Fei Liu, Tai-Peng Tsai, Ian Zwiers
In this note we study analytically and numerically the existence and stability of standing waves for one dimensional nonlinear Schrödinger equations whose nonlinearities are the sum of three powers. Special attention is paid to the curves of non-existence and curves of stability change on the parameter planes.
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We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.
We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of the region where the nonlinearity is localized that imply, respectively, existence or nonexistence of ground states. In order to obtain these results we adapt to the context of metric graphs some classical techniques from the Calculus of Variations.
Infinitely many normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
该研究针对带局部非线性的非紧度量图上的非线性薛定谔方程，在$L^2$-超临界 regime 下证明了对任意给定质量都存在无穷多归一化解，拓展了相关方程解的存在性结论。
NLS with mass-subcritical combined nonlinearities: small mass $L^2$-scattering
本文研究了具有聚焦主导项与低阶散焦扰动项的双重非线性NLS方程，利用伪共形变换与变分论证证明了仅需初始数据质量足够小即可实现质量次临界区域中的散射。
Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
本文针对非紧度量图上带局部非线性项的NLS方程，在$L^2$超临界情形下证明了任意给定质量的规范化解存在，所用方法可推广至更一般方程。
This paper proves existence and stability results of solitary-wave solutions to coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of solitary waves is obtained by solving a variational problem subject to two independent constraints and using the concentration-compactness method. The set of minimizers is shown to be stable and further information about the structures of this set are given. The paper extends the results previously obtained by Cipolatti and Zumpichiatti, Nguyen and Wang, and Ohta.
We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrödinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k, the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.
A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…
Scattering for non-radial 3D NLS with combined nonlinearities: the interaction Morawetz approach
本文针对具有聚焦主导非线性项与散焦扰动的三维质量-能量临界竞争非线性薛定谔方程，利用交互 Morawetz 估计与局部化函数的 Pohozaev 泛函新界，给出了基态能级以下散射的新证明，并推导了对称解的爆破速率。
We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) …
Scattering for the $2d$ NLS with inhomogeneous nonlinearities
本文研究了二维非均匀非线性薛定谔方程的大数据散射问题，通过集中-紧致性方法和Morawetz估计，证明了所有幂次p>0下的H^1空间散射存在性。
Blow-up criteria for the 1D NLS with combined nonlinearities
本文提出了一种用于一维NLS中结合非线性有限时间爆炸解的负能量和正能量准则，并给出了满足这些准则的初始数据示例。
We study the strong instability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction under the mass resonance condition in dimension $d=5$.
On 2D NLS on non-trapping exterior domains
二维非线性散焦薛定谔方程的整体存在性和散射性在星形障碍物外的区域和至少以五次幂增长的非线性中是已知的。在本文中，我们推广了所有非陷阱障碍和非线性项的幂严格大于大于四次的整体存在性结果。对于这样的非线性，我们也证明了散射一类所谓的几乎是星形的障碍。
In this paper we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is $L^{2}$-critical and supercritical.
Scattering for the $1d$ NLS with inhomogeneous nonlinearities
本文利用集中-紧致性与反证法及 Nakanishi 风格的 Morawetz 估计，证明了一维非齐次非线性 Schrödinger 方程在 $H^1$ 中的大数据散射，排除了紧致解的存在。
We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions $d\leq 3$. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension $d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.
Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case
该研究针对全空间ℝ^N上Sobolev临界的带组合幂次非线性项的非线性薛定谔方程，证明了不同类型扰动下归一化基态的存在性、不存在性及稳定性结果，是该领域的首个相关贡献。
Orbital stability of kinks in the NLS equation with competing nonlinearities
本文研究了具有竞争非线性的NLS方程中kinks的轨道稳定性，通过严格证明展示了其稳定性，并扩展了分析方法至一般情况。
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|…
We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
In this paper we consider the logarithmic Schrödinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the existence of several families of standing waves. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we show that the ground states are orbitally stable via a variational approach.
