Yung-Fu Fang, Chi-Kun Lin, Jun-ichi Segata
This paper is concerned with the quantum Zakharov system. We prove that when the ionic speed of sound goes to infinity, the solution to the fourth order Schrodinger part of the quantum Zakharov system converges to the solution to quantum modified nonlinear Schrodinger eqaution.
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We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a nonlinear Schrödinger equation.
We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t^{-1}, whereas the Schrödinger component decays almost at a rate of t^{-7/6}.
We study the eigenvalues of the self-adjoint Zakharov-Shabat operator corresponding to the defocusing nonlinear Schrodinger equation in the inverse scattering method. Real eigenvalues exist when the square of the potential has a simple well. We derive two types of quantization condition for the eigenvalues by using the exact WKB method, and show that the eigenvalues stay real for a sufficiently small non-self-adjoint perturbation when the potential has some PT-like symmetry.
We consider the vectorial Zakharov system describing Langmuir waves in a weakly magnetized plasma. In its original derivation the evolution for the electric field envelope is governed by a Schrödinger type equation with a singular parameter which is usually large in physical applications. Motivated by this, we study the rigorous limit as this parameter goes to infinity. By using some Strichartz type estimates to control separately the fast and slow dynamics in the problem, we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vec…
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3/4<k\leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for "Sc…
We investigate some focusing fourth-order coupled Schrodinger equations. Existence of ground state and global well-posedness are obtained. Moreover, the best constant of some Gagliardo-Nirenberg inequality is studied.
In this paper we establish a global Carleman estimate for the fourth order Schrödinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from boundary measurements.
Dispersive estimates of fourth order Schrödinger operators with scaling-critical magnetic potentials in dimension two
本文通过构造预解核并运用稳相方法，获得了二维中具有尺度临界磁势的四阶Schrödinger算子的色散估计。
Multilinear Weighted Estimates and Quantum Zakharov System
本文研究了一维量子Zakharov系统的多线性加权估计，通过Bourgain范数方法证明了低正则性局部适定性，并在无小性假设下证明了全局适定性，同时探讨了量子极限在不同时间区间的行为。
We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrodinger equation in high dimensions n>4, for general L^2 initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the Ground State in the focusing case.
Considered in this report is the one-dimensional fourth-order dispersive cubic nonlinear Schrödinger equation with mixed dispersion. Orbital stability, in the energy space, of a particular standing-wave solution is proved in the context of Hamiltonian systems. The main result is established by constructing a suitable Lyapunov function.
In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schrödinger equation $$ iu_t-μΔu+Δ^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $μ\ge 0$. As an application, we establish a dichotomy result on the existence of extremals to the symmetric Schrödinger Strichartz inequality.
For $n\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shrödinger equation in anisotropic Lebesgue spaces with $\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations with three order derivatives and obtain the global well posedness for this problem with small data in modulation space $M^{9/2}_{2,1}({\Real^{n}})$.
We investigate the cubic defocusing fourth order Schrödinger equation $iu_t + Δ^2u + |u|^2u=0$ in arbitrary space dimension $\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n \le 8$ and ill-posed when $n \ge 9$, with the additional important information that scattering holds true when $5 \le n \le 8$.
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.
We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $H^s$-spaces. We also analyze the scaling limit of fast dispersion management and the convergence to a model with averaged dispersions.
We consider the time local and global well-posedness for the fourth order nonlinear Schrodinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).
Modified Strichartz Estimate of the Periodic Fourth Order NLS
本文在一维环面上建立适配四阶色散关系的修正Strichartz估计，证明四阶非线性Schrödinger方程整体适定性，并揭示解对量子参数的连续依赖仅局部一致而非整体一致。
In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schrödinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$.
In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schrödinger equation. Using the $I$-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in $H^γ(\mathbb{R}^d), 5\leq d\leq 7$ with $γ(d)<γ<2$, where $γ(5)=\frac{8}{5}, γ(6)=\frac{5}{3}$ and $γ(7)=\frac{13}{7}$.
We study the fourth order Schrödinger type differential inequality $-Δ^2 u + λV(x)u \geq a(x)u^q$ with $a,V\in L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $λ>0$. We consider nonnegative solutions without making any assumptions about their asymptotic behavior, and choosing appropriate test functions, we find a priori estimates for some type of subharmonic solutions. Finally, applying these a priori estimates, we proof a Liouville type result.
In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.
Global dynamics above the ground state energy for the 3D Zakharov system
该研究对能量略高于基态能量的三维Zakharov系统径向初值，通过正则形式技术和新型局域virial估计，将初值划分为9个区域并分类解的全局动力学，拓展了此前基态能量以下的相关结果。
Stability of inverse boundary value problem for the fourth-order Schrödinger equation
本文研究第四阶薛定谔方程逆边界值问题的稳定性，采用复几何光学解法和傅里叶分析，基于边界测量建立稳定性结果。
We prove the local well-posedness for the nonlinear fourth-order Schrödinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global well-posedness above mass and energy spaces under some assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the super-critical range.
