Vector Higher Order Nonlinear Schr? Dinger Equation For The Corresponding Spectral Problem Of The Trace Formula And Its Application

In this paper, the n×n matrix eigenvalue problem that related to the vector higher order non-linear Schr(o|¨)dinger equation is studied. The isospectral hierarchy has been found by carrying out the AKNS procedures, and the third of them is the vector higher order non-linear Schr(o|¨)dinger equation. An infinite set of local conservation laws for the evolution equation have been derived by a method due to Konno et al. Using the approach given by Zakharov and faddeev, the trace formulae of the eigenvalue problem are established. Moreover, the eigenfunctions representations for the field variables are obtained by taking functional derivatives. Further, the completely integrable system in Liouville sense is given out explicitly, as the case n=3.