Abstract
In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size varepsilon > 0, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least O(varepsilon ^{-2}). To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.
Highlights
The Korteweg-de Vries (KdV) equation∂t u = −∂x3u + 6u∂x u (1.1)is one of the most important model equations for describing dispersive phenomena
In this paper we obtain the following stability result for periodic multisolitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size ε > 0, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least O(ε−2)
The aim of this paper is to study in the periodic setup the long time asymptotics of the solutions of Hamiltonian perturbations of (1.1) with initial data close to a periodic multisoliton of arbitrary large amplitude
Summary
Is one of the most important model equations for describing dispersive phenomena. It is named after the two Dutch mathematician Korteweg and de Vries [29] (cf. Boussinesq [14], Raleigh [40]) and originally was proposed as a model equation in one space dimension for long surface waves of water in a narrow and shallow channel. This means that we normalize the terms in the Taylor expansion of the Hamiltonian vector field XH which do not contain w and are homogeneous of order at most two This is achieved by a standard normal form procedure which consists in constructing a canonical transformation, given by the time one flow map F of a Hamiltonian vector field XF with a Hamiltonian F of the form. For Hamiltonian perturbations of linear integrable PDEs on T1, which satisfy nonresonance conditions, a standard normal form method has been developed allowing to prove the stability of the equilibrium solution u ≡ 0 of (Hamiltonian) perturbations for time intervals of large size—see e.g. A key ingredient are the normal form coordinates, constructed in [27]
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