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Register a new userParametrix problem for the Korteweg--de Vries equation with steplike initial data
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Authors
Mateusz Piorkowski
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In this paper we study the asymptotics of the Korteweg--de Vries (KdV) equation with steplike initial data, which leads to shock waves, in the middle region between the dispersive tail and the soliton region, as $t rightarrow infty$. In our previous work we have dealt with this question, but failed to obtain uniform estimates in $x$ and $t$ because of the previously unknown singular behaviour of the matrix model solution. The main goal of this paper is to close this gap. We present an alternative approach to the usual argument involving a small norm Riemann--Hilbert (R-H) problem, which is based instead on Fredholm index theory for singular integral operators. In particular, we avoid the construction of a global model matrix solution, which would be singular for arbitrary large $x$ and $t$, and utilize only the symmetric model vector solution, which always exists and is unique.
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This paper is a continuation of arXiv:17.01.02867. We give here rigorous solution of the parametrix problem for Toda rarefaction problem and complete asymptotic analysis, justifying the asymptotics obtained in arXiv:17.01.02867.
In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization using Z-transformation. We give some numerical benchmark examples from the literature to illustrate our findings.
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation begin{eqnarray*} u_t+u_{xxx}+epsilon |partial_x|^{2alpha}u+(u^2)_x=0, u(0)=phi, end{eqnarray*} where $0<epsilon,alphaleq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s (s>s_alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $epsilon in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $epsilon$ tends to 0.
Original Whithams method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg-de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained for each case. The essential difference between them is illustrated by an example of so-called `generalized Korteweg-de Vries equation. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In cite{IsazaLinaresPonce} it was established that if the initial datun $u_0in H^l((b,infty))$ for some $linmathbb Z^+$ and $bin mathbb R$, then the corresponding solution $u(cdot,t)$ belongs to $H^l((beta,infty))$ for any $beta in mathbb R$ and any $tin (0,T)$. Our goal here is to extend this result to the case where $,lin mathbb R^+$.
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