arXiv subjects

Binhua Feng

Publications and source records attributed to Binhua Feng.

7 recordsLinked to original sources

arXiv2021

On the standing waves for the X-ray free electron laser Schrödinger equation

In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equation $$i\partial_{t}ψ=-Δψ+b^2(x_1^2+x_2^2)ψ+\frac{λ_1}{|x|}ψ+ λ_2(|\cdot|^{-1}\ast |ψ|^2)ψ- λ_3|ψ|^p ψ,~~~ (t,x)\in \mathbb{R}^+\times \mathbb{R}^3,$$ where $0<p<4$. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved, (i.e. $b=0$ in the equation) and the second one is that $b\neq 0$. To be precise, in the first case, by considering a minimization problem on a suitable Pohozaev manifold we prove the existence of ground states, and show further that all ground state standing waves are strongly unstable by blow-up in finite time. Moreover, by making use of the ideas of their proofs, we are able to prove the existence and instability of normalized solutions, whose proofs seem to be new, compared with the studies of normalized solutions in the existing literature. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for $p\in (0,4)$ and with some assumptions on the coefficients, where solutions are obtained as global minimizers if $p\in (0,\frac{4}{3}]$, and as local minimizers if $p\in [\frac{4}{3}, 4)$. In the mass-critical and supercritical cases $p\in [\frac{4}{3}, 4)$, we establish the variational characterization of the ground states on a suitable manifold which is different from the one neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on $ω$ and $p$, we prove that the ground state standing waves are strongly unstable.

math.AP↗

arXiv2019

Boson Stars with Long-range Perturbations

We consider the Boson star equation with long-range perturbation given by $$i\partial_t ψ=\sqrt{-\triangle+m^2}\,ψ+β(\frac{1}{|x|^α}\ast |ψ|^2)ψ-(\frac{1}{|x|}\ast |ψ|^2)ψ \ \ \text{on $\mathbb{R}^3$,}$$ where $\frac{1}{|x|^α} (0<α<1)$ denotes the long-range potential. In contrast to the well known fact that for $β=0$ no maximal ground state solitary wave exists when the partical number $N=N_c$ (Chandrasekhar limiting mass) [E.H. Lieb, H.T. Yau, \emph{Commun. Math. Phys.}, 112 (1987), pp: 147-174 ], we show that for $β>0$ and small enough, there exists at least one maximal ground state at $N=N_c$. Moreover, for $β>0$, we find that for initial value $\|ψ_0\|^2_2=N_c$, the solution $ψ(t)$ is global well-posedness, and we obtain an "orbital stability" of those maximal ground state solitary waves in some sense, which implies that such long-range perturbation pushes the Boson star system more stable. Finally, we analyse blow-up behaviours of maximal ground states when $β\rightarrow 0^+$.

math.AP↗

arXiv2019

Normalized ground states for the fractional nonlinear Schrödinger equations

In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schrödinger equation \begin{equation} i\partial_{t}ψ=(-Δ)^{s}ψ-f(ψ), \qquad (0.1)\end{equation} where $0<s<1$, $f(ψ)=|ψ|^{p}ψ$ with $\frac{4s}{N}<p<\frac{4s}{N-2s}$ or $f(ψ)=(|x|^{-γ}\ast|ψ|^2)ψ$ with $2s<γ<\min\{N,4s\}$. To this end, we look for normalized solutions of the associated stationary equation \begin{equation} (-Δ)^s u+ωu-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.

math.AP↗

arXiv2018

On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities \[ iu_{t}+Δu=λ_1|u|^{p_1}u+λ_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and $0<p_2<\frac{4}{N}$, we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Comm. Partial Differential Equations 32: 1281-1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rates and limiting profile. When $\frac{4}{N}<p_1<\frac{4}{N-2}$($4<p_1<\infty$ if $N=1$, $2<p_1<\infty$ if $N=2$), we prove that the blow-up solution with bounded $\dot{H}^{s_c}$-norm must concentrate at least a fixed amount of the $\dot{H}^{s_c}$-norm and, also, its $L^{p_c}$-norm must concentrate at least a fixed $L^{p_c}$-norm.

math.AP↗

arXiv2018

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities \[ i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rate and limiting profile.

math.AP↗

arXiv2015

The obstacle problem for nonlinear degenerate equations with $L^{1}$-data

The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data. Meanwhile, we prove that every entropy solution belongs to some Sobolev space $W^{1,q}(Ω)$.

math.AP↗

arXiv2013

Optimal bilinear control of nonlinear Schrödinger equations with singular potentials

In this paper, we consider an optimal bilinear control problem for the nonlinear Schrödinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in \cite{Sp} in several aspects.

math.AP↗