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 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.

The short pulse equation: Bäcklund transformations and applications

“…12 While the last approach adopted by Boutet de Monvel et al is of analytic nature, we have taken an essentially algebraic route which is more closer to the first two works. 11,14 Through a novel hodograph between the SP equation and the sine-Gordon equation, Matsuno obtained the multisoliton solutions of the SP equation which was given in summation forms. Both multiloop solitons and multibreather solutions were exhibited.…”

“…10) and the Riemann‐Hilbert approach 12 . While the last approach adopted by Boutet de Monvel et al is of analytic nature, we have taken an essentially algebraic route which is more closer to the first two works 11,14 …”

“…As for the Darboux transformation method, 11 it is noted that the solution formulae obtained therein were implemented in intermediate variables rather than the original coordinates (cf. (2.13) of Ref.…”

“…Within the framework of Hirota bilinear method, Kuetche et al calculated the two‐loop solution 8 and Matsuno obtained the ‐loop solution and ‐breather solution 9 . Saleem and Hassan considered the Darboux transformation in terms of quasi‐determinants 10 and Liu et al showed that the quasi‐determinants can be avoided and the ‐fold Darboux transformation may be given by determinants 11 . Recently, a Riemann‐Hilbert approach was applied to the SP equation 12,13 and the long‐time behavior of the solution was studied.…”

“…9 Saleem and Hassan considered the Darboux transformation in terms of quasi-determinants 10 and Liu et al showed that the quasi-determinants can be avoided and the -fold Darboux transformation may be given by determinants. 11 Recently, a Riemann-Hilbert approach was applied to the SP equation 12,13 and the long-time behavior of the solution was studied. It is interesting to note that Matsuno obtained the periodic solutions for the SP equation.…”