We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i\partial_t u=-\epsΔu+Δ^2 u+P((\partial_x^αu)_{\absα\ls 2}, (\partial_x^α\bar{u})_{\absα\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces $B^{s}_{2,1}(\R^n)$ are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces $\dot{B}^{s}_{2,1}$. As a by-product, the existence of the scattering operators with small data is also obtained. In order to show these results, the global version of the estimates for the maximal functions and the loc…
We prove a weak form of the soliton resolution conjecture of bounded solutions of high-dimensional fourth-order Schrodinger equations. The result relies upon two properties to be proved: the asymptotic frequency localization and the asymptotic spatial localization. In order to prove the asymptotic frequency localization we use the fact that the high frequency pieces of the free solution have better dispersive properties than the lower ones. In order to prove the asymptotic spatial localization, we use the symmetries of the phase of the fundamental solution.
Sharp Decay estimates for the discrete Fourth-Order Schrödinger Equation
本文在任意维数格点上建立离散四阶薛定谔方程的尖锐时间衰减估计，通过标量傅里叶分解与牛顿多面体方法，得到双调和传播子的衰减率$|t|^{-d/4}$。
Unconditional well-posendness for the fourth order nonlinear Schrodinger type equations on the torus
该研究证明了s≥1时环面H^s空间中四阶非线性薛定谔型方程（含非可积情形）的无条件适定性，通过正规形约化与抵消性质处理导数损失，且确定s≥1为最优正则性阈值。
In this paper, we first deduce the asymptotic expansions of resolvent of $H=(-Δ)^2+V$ with the presence of resonance or eigenvalue at the degenerate zero threshold for $d\geq5$. In particular, we identify these resonance spaces for full kinds of zero resonances. As a consequence, we then establish the {\it time asymptotic expansions} and {\it Kato-Jensen estimates} for the solution of fourth-order Schrödinger equation under the presence of zero resonance or eigenvalue.
In dimension N=3 the cubic nonlinear Schrodinger equation has solutions which become singular, i.e. at a spatial point they blow up to infinity in finite time. In 1972 Zakharov famously investigated finite time singularity formation in the cubic nonlinear Schrodinger equation as a model for spatial collapse of Langmuir waves in plasma, the most abundant form of observed matter in the universe. Zakharov assumed that (NLS) blow up of solutions is self-similar and radially symmetric, and that singularity formation can be modeled by a solution of an associated self-similar, complex ordinary differ…
In this paper we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology $L^2(\mathbb{R})\times L^2(\mathbb{R})$ and with an approximation in the energy space $H^1(\mathbb{R})\times L^2(\mathbb{R})$. In the case of the Zakharov system this is achieved without the condition $\partial_t n(x,0) \in \dot H^{-1}(\mathbb…
In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.
We consider the fourth-order Schrödinger equation $$ i\partial_tu+Δ^2 u+μΔu+λ|u|^αu=0, $$ where $α>0,μ=\pm1$ or $0$ and $λ\in\mathbb{C}$. Firstly, we prove local well-posedness in $H^4\left(\R^N\right)$ in both $H^4$ subcritical and critical case: $α>0$, $(N-8)α\leq8$. Then, for any given compact set $K\subset\mathbb{R}^N$, we construct $H^4(\R^N)$ solutions that are defined on $(-T, 0)$ for some $T>0$, and blow up exactly on $K$ at $t=0$.
Remark on the Adiabatic Limit of Quantum Zakharov System
本文研究量子 Zakharov 系统绝热极限所得修正 NLSE 的低正则适定性，并严格证明量子参数趋于零时其收敛于标准 NLSE。
We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^\infty$ dispersive bounds. In all cases, we show that the natural $|t|^{-\frac34}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator $-Δ+V$.
We consider the fourth order Schrödinger operator $H=Δ^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=Δ^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ then the wave operators $W_{\pm}= s\,-\,\lim_{t\to \pm \infty} e^{itH}e^{-itH_0}$ extend to bounded operators on $L^p(\mathbb R^3)$ for all $1<p<\infty$.
