On zeros of the scattering data of the initial value problem for the short pulse equation

Abstract The short pulse equation is an important integrable equation, which can be solved by the so-called inverse scattering method with the help of a Wadati–Konno–Ichikawa type Lax pair, in nonlinear optical field. In a recent paper Xu (2018), we obtain the leading order long-time asymptotics of the initial value problem for the short pulse equation under the assumption that the scattering data a ( k ) has no zero on the above half-plane for the complex variable k . In this paper, we show two different results for the zeros of the scattering data a ( k ) with the finite compact support initial value. Firstly, in the linear initial value case, a ( k ) has no zeros. Secondly, in the even box-type linear initial value, a ( k ) has infinite simple zeros with the same image part on the upper-half plane.