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Register a new userInstability of Standing Waves to the Inhomogeneous Nonlinear Schrodinger Equation with Harmonic Potential
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We study the instability of standing-wave solutions $e^{iomega t}phi_{omega}(x)$ to the inhomogeneous nonlinear Schr{o}dinger equation $$iphi_t=-trianglephi+|x|^2phi-|x|^b|phi|^{p-1}phi, qquad inmathbb{R}^N, $$ where $ b > 0 $ and $ phi_{omega} $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
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In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.
We are going to study the standing waves for a generalized Choquard equation with potential: $$ -ipartial_t u-Delta u+V(x)u=(|x|^{-mu}ast|u|^p)|u|^{p-2}u, hbox{in} mathbb{R}timesmathbb{R}^3, $$ where $V(x)$ is a real function, $0<mu<3$, $2-mu/3<p<6-mu$ and $ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $omega$ , the stability and instability of the standing waves $u=e^{i omega t}varphi(x)$ are investigated .
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesnt enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4omega>c^2$ and the critical parameters $4omega=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4omega=c^2, cleq 0$ and the supercritical parameters $4omega<c^2$, there is no nontrivial solitary wave for (DNLS). At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for (DNLS) with initial data in the invariant set $mathcal{K}^+_{omega,c}subseteq H^1(R)$, with $4omega=c^2, c>0$ or $4omega>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in cite{Dod:NLS_sct}, the scattering result of (DNLS) doesnt hold for initial data in $mathcal{K}^+_{omega,c}$ because of the existence of infinity many small solitary/traveling waves in $mathcal{K}^+_{omega,c},$ with $4omega=c^2, c>0$ or $4omega>c^2$. On the other hand, our global result improves the global result in cite{Wu-DNLS, Wu-DNLS2} (see Corollary ref{cor:gwp}).
In this paper we consider the inhomogeneous nonlinear Schrodinger equation $ipartial_t u +Delta u=K(x)|u|^alpha u,, u(0)=u_0in H^s({mathbb R}^N),, s=0,,1,$ $Ngeq 1,$ $|K(x)|+|x|^s| abla^sK(x)|lesssim |x|^{-b},$ $0<b<min(2,N-2s),$ $0<alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $alpha_0(b)<alpha<(4-2b)/N$. The value $alpha_0(b)$ is the positive root of $Nalpha^2+(N-2+2b)alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=mu|x|^{-b}$, $muin mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.
In this paper, we establish the existence of ground state solutions for a fractional Schrodinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrodinger equations. Our results explain the effect of each term of the Schrodinger equation : The fractional power, the power of the nonlinearity and the harmonic potential.
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