In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…
Continuum limit of fourth-order Schrödinger equations on the lattice
研究格点上的四阶薛定谔方程，通过驻定相位法和Littlewood-Paley不等式建立均匀Strichartz估计，并证明连续极限下解的L2收敛速率。
Global Well-posedness for The Fourth-order Nonlinear Schrödinger Equations on $\mathbb{R}^{2}$
本文利用$I$-方法证明了二维散焦四阶非线性Schrödinger方程在$H^{s}(\mathbb{R}^{2})$中当$2-\frac{3}{4k}<s<2$时的全局适定性。
On the decay property of the cubic fourth-order Schrödinger equation
本文证明了$5 \leq d \leq 8$维空间上三次四阶Schrödinger方程（4NLS）的解具有与线性解相同的逐点衰减性质，该结果通过基于全局结果的bootstrap argument得出，并可推广至具有散射渐近的一般色散方程。
Random data theory for the cubic fourth-order nonlinear Schrödinger equation
该研究针对N≥5维实空间上带随机初值的三次四阶非线性薛定谔方程，证明了低正则性下的几乎必然局部适定性、小初值概率全局适定性与散射，以及膨胀立方体随机初值的大概率全局适定性与散射。
The fourth-order Schrödinger equation on lattices
该研究针对一维、二维格点上含实参数γ的四阶薛定谔方程，通过分析振荡积分相位函数的牛顿多面体建立精确色散估计，证明Strichartz估计并得到小初值非线性方程全局解的存在性。
The energy-critical stochastic Zakharov system
本文研究四维能量临界随机Zakharov系统，证明了能量空间中的局部适定性与爆破二择一、大初值下低于基态阈值的全局存在性，以及非守恒噪声强度趋于无穷时全局存在且散射的概率趋于1的噪声正则化现象。
In this paper, we consider a class of nonlinear fourth-order Schrödinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +Δ^2 u &=&-|u|^{ν-1} u, \quad 1+ \frac{8}{d}<ν<1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^γ(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] Using the $I$-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in $H^γ(\mathbb{R}^d)$ with $γ(d,ν)<γ<2$ for some value $γ(d,ν)>0$.
We consider the Cauchy problem for the fourth order nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R}…
In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schrödinger type operator with unbounded coefficients of the form $$A=a^{2} Δ^2+V^{2}$$ where $a(x)=1+|x|^α$ and $V=|x|^β$ with $α>0$ and $β>(α-2)^+$. We obtain that $(-A,D(A))$ generates an analytic strongly continuous semigroup in $L^2(\mathbb{R}^N)$ for $N\geq5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2α-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]
Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case
本文研究了带有势能的第四阶非线性薛定谔方程在n≥5维空间中的全局良态性，证明了径向初值在H²空间下的存在性和散射性，采用Strichartz估计、Sobolev范数等方法扩展了无势能情况下的结果。
We prove that the solution map, associated to the quintic fourth order nonlinear Schrödinger equation, exhibits the norm inflation phenomenon at every point in the Sobolev spaces of super-critical regularity. Indeed, we prove this result separately in the cases of negative and of positive regularity. In the negative regularity case, we prove the result for both the defocusing and focusing equations: in the one dimensional case, the associated solution map exhibits norm inflation in Sobolev spaces of super-critical regularity (including the critical index); in the higher dimensional case, the s…
Blow-up results for Inhomogeneous fourth-order nonlinear Schrödinger Equation
本文研究非齐次双调和薛定谔方程解的H²范数爆破现象，通过凸性方法和virial恒等式，分别分析负能量及低于基态能量且动能超基态的正能量下径向与非径向解的爆破情况。
Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation
该研究在调制空间$M^{s}_{2,q}$中建立了一维四阶非线性薛定谔方程的局部与全局适定性，借助$U^p-V^p$空间、双线性估计及完全可积PDE的微扰行列式幂级数展开得到先验估计，推进了相关方程适定性理论研究。
In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=Δ^2-V=Δ^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $ΔA^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.
Solutions for fourth order anisotropic nonlinear Schrödinger equations in $\R^2$
本文研究二维空间中四阶各向异性非线性薛定谔方程，证明其柯西问题的适定性与爆破性，建立驻波的存在性、对称性、衰减性及轨道稳定性结论，且结果可推广至高维。
We consider the fourth order Schrödinger operator $H=Δ^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t|^{-\frac54}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of r…
A note on the fourth-order Schrodinger equation with spatially growing inhomogeneous source term
本文研究了一个非线性双调和薛定谔方程，考虑了非均匀项的影响，发展了小数据的局部和全局理论，并在更低正则性下发展了局部理论。
We study time decay estimates of the fourth-order Schrödinger operator $H=(-Δ)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze the low energy and high energy behaviour of resolvent $R(H; z)$, and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for $e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on Jensen-Kato decay estimate and local decay estimate, we obtain the $L^1\rightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{-itH}…
Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations
本文证明5维二次四阶薛定谔方程在小初值下解的散射性，推广时空共振方法至双拉普拉斯情形，并建立 $t^{-5/4}$ 衰减估计。
We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different …
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