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Scipio Cuccagna

Publications and source records attributed to Scipio Cuccagna.

At least 37 records · Page 2Linked to original sources
arXiv2018

Long time oscillation of solutions of nonlinear Schrödinger equations near minimal mass ground state

In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schrödinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time. More precisely, we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator. Then, we show that the finite dimensional part, two dimensional, approximately obeys Newton's equation of motion for a particle in an anharmonic potential well. Showing that the infinite dimensional part is well separated from the finite dimensional part, we will have long time oscillation.

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arXiv2018

On nonlinear profile decompositions and scattering for a NLS-ODE model

In this paper, we consider a Hamiltonian system combining a nonlinear Schr\" odinger equation (NLS) and an ordinary differential equation (ODE). This system is a simplified model of the NLS around soliton solutions. Following Nakanishi \cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on $L^4$ in time norms.

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arXiv2016

On asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation

We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t \to \infty$ for a general class of initial data whose difference from the non-vanishing background possess's a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).

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arXiv2015

On small energy stabilization in the NLKG with a trapping potential

We consider a nonlinear Klein Gordon equation (NLKG) with short range potential with eigenvalues and show that in the contest of complex valued solutions the small standing waves are attractors for small solutions of the NLKG. This extends the results already known for the nonlinear Schrödinger equation and for the nonlinear Dirac equation. In addition, this extends a result of Bambusi and Cuccagna (which in turn was an extension of a result by Soffer and Weinstein) which considered only real valued solutions of the NLKG.

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arXiv2015

On small energy stabilization in the NLS with a trapping potential

We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential. In particular we generalize work of Soffer and Weinstein, and of Tsai et. al. The novelty is that we allow generic spectra associated to the potential. This is yet a new application of the idea to interpret the nonlinear Fermi Golden Rule as a consequence of the Hamiltonian structure.

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arXiv2012

Decay and scattering of small solutions of pure power NLS in $\R$ with $p>3$ and with a potential

We prove decay and scattering of solutions of the Nonlinear Schröding-er equation (NLS) in ${\mathbf R}$ with pure power nonlinearity with exponent $3<p<5$ when the initial datum is small in $Σ$ (bounded energy and variance), in the presence of a linear inhomogeneity represented by a linear potential which is a real valued Schwarz function. We assume absence of discrete modes. The proof is analogous to the one for the translation invariant equation. In particular we find appropriate operators commuting with the linearization.

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arXiv2012

On stability of standing waves of nonlinear Dirac equations

We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schrödinger equation, because of the strong indefiniteness of the energy.

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arXiv2012

On asymptotic stability of moving ground states of the nonlinear Schrodinger equation

We extend to the case of moving solitons, the result on asymptotic stability of ground states of the NLS with a short range linear potential obtained by the author in a previous paper. Now we drop the potential and allow moving solitons. For technical reasons we consider only smooth solutions. The proof is similar to the earlier paper. However now the flows required for the Darboux Theorem and the Birkhoff normal forms, instead of falling within the framework of standard theory of ODE's, are related to quasilinear hyperbolic symmetric systems. So there is an issue about well posedness and regularity with respect of the initial data of the flows. It is also not obvious that Darboux Theorem can be applied, since we need to compare two symplectic forms in a neighborhood of the ground states not in in energy space, but rather in a smaller space where also the variance is bounded. Then the validity of Darboux Theorem is not obvious because, since the NLS does not preserve small neighborhoods of the ground states in this smaller space, we are comparing the to symplectic forms in a set which, in some respects, is large.

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arXiv2011

On instability of some approximate periodic solutions for the full nonlinear Schrödinger equation

Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrödinger operator with symmetric potential. This goes in the direction of proving that the approximate periodic solutions of the NLS shadowed in a recent work of the second author with Michael Weinstein do not persist for the full NLS.

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arXiv2011

The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states

In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M.Weinstein, are also asymptotically stable, for seemingly generic equations. Here we assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.

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arXiv2010

On scattering of small energy solutions of non autonomous hamiltonian nonlinear Schrödinger equations

We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schr\" odinger equation (NLS) with an attractive localized potential and a time-dependent factor in the nonlinearity. We show that, under generic hypotheses on the linearization at 0 of the equation, small energy solutions are asymptotically free. This is yet a new application of the hamiltonian structure, continuing a program initiated in a paper by Bambusi and Cuccagna.

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