Spectral stability of elliptic solutions to the short-pulse equation

The short pulse (SP) equation describes the propagation of ultrashort optical pulses in nonlinear media and possesses a Lax pair of the Wadati–Konno–Ichikawa (WKI) type. In this paper, using integrability, we examine the spectral stability of elliptic solutions to the SP equation. Firstly, we analytically give an explicit description of the spectrum of the WKI-Lax operator to the SP equation for elliptic potentials. Then, by constructing the squared-eigenfunction connection between the non-standard linear stability problem (LZ=ΛZ′) and the Lax spectral problem, we prove that the elliptic solutions are spectrally stable with respect to subharmonic perturbations.