Multi-Cuspon Solutions of the Wadati-Konno-Ichikawa Equation by Riemann-Hilbert Problem Method

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 Rie-mann-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. Fur-ther, the numerical simulations are given to show the dynamic behaviors of these soliton solutions.


Introduction
The initial value problem for the nonlinear integrable equation , , where we assume ( ) 0 q x decays to 0 sufficiently fast, was derived Wadati, Konno and Ichikawa (WKI) in [1] [2]. This equation can be used to describe nonlinear transverse oscillations of elastic beams under tension [3] [4] and the motion of curves in E 3 [5]. If beam is flexible enough, it could be deformed into a shape of loop, of which upper half portion takes the negative curvature. One can easily realize such a situation on a stretched rope. Compared with the already known systems such as the nonlinear Schrödinger (NLS) equation and the derivative nonlinear Schrödinger (DNLS) equation, Equation (1) is highly nonlinear and even has the saturation effects [1]. Therefore its analysis is quite interesting mathematically and physically.
For the WKI Equation (1), many researchers have done a lot of studies and obtained many classical conclusions. There are so many interesting works on this equation in these years. Wadati, Konno and Ichikawa presented two types of nonlinear equations, and showed the equations have an infinite number of conservation laws and can be expressed in the Hamiltonian form in [1]. The authors of [6] studied the WKI equation by the inverse scattering transform (IST) method [7], obtained a one-soliton solution and a two-soliton solution, and analyzed some properties. In [8], the authors showed that two inverse scattering formalisms by Ablowitz, Kaup, Newell and Segur and by Wadati, Konno and Ichikawa are connected through a gauge transformation, and one-soliton solutions of equations associated with the W-K-I scheme are also examined. The authors derived the WKI equation from motion of curves in E 3 and gave the corresponding group-invariant solutions [5]. However, the three-soliton solution of the WKI equation hasn't been discussed via the Riemann-Hilbert problem and given relevant numerical simulations.
In this paper, the one-cuspon solution, two-cuspon solutions and three-cuspon solution of the WKI equation were discussed via the Riemann-Hilbert problem. And the numerical simulations were given to show the dynamic behaviors of these cuspon solutions. Besides, the novelty of this paper is twofold. First, the solution of the WKI Equation (1) is reconstructed via a 2 2 × matrix Riemann-Hilbert problem at 0 λ → instead of λ → ∞ , although 0 λ = is not the singularity point of the Lax pair (2). Second, we obtain multi-cuspon solutions, which were not discussed before. This paper is organized as follows. In Section 2, we perform the spectral analysis at λ = ∞ and 0 λ = , respectively. Then reconstruct the solution ( ) , q x t in terms of the solution of the associated Riemann-Hilbert problem for the WKI equation in variable ( ) , y t instead of ( ) , x t via 0 λ → . In Section 3, assuming ( ) a λ only has simple zeros, we obtain the algebraic system of N-soliton solutions. We give the details for 1, 2, 3 N = and the numerical simulations of these cuspon solutions.

Lax Pair
The Lax pair of the WKI equation is , , where

For λ = ∞
Firstly, we define a matrix-value function ( ) , G x t and a scalar function , Similarly to [9], we introduce a transformation , , e , , , , d , , , e , , , , d , . Therefore, similarly to [9], we can obtain some properties , which are useful in the following analysis, such as, the first column of ( ) , where the off-diagonal en- Next, we define the scattering matrix by ∫ is a conserved quantity.

The Riemann-Hilbert Problem in Variable (y, t)
Defining We assume that the initial value , , q x y t t . , , e 1 , , . 0 , , , ,

Algebraic System of N-Solitons
Combining the symmetry conditions and taking