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.

We consider the direct scattering problem for a highly nonlinear complete integrable equation introduced by Wadati–Konno–Ichikawa (WKI), which describes nonlinear transverse oscillations of elastic beams under tension, with box‐like initial value. We show that in some cases of the initial value, the scattering data may have no zeros in upper half plane of the spectral variable. And furthermore in other cases, the scattering data may have infinitely many simple zeros, which seems not to be found before in the literature of the WKI equation.

In this paper, we consider the initial value problem for a complete integrable equation introduced by Wadati-Konno-Ichikawa (WKI). The solution () , q x t is reconstructed in terms of the solution of a 2 2 × matrix Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at one non-singularity point, i.e., 0 λ =. Then, the one-cuspon solution, two-cuspon solutions and three-cuspon solution are discussed in detail. Further, the numerical simulations are given to show the dynamic behaviors of these soliton solutions.