Asymptotic Behavior Analysis of Solutions for the Coupled WKI Type Equation With the Schwartz Initial Data

For the Cauchy problem of the coupled WKI type equation associated with a 4x4$$ 4\times 4 $$ matrix Lax pair with the Schwartz initial data, the long-time asymptotics behavior of solutions is studied by using the nonlinear steepest descent method. First, based on the inverse scattering transformation, a 4x4$$ 4\times 4 $$ matrix Riemann-Hilbert problem is constructed, and the potentials are exactly reconstructed by resorting to the asymptotic behavior of the eigenfunctions near k=0$$ k=0 $$ and k=infinity$$ k=\infty $$. Then, the multisoliton solutions of the coupled WKI type equation are obtained and the interaction dynamics of various soliton solutions are analyzed by selecting suitable parameters. Finally, the basic Riemann-Hilbert problem is transformed into a solvable model Riemann-Hilbert problem by a series of deformations, from which the long-time asymptotics of the Cauchy problem of the coupled WKI type equation is obtained in the solitonless sector.