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Register a new userOrbital stability of KdV multisolitons in $H^{-1}$
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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 $H^1$.
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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 prove 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.
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