Abstract
In this paper, we consider the initial value problem for a complete integrable equation introduced by Wadati-Konno-Ichikawa (WKI). The solution is reconstructed in terms of the solution of a matrix Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at one non-singularity point, i.e., . 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.
Highlights
The initial value problem for the nonlinear integrable equation ( ) qt + qx 1+ q2 3 = 2 xx 0, t > 0, − ∞ < x < +∞, (1)= q( x,0) q0 ( x), − ∞ < x < +∞, where we assume q0 ( x) decays to 0 sufficiently fast, was derived Wadati, Konno and Ichikawa (WKI) in [1] [2]
We consider the initial value problem for a complete integrable equation introduced by Wadati-Konno-Ichikawa (WKI)
The one-cuspon solution, two-cuspon solutions and three-cuspon solution are discussed in detail
Summary
The three-soliton solution of the WKI equation hasn’t been discussed via the Riemann-Hilbert problem and given relevant numerical simulations. To [9], we can obtain some properties of μ j ( x,t,λ )( j = 1,2) , which are useful in the following analysis, such as, the first column of μ1 ( x,t,λ ) and μ2 ( x,t,λ ) (denote by [μ1]1 and [μ2 ]1 ) is bounded and analytic in upper and lower half-plane (denote by D1 and D2 ) of λ , respectively; the symmetry condition ( ) μ j x,t,λ = σ2μ j ( x,t,λ )σ2 , μ j ( x,t,−λ ) = σ1μ j ( x,t,λ )σ1 ; and the asymptotic behavior as λ →∞, μj.
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