I. Rodnianski, W. Schlag, A. Soffer
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time goes to infinity) of a linear combination of N non-colliding solitons.
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We consider soliton-like solutions of a variable coefficients, subcritical nonlinear Schrodinger equation (NLS). In a previous result, we proved the existence of a pure, global-in-time, generalized soliton with prescribed asymptotic as t tends to -infinity. In addition, we described the corresponding dynamics for all time, up to a small error term. In this paper we prove that the soliton is never pure as t tends to infinity. Indeed, we give a precise sharp lower bound on the defect induced by the potential on the soliton. This result shows the existence of nontrivial dispersive effects acting …
We use the inverse scattering transform, the auto-Backlund transformation and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrodinger) equation.
We extend the global existence result for the derivative NLS equation to the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Bäcklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann--Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently.
The monotonicity conjecture and stability of solitons for the Cubic-Quintic NLS on R^3
本文通过交叉约束变分问题、能量极小元唯一性及几何谱分析，解决 R^3 上三次–五次 NLS 的单调性猜想，并分类各频率孤子的稳定性。
We review some recent results concerning Gibbs measures for nonlinear Schroedinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.
Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
Stationary Solitons in discrete NLS with non-nearest neighbour interactions
本文研究了扩展的一维离散NLS模型中非最近邻相互作用下的稳态离散孤子构造，通过动力系统方法在参数空间大区域内精确构建了稳定孤子。
We address stability of multi-solitons in the cubic NLS (nonlinear Schrödinger) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we obtain the orbital stability of multi-solitons in the $L^2(\mathbb{R})$ space when the initial data is in a weighted $L^2(\mathbb{R})$ space.
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schrö\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.
A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton
该研究针对聚焦五次非线性薛定谔方程，证明了质量与孤子质量相等的爆破解仅为两类：重标度孤子及其伪共形变换，明确了该方程对应质量条件下爆破解的形式。
Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrödinger equation in $\R^{N}, N\ge 3$, in presence of a singular external potential.
Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
该研究证明耦合NLS和修正KdV方程的多孤子（含呼吸子）在特定Sobolev空间中非线性稳定，并给出Lyapunov泛函负特征值个数，方法基于可积性，同时推广至经典方程及李代数框架。
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.
We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which was recently developed by D. Pelinovsky and Y. Shimabukuro, and an auto-Bäcklund transform in order to include solitons.
We study the two dimensional non-linear Schrödinger equation with two types of exponential non-linearities. It is well-known by a work of Ruf - Sani, that such models support solitary wave solutions, which are solutions of some constrained minimization problem. We show that these Ruf - Sani solitons, are spectrally unstable as well as unstable by blow-up.
Asymptotic Stability of multi-solitons for $1$d Supercritical NLS
研究一维\(L^2\)超临界非线性薛定谔方程多孤子渐近稳定性。结合调制分析、线性理论和精心设计的范数，证明在有限余维中心稳定流形上，速度不同且充分分离的多孤子的渐近稳定性。
Transmission of fast solitons for the NLS with an external potential
该研究探讨含外势的三次NLS方程中加速孤子的动力学，证明足够大速度下孤子可有效传输，拓展了前人关于δ势两类情形的相关研究成果。
Modeling soliton-potential interactions for the 1-D cubic NLS via an invariant-preserving finite-difference method
该研究针对带光滑快速衰减空间依赖势的一维三次NLS，采用保质量和能量不变量的有限差分法，对不同入射速度下的孤子-势相互作用进行建模。
Monotonicity Conjectures and Sharp Stability for Solitons of the Cubic-Quintic NLS on R^3
本文研究了R^3上的立方-五次非线性Schrödinger方程，完全解决了关于孤子频率与质量单调性的两个猜想，证明了能量极小元的唯一性，建立了孤子的锐稳定性，并首次给出了规范化解的分类。
We consider a Nonlinear Schrödinger Equation with a very general non linear term and with a trapping $δ$ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian symmetric spaces $G/SO(N)$. The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group $G$. This is shown to yield the two known $U(N-1)$-invariant vector generalizations of the nonlinear Schrodinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as $U(N-1)$-invariant vector…
In this paper we give the complete classification of solitons for a cubic NLS equation on the simplest network with a non-trivial topology: the tadpole graph, i.e. a ring with a half-line attached to it and free boundary conditions at the junction. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: 1) a denumerable family of continuous branches of embedded solitons bifurcating from linear eigenstates and threshold resonances of the system; 2) a continuous branch of edge solitons displaying a pitchfork symmetry breaking bifurcation at the…
Confinement and orbital stability of solitons of the NLS equation on metric graphs
研究亚临界聚焦NLS方程在非紧度量图上的孤子行为，证明孤子约束与轨道稳定性，并推广到含势或δ相互作用的情形。
Two-solitons with logarithmic separation for 1D NLS with repulsive delta potential
研究一维带排斥δ势的聚焦非线性薛定谔方程，证明弱势下可构造对数分离双孤子解，强势下则不存在此类解。
Sharp Stability of Solitons for the Cubic-Quintic NLS on R^2
本文研究了R^2上的立方-五次非线性Schrödinger方程，通过引入并求解一族新变分问题，获得关键单调性与唯一性结果，进而证明各频率下孤子的轨道稳定性并首次对归一化基态进行分类。
Global dynamics below excited solitons for the non-radial NLS with potential
该研究针对带外势的三维三次非线性薛定谔方程，将Nakanishi关于径向情形下小质量激发孤子以下解的全局动力学结果，推广至非径向情形，证明此类解要么散射到基态，要么H¹范数随时间增长。
We construct a local Lipschitz graph around a soliton of the cubic focusing NLS in three dimensions on which global solutions exist, and asymptotic stability as well as scattering holds.
