The determinant representation of an N-fold Darboux transformation for the short pulse equation

Abstract

We present an explicit representation of an N-fold Darboux transformation T̃N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of T̃N, 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. T̃N 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.