We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subc…
Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
本文研究了星图上具有幂非线性与顶点delta型非线性的NLS方程基态存在性，发现两者竞争导致质量阈值现象，并证明了径向定态分支的轨道稳定性。
We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction app…
We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stabl…
Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$
本文研究波导流形上组合幂次非线性NLS的长时间动力学，通过构造两个重标度变分问题族，建立了孤立子、散射与爆破解的定性定量性质及周期性依赖结果。
Variational properties of space-periodic standing waves of nonlinear Schr{ö}dinger equations with general nonlinearities
本文研究一般非线性薛定谔方程空间周期驻波的变分刻画，通过固定质量与动量最小化能量及在Nehari流形上最小化作用量，并讨论周期、反周期及聚焦、散焦情形。
Existence and orbital stability of standing-wave solutions of the NLS-log equation on a tadpole graph
本文首次在蝌蚪图上研究 NLS-log 方程，利用 Neumann-Kirchhoff 边界条件和分裂特征值方法，证明了正单瓣态驻波解的存在性与轨道稳定性，并给出相关线性化算子的 Morse 指数与零化指数。
Existence and instability of standing waves for the biharmonic nonlinear Schroedinger equation with combined nonlinearities
该研究针对带组合非线性项的双调和薛定谔方程，用新方法证明了归一化基态解的存在性、基态对应限制能量泛函的局部极小值，且驻波会因爆破强不稳定，结果涵盖临界情形。
Large data local solutions for the derivative NLS equation
考虑具有一般二次非线性项的导数NLS方程。在\cite{be2}中，第一作者证明了在$n = 2$维无穷远处具有衰减结构的索伯列夫空间中的一个小数据局部适定性结果。在这里，我们证明了一个类似的结果，在所有维度的大初始数据$n \geq 2$。
Spectra of Linearized Operators for NLS Solitary Waves
具有聚焦幂非线性项的非线性薛定谔方程存在孤波解。这些孤波周围的线性化算子的谱与孤波的稳定性以及（NLS）解的长时间动力学密切相关。我们研究这些光谱的详细，分析和数值。
Ground states for the NLS equation with combined nonlinearity on periodic metric graphs
研究周期度量图上非线性薛定谔方程带混合非线性的基态存在性，扩展了同质NLS方程在正方形和蜂窝网格上的结果，并改进了非紧致图上的相关研究。
Orbitally stable standing waves for the asymptotically linear one-dimensional NLS
本文研究了一维渐近线性、非线性薛定谔方程（NLS）。我们证明了存在一个全球性的平滑曲线的驻波这个问题，我们证明了这些驻波是轨道稳定的。据我们所知，这是渐近线性NLS的第一个严格稳定性结果。我们还讨论了我们的结果的应用到自聚焦波导与饱和折射率。
We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrödinger equation with $δ$-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.
We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, …
We study a boundary value problem related to the search of standing waves for the nonlinear Schrödinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the {\it double-bridge graph}, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-called Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$, and we are interested in the existence and properties of standing waves of given frequency $ω$. For every $ω>0$ on…
We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with $δ$-interaction on a star graph $Γ$. The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines $δ$-interaction. This permits to prove virial identity for the $H^1$- solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-$δ'$ e…
Standing waves with prescribed mass for biharmonic NLS with positive dispersion and Sobolev critical exponent
本文针对具有正色散和Sobolev临界指数的双调和NLS方程，利用直接极小化方法证明存在两个指定质量驻波解，并分析了其稳定性与全局存在性。
We consider the following triple power nonlinear Schr{ö}dinger equation: iut + $Δ$u + a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0. We are interested in algebraic standing waves i.e standing waves with algebraic decay above equation in dimensions n (n = 1, 2, 3). We prove the instability of these solutions in the cases DDF (we use abbreviation D: defocusing (ai < 0), F:focusing (ai > 0)) and DFF when n = 2, 3 and in the case DFF with a1 = --1, a3 = 1 and a2 < 32/15$\sqrt$6 when n = 1. Under these assumptions, the standing waves are orbitally unstable in the case of small positive frequency. When t…
On a star graph made of $N \geq 3$ halflines (edges) we consider a Schrödinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or $L^2$-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then…
An explicitly solvable NLS model with discontinuous standing waves
本文研究带点相互作用的NLS方程，证明次临界和临界情形下基态与激发态的存在性及唯一性，给出最优常数并显式描述驻波分支，构成可求解模型。
On (in)stability of discontinuous standing waves for the NLS with a delta-prime on star graphs
研究具有幂次非线性的一维非线性薛定谔方程在含δ′相互作用的星型图上顶点处不连续驻波的存在性与（不）稳定性，用非变分方法刻画驻波轮廓，分析其稳定性。
On the nonexistence of NLS breathers
本文利用位力泛函，仅凭内部能量与非线性幂次严格证明 NLS 呼吸子解不存在，并将结论推广到导数型、对数型等多种 NLS 模型。
Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schrödinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic nonlinearities, and get polynomial in time bounds for any higher order nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most exponentially, while for sub-cubic NLS we get polynomial bounds on the growth of the $H^2$-norm.
We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schrödinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space $H^1(\mathbb{R})$ only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models as in Kowalczyk-Martel-Muñoz, a…
We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.