Dirac solitons in one-dimensional nonlinear Schrödinger equations
本文通过构造狄拉克孤子，证明了非线性狄拉克方程可作为非线性薛定谔方程的有效模型。
We consider the one dimensional focusing (cubic) Nonlinear Schrödinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${\bf M}_N$ of pure $N$-soliton states, and their associated multisoliton solutions. We prove that (i) the set ${\bf M}_N$ is a uniformly smooth manifold, and (ii) the ${\bf M}_N$ states are uniformly stable in $H^s$, for each $s>-\frac12$. One main tool in our analysis is an iterated Backlund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove…
Threshold solutions for the 3d cubic-quintic NLS
本文研究了三维立方-五次 NLS 在正 virial 区域边界（仅由孤子达到部分）上解的行为，证明非孤子解要么双向散射，要么与单向散射且另一向指数收敛于孤子的特解重合。
We study instability of a vortex soliton $e^{i(mθ+ωt)}ϕ_{ω,m}(r)$ to $$iu_t+Δu+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$$ where $n=2$, $m\in\N$ and $(r,θ)$ are polar coordinates in $\R^2$. Grillakis \cite{Gr} proved that every radially standing wave solutions are unstable if $p>1+4/n$. However, we do not have any examples of unstable standing wave solutions in the subcritical case $(p<1+n/4)$. Suppose $ϕ_{ω,m}$ is nonnegative. We investigate a limiting profile of $ϕ_{ω,m}$ as $m\to\infty$ and prove that for every $p>1$, there exists an $m_*\in \N$ such that for $m\ge m_*$, a vortex soli…
We study the one-dimensional nonlinear Schrödinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound sta…
Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to addition of a small decaying potential and remains spectrally stable when it is pinned to t…
Transverse Instability and Bifurcation Analysis of the line soliton for the NLS equation on a Fractured Strip
针对断裂带区域上带吸引δ相互作用与幂次非线性的二维NLS方程，确定线孤子横向不稳定性的临界宽度，构造分岔解并给出其稳定性准则。
Collision of two solitons for $1d$ Nonlinear Schrodinger Equation with the same mass
本文研究一维多幂次非线性薛定谔方程中两个等质量孤子的碰撞，证明当入射速度足够小时，碰撞后出射速度与入射速度之差为 O(v^{k})，并应用于三次及三次-五次 NLS 等模型。
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. This paper proves a "statistical version" of this conjecture at mass-subcr…
We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ Recently, for the space-time region $|x/(2t)|<1$ which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time a…
Large-time existence results for the nonlocal NLS around ground state solutions
本文研究非局部非线性 Schrödinger 方程在基态孤子附近的动力学，利用基态附近的双曲动力学证明初始数据接近基态孤子时，解可在大时间内保持接近孤子轨道，并揭示其局部结构稳定性。
Asymptotic stability of solitons for near-cubic NLS equation with an internal mode
该文研究近立方非线性薛定谔方程的孤波摄动问题，证明小孤波线性化问题存在唯一内模，并在费米黄金定则假设下证得小频率孤波的渐近稳定性。
Dynamics of the black soliton in a regularized nonlinear Schrodinger equation
本文研究正则化散焦非线性薛定谔方程中黑孤子的动力学，证明了其时间演化的适定性与随正则化参数阈值变化的谱稳定性，并通过数值有限差分法展示了相关动力学，指出轨道稳定性问题因函数空间不匹配而未决。
We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations $$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz e…
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。
A note on averaging for the dispersion-managed NLS
该注记研究快色散管理 regime 下色散管理非线性薛定谔方程的平均化方法，给出其在构造类孤子解问题中的应用。
The Peregrine soliton $Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$ is an exact solution of the 1d focusing nonlinear schrödinger equation (NLS) $iB_t+B_{xx}=-2|B|^2B$, having the feature that it decays to $e^{it}$ at the spatial and time infinities, and with a peak and troughs in a local region. It is considered as a prototype of the rogue waves by the ocean waves community. The 1D NLS is related to the full water wave system in the sense that asymptotically it is the envelope equation for the full water waves. In this paper, working in the framework of water waves which decay non-tangentia…
Approximate Peregrine Solitons in Dispersive Nonlinear Wave Equations
本文将NLS近似的有效性从Sobolev空间推广到周期部分与实轴部分之和的函数空间，从而能利用Peregrine解在更复杂系统中研究怪波动力学。
In this paper, we consider a Hamiltonian system combining a nonlinear Schr\" odinger equation (NLS) and an ordinary differential equation (ODE). This system is a simplified model of the NLS around soliton solutions. Following Nakanishi \cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on $L^4$ in time norms.
We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schrödinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main to…
New logarithmic power nonlinear Schrödinger equations with super-Gaussons
该研究提出了一种新的对数幂非线性薛定谔方程，允许超高斯形局域解，并通过数值模拟探讨了其在非线性光学和凝聚态物理中的应用。
We give a unified proof of the step to find Darboux coordinates and of the ensuing Birkhoff normal forms procedure, developed in the course of the proof of asymptotic stability of solitary waves for NLS and special cases of Nonlinear Dirac equations
We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type. In stark contrast to the standard soliton solutions for non-degenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $1$ dimension, ODE methods apply, however the models considered have formally conserved Hamiltonian, Mass and Momentum functionals, which allow for variational analysis as well.
We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.
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