Soliton Resolution for the Wadati-Konno-Ichikawa Equation with Weighted Sobolev Initial Data

In this work, we employ the partial derivative-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space H(R). The long time asymptotic behavior of the solution q(x, t) is derived in a fixed spacetime cone S(y(1),y(2),v(1),v(2)) = {(y, t) is an element of R-2 : y = y(0) + vt, y(0) is an element of [y(1),y(2)], v is an element of [v(1), v(2)]}. Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by N(I)-soliton on discrete spectrum and the t(-1/2) order term on continuous spectrum with residual error up to O(t(-3/4)).

Automated summary

In this study, we employ the partial derivative-steepest descent method to investigate the Cauchy problem of the WKI equation with initial conditions in weighted Sobolev space H(R). The long time asymptotic behavior of the solution is derived, and based on this behavior, we prove the soliton resolution conjecture of the WKI equation.