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Register a new userOn vector and matrix Riemann-Hilbert problems for KdV shock waves
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This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg-de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann-Hilbert problems for the KdV case. To the best of our knowledge this is the first time such ill-posedness is discussed in applications of Riemann--Hilbert theory. Furthermore, we rigorously justify the asymptotics for the shock wave in the elliptic zone derived previously.
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We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painleve region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations.
The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R simeq mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $mathbb{D}_h$ symmetries are presented.
We show that the KdV flow evolves any real singular initial profile q of the form q=r+r^2, where rinL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.
We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature.
We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles qs which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{{epsilon}}}),xrightarrow+infty, with some positive c and {epsilon}. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if {epsilon}>1/2, (2) meromorphic on a strip around the real line if {epsilon}=1/2, and (3) Gevrey regular if {epsilon}<1/2. Note that qs need not have any decay or pattern of behavior at -infty.
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