On the double-pole solutions of the complex short-pulse equation

In non-linear optics, it is well known that the non-linear Schrödinger (NLS) equation was always used to model the slowly varying wave trains. However, when the width of optical pulses is in the order of femtosecond ([Formula: see text] s), the NLS equation becomes less accurate. Schäfer and Wayne proposed the so-called short pulse (SP) equation which provided an increasingly better approximation to the corresponding solution of the Maxwell equations. Note that the one-soliton solution (loop soliton) to the SP equation has no physical interpretation as it is a real-valued function. Recently, an improvement for the SP equation, the so-called complex short pulse (CSP) equation, was proposed in Ref. 9. In contrast with the real-valued function in SP equation, [Formula: see text] is a complex-valued function. Since the complex-valued function can contain the information of both amplitude and phase, it is more appropriate for the description of the optical waves. In this paper, the new types of solutions — double-pole solutions — which correspond to double-pole of the reflection coefficient are obtained explicitly, for the CSP equation with the negative order Wadati–Konno–Ichikawa (WKI) type Lax pair by Riemann–Hilbert problem method. Furthermore, we find that the double-pole solutions can be viewed as some proper limits of the soliton solutions with two simple poles.