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Register a new userMultisolitons for the cubic NLS in 1-d and their stability
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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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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 $H^1$.
We consider a Nonlinear Schrodinger Equation with a very general non linear term and with a trapping $delta $ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.
We consider the nonlinear Schrodinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper we prove scattering in a larger region where the virial quantity is no longer guaranteed to be sign definite.
We prove that the cubic nonlinear Schrodinger equation (both focusing and defocusing) is globally well-posed in $H^s(mathbb R)$ for any regularity $s>-frac12$. Well-posedness has long been known for $sgeq 0$, see [51], but not previously for any $s<0$. The scaling-critical value $s=-frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 38, 46]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(mathbb R)$ for any $s>-frac12$. The best regularity achieved previously was $sgeq tfrac14$; see [15, 24, 32, 38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a one-parameter family of microscopic conservation laws that remain meaningful at this low regularity.
We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
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