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

In non-linear optics, it is well known that the non-linear Schrodinger (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 (10-15 s), the NLS equation becomes less accurate. Schafer 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, u(x,t) 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.