اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOrbital stability of KdV multisolitons in $H^{-1}$
80
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove that multisoliton solutions of the Korteweg--de Vries equation are orbitally stable in $H^{-1}(mathbb{R})$. We introduce a variational characterization of multisolitons that remains meaningful at such low regularity and show that all optimizing sequences converge to the manifold of multisolitons. The proximity required at the initial time is uniform across the entire manifold of multisolitons; this had not been demonstrated previously, even in $H^1$.
قيم البحث
اقرأ أيضاً
For both the cubic Nonlinear Schrodinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${bf M}_N$ of pure $N$-soliton states, and their associated multisoliton solutions. We prov
e that (i) the set ${bf M}_N$ is a uniformly smooth manifold, and (ii) the ${bf M}_N$ states are uniformly stable in $H^s$, for each $s>-frac12$. One main tool in our analysis is an iterated Backlund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(mathbb{R}/mathbb{Z})$.
In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $Omega$ and on the domain $phi(Omega)$ resulting from $Omega$ by means of a bi-Lipschitz map $phi$. We c
onsider the solutions $u$ and $tilde u$ of the corresponding elliptic equations with the same right-hand side $fin L^2(Omegacupphi(Omega))$. Under certain assumptions we estimate the difference $| ablatilde u- abla u|_{L^2(Omegacupphi(Omega))}$ in terms of certain measure of vicinity of $phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $phi(Omega)$ and $Omega$, that is $|phi(Omega)triangle Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.
The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the $L^2cap L^infty$ orbital stability of a wave train c
ontaining $N$ smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce textit{a priori } estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy ($L^2$-norm) is studied. It is proved that the interaction
of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately satisfies Airy equation.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
