of the forefront topics in the studies of the nonlinear physics.In the field of finding the explicit solutions of nonlinear differential equations, Darboux transformation is regarded as one of the most effective ways. But in many studies, the attention is mainly paid to a family of equations about 2×2 spectral problem and its characters. The main deficiency lies in that it can't include such relatively complex high-order nonlinear differential equations such as HNLS. Suggested by previous work, this thesis constructs a new Lax pair related to 3 × 3 spectral problem. Through the intensive study of this Lax pair, its practical value is also discovered. By studying this Lax pair, this thesis educes an hierarchy. This hierarchy includes not only KdV, mKdV, NLS and other common equations, but also some complex high-order equations just like HNLS. With the help of the gauge transformation of spectral problem,this paper succeeds in constructing the Darboux transformation of this 3×3 spectral problem.Besides, it gets the explicit solutions of the nonlinear evolution equations which correspond to this Lax pair with the help of computer algebraic system Maple.