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Register a new userOn small energy stabilization in the NLKG with a trapping potential
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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 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.
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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 show 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.
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 study the long time behavior of small (in $l^2$) solutions of discrete nonlinear Schrodinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrodinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.
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.
We prove scattering for the radial nonlinear Klein-Gordon equation $ partial_{tt} u - Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) in H^{s} times H^{s-1} $, $ 1 > s > 1- frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4 geq p > 3 $ and $ 1 > s > 1 - frac{(5-p)^{2}}{2(p-1)(6-p)}$ if $ 5> p geq 4$. First we prove Strichartz-type estimates in $ L_{t}^{q} L_{x}^{r} $ spaces. Then by using these decays we establish some local bounds. By combining these results with a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrarily large intervals. Once we have showed that this quantity is controlled, we prove that some of these local bounds can be upgraded to global bounds. This is enough to establish scattering. All the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling.
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