The initial value problem for the nonlinear integrable equation
where we assume decays to 0 sufficiently fast, was derived Wadati, Konno and Ichikawa (WKI) in  . This equation can be used to describe nonlinear transverse oscillations of elastic beams under tension   and the motion of curves in E3 . 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 . 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 . The authors of  studied the WKI equation by the inverse scattering transform (IST) method , obtained a one-soliton solution and a two-soliton solution, and analyzed some properties. In , 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 E3 and gave the corresponding group-invariant solutions . 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 matrix Riemann-Hilbert problem at instead of , although 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 and , respectively. Then reconstruct the solution in terms of the solution of the associated Riemann-Hilbert problem for the WKI equation in variable instead of via . In Section 3, assuming only has simple zeros, we obtain the algebraic system of N-soliton solutions. We give the details for and the numerical simulations of these cuspon solutions.
2. Spectral Analysis and Riemann-Hilbert Problem
2.1. Lax Pair
The Lax pair of the WKI equation is
Firstly, we define a matrix-value function and a scalar function as
Similarly to , we introduce a transformation
where . Then defining two Jost solutions via
where . Therefore, similarly to , we can obtain some properties
of , which are useful in the following analysis, such as, the first column of and (denote by and ) is bounded and analytic in upper and lower half-plane (denote by and ) of , respectively; the symmetry condition
, ; and the asymptotic behavior as , , where the off-diagonal entries of are and .
Next, we define the scattering matrix by
here, , with and .
Then, we obtain (where means the determinate of a matrix A), and is analysis in .
Introducing another transformation
Similarly, we define two Jost solutions
where . Then, and are bounded and analytic in and , respectively. And as
2.4. The Relation between and
The function and related by
Then, we have , where is a conserved quantity.
2.5. The Riemann-Hilbert Problem in Variable (y, t)
As , . Then introducing the new scale and defining .
We assume that the initial value satisfies that it can make has the finite simple zeros . Therefore satisfies the following the Riemann-Hilbert problem:
• Analytic property: is analytic in and , and continuous up to the boundary, ;
• Jump condition: , where , ;
• Normalization: ;
• Residue conditions: for some constant ,
Letting satisfies the above conditions, then this Riemann-Hilbert problem has a unique solution. Assume that the WKI equation satisfies the initial value conditions. We obtain the solution for the initial value problem as follows:
then calculating derivative of Equation (14) with respect to y, we can obtain the solution .
3. Cuspon Solutions
3.1. Algebraic System of N-Solitons
From residue conditions (13), we have
Combining the symmetry conditions and taking , then, by solving the algebraic system for and , and recall the symmetry conditions, we can obtain .
3.2. One-Cuspon Solution
When , assuming , and letting , where d and are real constants. Denoting by , we can obtain one-cuspon solution
For convenience to obverse solution figure, a numerical simulation example of such a one-cuspon solution is given via Equation (17) in Figure 1.
3.3. Two-Cuspon Solutions
3.3.1. Two-Cuspon Solution
When , assuming and , then letting and , taking and , here and are real constants. We obtain two-cuspon solution as follows:
Figure 1. One-cuspon solution (17) with parameters and . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at .
For convenience to obverse solution figure, a numerical simulation example of such a two-cuspon solution is given via Equation (18) at Figure 2.
Figure 2. Two-cuspon solution (18) with parameters and . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at .
3.3.2. Breather Solution
Assuming that and , taking , and , here and are real constants, we can obtain the breather cuspon as follows:
For convenience to obverse solution figure, a numerical simulation example of such a breather solution is given via Equation (19) at Figure 3.
Figure 3. Breather solution (19) with parameters and . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at .
3.4. Three-Cuspon Solution
When , assuming and , mean while taking , , , and , , , here and are real constants. The expression of three-soliton solution is very complicated, we don’t write it in detail for space reason.
For convenience to obverse solution figure, a numerical simulation example of such a three-cuspon solution is given in Figure 4.
In this paper, the solutions of the WKI equation are recovered in terms of the solution of the matrix Riemann-Hilbert problem from the order at , like the case of the SP equation . However, the novelty of our paper
Figure 4. Three-cuspon solution with parameters and . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at .
is that is non-singularity of the WKI equation. Then, one-cuspon solution, two types of two-cuspon solutions and three-cuspon solution are given via Riemann-Hilbert approach; one type of two-cuspon solutions is the breather, a novel solution of the WKI equation, which was not shown before. The numerical simulations of these solutions are given by choosing suitable parameters. Compared with the inverse scattering transform method, the calculation processes of the Riemann-Hilbert problem are more concise and efficient, and the most important advantage of the Riemann-Hilbert problem is analyzing the long-time asymptotic behavior of the solutions. The work of the analyzing long-time asymptotic behavior is beyond our aim in this paper, but we plan to complete this question in the future.
 Ichikawa, Y., Konno, K. and Wadati, M. (1981) Nonlinear Transverse Oscillation of Elastic Beams under Tension. Journal of the Physical Society of Japan, 50, 1799-1802.
 Ichikawa, Y. (1982) A Relationship between the Ablowitz-Kaup-Newell-Segur and Wadati-Konno-Ichkawa Schemes of the Inverse Scattering Method. Journal of the Physical Society of Japan, 51, 3036-3041.