Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

Abstract

In this work, we employ the $\bar{\partial }$-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. The long time asymptotic behavior of the solution $q(x,t)$ is derived in a fixed space-time cone $S(y_1,y_2,v_1,v_2)=\{(y,t)\in {\mathbb{R}}^2:y=y_0+vt,y_0\in [y_1,y_2],v\in [v_1,v_2]\}$. Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by $N(\mathcal{I})$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.

50 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:2109.06384; text overlap with arXiv:2101.12697, arXiv:2012.11928

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