The Darboux Transformation Of Short Pulse Equation And Its Application

We present an explicit representation of an N-fold Darboux transformation ???_N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of ???_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. ???_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. The soliton and positon solutions of the hodograph equivalent short pulse equations are obtained by using the Darboux transformation from different "seed" solutions,and the decomposition of the lower-order positons into single-solitons is given analytically when time T is sufficiently large.