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 $$$ 4\times 4 $$$ 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 $$$ 4\times 4 $$$ matrix Riemann–Hilbert problem is constructed, and the potentials are exactly reconstructed by resorting to the asymptotic behavior of the eigenfunctions near $$$ k=0 $$$ and $$$ k=\infty $$$. 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.