Joining together virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we s
how how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associate to a repulsive potential by a series of Darboux transformations. This a simpler model than the kink stability for wave equations, but is still a classical one and retains some of the main difficulties.
In this note, we give an alternative proof of the theorem on soliton selection for small energy solutions of nonlinear Schrodinger equations (NLS) which we studied in Anal. PDE 8 (2015), 1289-1349 and more recently in Annals of PDE (2021) 7:16. As
in in the latter paper we use the notion of Refined Profile, with the difference that here we do not modify the modulation coordinates and we do not search for Darboux coordinates. This shortens considerably the proof.
We give short survey on the question of asymptotic stability of ground states of nonlinear Schrodinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with
several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
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.
In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schrodinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass g
round 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 Newtons 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.
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{NakanishiJM
SJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishis framework and Fermi Golden Rule estimates on $L^4$ in time norms.
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 result
s already known for the nonlinear Schrodinger 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.
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 regio
n 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 possesss 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).
We study the interaction of a ground state with a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.
