We revisit the scattering problems for the 2D mass super-critical Schr{o}dinger and Klein-Gordon equations with radial data below the ground state in the energy space. We give an alternative proof of energy scattering for both defocusing and focusing cases using the ideas of Dodson-Murphy citep{dodson2017new-radial}. Our results also include the exponential type nonlinearities which seems to be new for the focusing exponential NLS.
In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr{o}dinger equation begin{equation} ipartial_{t}psi=(-Delta)^{s}psi-f(psi), qquad (0.1)end{equation} where $0<s<1$, $f(psi)=|psi|^{p}psi$ with $frac{4s}{N}<p<frac{4s}{N-2s}$ or $f(psi)=(|x|^{-gamma}ast|psi|^2)psi$ with $2s<gamma<min{N,4s}$. To this end, we look for normalized solutions of the associated stationary equation begin{equation} (-Delta)^s u+omega u-f(u)=0. qquad (0.2) end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.
We study the scattering problems for the quadratic Klein-Gordon equations with radial initial data in the energy space. For 3D, we prove small data scattering, and for 4D, we prove large data scattering with mass below the ground state.
In this paper, we study the nonlinear Schr{o}dinger equation $ -Delta u+V(x)u=f(x,u) $on the lattice graph $ mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from $mathbb{Z}^{N}$ to quasi-transitive graphs.
We show that for a one-dimensional Schrodinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schrodinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.
In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schrodinger equations in higher spatial dimensions ($ngeq 2$) and some global well-posedness results with small initial data in critical Besov spaces $B^s_{2,1}$ are obtained. As by-products, the scattering results with small initial data are also obtained.
Zihua Guo
,Jia Shen
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(2019)
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"Scattering below the ground state for the 2D non-linear Schr{o}dinger and Klein-Gordon equations revisited"
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Zihua Guo
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