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Register a new userOn the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I
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Authors
Fabrice Bethuel
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The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature. In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.
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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 $,lin mathbb R^+$.
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.
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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.
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.
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