The determinant representation of an <i>N</i>-fold Darboux transformation for the short pulse equation

Abstract

We present an explicit representation of an N-fold Darboux transformation T N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of T N , we show that the quasi-determinant is avoidable, and it is contrast to a recent paper (J. Phys. Soc. Jpn. 81 (2012), 094008) by using this relatively new tool which was introduced to study noncommutative mathematical objectives. T N produces new solutions u [N] and x [N] which are expressed by ratios of two corresponding determinants. We also obtain the soliton solutions, which have a variable trajectory, of the short pulse equation from new "seed" solutions.