ﻻ يوجد ملخص باللغة العربية
In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.
We prove that the Cauchy problem for the Schrodinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(R)times H^{-{3/4}}(R)$. The new ingredient is that we use the $bar{F}^s$ type space, introdu
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
We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(R)$. The new ingredient is that we use directly the contract
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable p
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