cosh 2 ( ϕ 2 ) ) 2 8 m 2 n 2 sin 2 ( ϕ 1 ) cosh 2 ( ϕ 2 ) , x ( y , t ) = y + m n m 2 + n 2 ( n sin ( 2 ϕ 1 ) m sinh ( 2 ϕ 2 ) n 2 sin 2 ( ϕ 1 ) + m 2 cosh 2 ( ϕ 2 ) + 2 m ) . (19)

For convenience to obverse solution figure, a numerical simulation example of such a breather solution q ( x , t ) is given via Equation (19) at Figure 3.

Figure 3. Breather solution (19) with parameters m = 0.2 , n = 1.2 and θ 0 = 0 . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at t = 1 .

3.4. Three-Cuspon Solution

When N = 3 , assuming λ 1 = i a , λ 2 = i b and λ 3 = i c ( a > 0 , b > 0 , c > 0 ) , mean while taking C 1 = a e 2 θ 1 , C 2 = b e 2 θ 2 , C 3 = c e 2 θ 3 , and ϕ = 2 a ( y 4 a 2 t ) + 2 θ 1 , ψ = 2 b ( y 4 b 2 t ) + 2 θ 2 , φ = 2 c ( y 4 c 2 t ) + 2 θ 3 , here a , b , c , θ 1 , θ 2 and θ 3 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 q ( x , t ) is given in Figure 4.

4. Conclusion

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 O ( λ ) at λ 0 , like the case of the SP equation [9]. However, the novelty of our paper

Figure 4. Three-cuspon solution with parameters a = 1 , b = 2 , c = 3 , θ 1 = 0 , θ 2 = 0 and θ 3 = 0 . (a) The perspective view of the wave; (b) The corresponding contour pattern; (c) The pattern at t = 0.2 .

is that λ = 0 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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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