arXiv subjects

Shihui Zhu

Publications and source records attributed to Shihui Zhu.

13 recordsLinked to original sources

arXiv2026

Sharp Exponent of Stable Standing Waves for the Perturbated Hartree Equation

This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital stability, and strong instability of standing waves without any frequency constraint are obtained. This improves the cross constrained variational argument proposed by Zhang (2005).

math.AP↗

arXiv2025

Monotonicity Conjectures and Sharp Stability for Solitons of the Cubic-Quintic NLS on R^3

This paper deals with the cubic-quintic nonlinear Schr\"{o}dinger equation on R^3. Two monotonicity conjectures for solitons posed by Killip, Oh, Pocovnicu and Visan are completely resolved: one concerning frequency monotonicity, and the other concerning mass monotonicity. Uniqueness of the energy minimizer is proved. Then sharp stability of the solitons is established. And classification of normalized solutions is first presented.

math.AP↗

arXiv2025

Sharp Stability of Solitons for the Cubic-Quintic NLS on R^2

This paper concerns with the cubic-quintic nonlinear Schr\"{o}dinger equation on R^2. A family of new variational problems related to the solitons are introduced and solved. Some key monotonicity and uniqueness results are obtained. Then the orbital stability of solitons at every frequency are proved in terms of the Cazenave and Lions' argument. And classification of normalized ground states is first presented. Our results settle the questions raised by Lewin and Rota Nodari as well as Carles and Sparber.

math.AP↗

arXiv2025

Uniqueness and stability of normalized ground states for Hartree equation with a harmonic potential

The dynamic properties of normalized ground states for the Hartree equation with a harmonic potential are addressed. The existence of normalized ground state for any prescribed mass is confirmed according to mass-energy constrained variational approach. The uniqueness is shown by the strictly convex properties of the energy functional. Moreover, the orbital stability of every normalized ground state is proven in terms of the Cazenave and Lions' argument.

math.AP↗

arXiv2022

Small Solitons and Multi-Solitons in Generalized Davey-Stewartson System

This paper is concerned with the generalized Davey-Stewarston system in two dimensional space. Existence and stability of small solitons are proved by solving two correlative constrained variational problems and spectrum analysis. In addition, multi-solitons with different speeds are constructed by bootstrap argument.

math.AP↗

arXiv2021

Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $\lambda$-family equations

In this paper, we prove the global existence of H\"older continuous solutions for the Cauchy problem of a family of partial differential equations, named as $\lambda$-family equations, where $\lambda$ is the power of nonlinear wave speed. The $\lambda$-family equations include Camassa-Holm equation ($\lambda=1$) and Novikov equation ($\lambda=2$) modelling water waves, where solutions generically form finite time cusp singularities, or in another word, show wave breaking phenomenon. The global energy conservative solution we construct is H\"older continuous with exponent $1- \frac{1}{2\lambda}$. The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

math.AP↗

arXiv2020

Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrödinger equation with a $μ$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denote by $Q_p$ the ground state for the BNLS with $μ=0$. We prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $μ\in ( -λ_0, \iy)$ for some $λ_0=λ_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$\,, we prove the orbital stability on certain mass level below $\|Q^*\|_{L^2}$, provided $μ\in (-\lam_1,0)$, where $\lam_1=\dfrac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows to fill the gap concerning existence of the ground states for the BNLS when $μ$ is negative and $p\in (1,1+\frac8d]$.

math.AP↗

arXiv2019

Orbital Stability of Standing Waves for Fractional Hartree Equation with Unbounded Potentials

We prove the existence of the set of ground states in a suitable energy space $Σ^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-Δ+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set of standing waves. The main ingredient is the observation that $Σ^s$ is compactly embedded in $L^2$. This enables us to apply the concentration compactness argument in the works of Cazenave-Lions and Zhang, namely, relative compactness for any minimizing sequence in the energy space.

math.AP↗

arXiv2018

Blow-up for the nonlinear Schrödinger equation with combined nonlinearities

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schrödinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the correspondng solution of which blows up. This result partially gives a positive answer to the open problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we obtain the lower blow-up rate and concentration rate for the blow-up solutions in the same case. Finally, we consider the mass-critical combined with the mass-supercritical power type case, studying the blow-up criteria, blow-up rate and concentration of the blow-up solutions in this case.

math.AP↗

arXiv2018

On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential

We consider the focusing nonlinear Schrödinger equation with inverse square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^αu, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq 3$, $c\ne 0$, $c<λ(d)=\left(\frac{d-2}{2}\right)^2$ and $0<α\leq \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<α<\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $α=\frac{4}{d}$, we show that ground state standing waves are strongly unstable by blow-up.

math.AP↗

arXiv2017

Sharp Criteria of Scattering for the Fractional NLS

In this paper, the sharp threshold of scattering for the fractional nonlinear Schrödinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N} M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s},$$ then the corresponding solution $u(t)$ blows up in finite time, according to Boulenger, Himmelsbach, and Lenzmann's results in [2].

math.AP↗

arXiv2017

Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0} M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, $Q$ is the ground-state solution for the fractional Hartree equation.

math.AP↗

arXiv2016

Existence and Uniqueness of Global Weak solutions of the Camassa-Holm Equation with a Forcing

In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.

math.AP↗