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Register a new userLong-time asymptotics for the Wadati-Konno-Ichikawa equation with the Schwartz initial data
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In this work, we investigate the long-time asymptotic behavior of the Wadati-Konno-Ichikawa equation with initial data belonging to Schwartz space at infinity by using the nonlinear steepest descent method of Deift and Zhou for the oscillatory Riemann-Hilbert problem. Based on the initial value condition, the original Riemann-Hilbert problem is constructed to express the solution of the Wadati-Konno-Ichikawa equation. Through a series of deformations, the original RH problem is transformed into a model RH problem, from which the long-time asymptotic solution of the equation is obtained explicitly.
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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 $S(y_{1},y_{2},v_{1},v_{2})={(y,t)inmathbb{R}^{2}: y=y_{0}+vt, ~y_{0}in[y_{1},y_{2}], ~vin[v_{1},v_{2}]}$. 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 $O(t^{-frac{3}{4}})$.
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $bar{partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asymptotic behavior of the solution $q(x,t)$ in space-time soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.
The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.
We investigate the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data in different solitonic regions begin{align*} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, quad (x,t)inmathbb{R}times mathbb{R}^{+}, &q(x,0)=q_{0}(x), quad lim_{xrightarrowpminfty}q_{0}(x)=pm 1, end{align*} where $q_0mp 1in H^{4,4}(mathbb{R})$.Based on the spectral analysis of the Lax pair, we express the solution of the mKdV equation in terms of a Riemann-Hilbert problem. In our previous article, we have obtained long time asymptotics and soliton resolutions for the mKdV equation in the solitonic region $xiin(-6,-2)$ with $xi=frac{x}{t}$.In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for the solitonic region $xiin(-varpi,-6)cup(-2,varpi)$ with $ 6 < varpi<infty$ being an arbitrary constant.For $-varpi<xi<-6$, there exist four stationary phase points on jump contour, and the asymptotic approximations can be characterized with an $N$-soliton on discrete spectrums and a leading order term $mathcal{O}(t^{-1/2})$ on continuous spectrum up to a residual error order $mathcal{O}(t^{-3/4})$. For $-2<xi<varpi$, the leading term of asymptotic expansion is described by the soliton solution and the error order $mathcal{O}(t^{-1})$ comes from a $bar{partial}$-problem. Additionally, asymptotic stability can be obtained.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.
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