The aim of this work is to demonstrate the effectiveness of the extension theory of symmetric operators in the investigation of the stability of standing waves for the nonlinear Schrödinger equations with two types of non-linearities (power and logarithmic) and two types of point interactions ($δ$- and $δ'$-) on a star graph. Our approach allows us to overcome the use of variational techniques in the investigation of the Morse index for self-adjoint operators with non-standard boundary conditions which appear in the stability study. We also demonstrate how our method simplifies the proof of th…
We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph $Γ$ ($N$ half-lines glued at the common vertex) with a $δ$ interaction at the vertex. The strength of the interaction is defined by a fixed value $α\in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $α\neq 0$ the NLS equation on $Γ$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $α<0$ that, in the NLS equation with a subcritical power-type nonl…
Global dynamics of the $2d$ NLS with white noise potential and generic polynomial nonlinearity
本文利用 Hairer-Labbé 方法，为 $\T^2$ 上带白噪声势和任意多项式非线性的 NLS 构建了唯一全局动力学，将解作为平滑势方程经典解的极限，改进了此前仅限于三次和四次非线性的研究。
Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
该研究针对超临界区二维带点相互作用的NLS方程，推导非标准位力公式，构造出爆破初值集，并证明基态对应的驻波在ω足够大时具有强不稳定性。
The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit
本文研究了一维空间中的尺度非线性薛定谔方程：\[ i\frac{d}{dt} ψ^{\varepsilon}(t) =-Δψ^{\varepsilon}(t) + \frac{1}εV\left(\frac{x}ε\right)|ψ^{\varepsilon}(t)|^{2μ}ψ^{\varepsilon}(t) \quad \quad ε>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ . \]该方程表示具有空间集中非线性的非线性薛定谔方程。我们发现，在极限$ε\to 0$，弱（积分）动力学收敛在$H^1(\mathbb{R})$的弱动力学的NLS与点集中的非线性：\[ i\frac{d}{dt} ψ(t) =H_αψ(t) . \]其中$H_α$是拉普拉斯算子的非线性边界条件在原点$ψ'(t,0+)-ψ'(t,0-)=α|ψ(t,0)|^{2μ}ψ(t,0)$和$α=\int_{\mathbb{R}}Vdx$。如果$V \geq 0$，则每个$μ\in \mathbb{R}^+$都发生收敛，否则每个$μ\in (0,1)$都发生收敛。对于具有任意数量$N$集中点的非线性，同样的结果也适用
The nonlinear Schrödinger equations with nonlinearities $|u|^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_tH^s_x$ for a certain range of scale-subcritical regularities $s$, which is almost optimal in the case $d\geq 4$ or $k\geq 2$. The proof is based on various multilinear estimates and the infinite normal form reduction argument.
Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities
本文研究了具有混合分数阶拉普拉斯算子和组合非线性项的NLS方程的归一化解，证明了在次临界和临界指数情形下基态解及山路型解的存在性，推广了已有结果。
Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D
本文提出一种新的坐标系来研究二维双曲或非椭圆三次非线性薛定谔方程。基于这些坐标，我们研究了有界连续双曲径向驻波的存在性，以及双曲径向自相似解的存在性。许多参数可以很容易地适用于更一般的非线性。
Standing Ring Blowup Solutions for Cubic NLS
证明了三维聚焦立方非线性薛定谔方程在环上L^2集中、环外H^1范数有界以及整体H^1范数以对数-对数速率增长的意义下存在解在圆上爆破.类似的行为发生在任何更高的维度。也就是说，存在这样的数据，对于这些数据，立方NLS的相应演化在一组余维2上爆炸。为了简化说明，证明是在三维，与注释，以表明在更高的维度的适应。
Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$
本文研究实数轴上KDNLS方程的整体适定性与长期行为，通过能量方法和频率局部化规范变换，建立了加权Sobolev空间中小初值解的整体存在性、最优衰减及修正散射的精确渐近轮廓。
In this paper we consider Schr{ö}dinger equations with nonlinearities of odd order 2$σ$ + 1 on T^d. We prove that for $σ$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{ö}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3.
Sign-changing multi-peak standing waves of the NLSE with a point interaction
本文通过 Lyapunov--Schmidt reduction 方法，证明了在特定参数条件下具有点相互作用的 NLSE 存在变号多峰驻波解。
We consider the periodic standing waves in the derivative nonlinear Schrodinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup-Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented with the numerical approximation of the spectral bands, this enables us to fully characterize the modulational instability of the periodic standing waves in the DNLS equation.
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