Wilhelm Schlag
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.
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In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.
We consider examples of discrete nonlinear Schroedinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schroedinger equations in eucliedean spaces. We do not investigate connections with work on quasi periodic solutions as in M.Johansson & S.Aubry and of Bamb…
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.
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…
We explain how spectrally stable vortices of the Nonlinear Schrödinger Equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete and continuous modes of the NLS.
We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler--Lagrange equation. As a consequence, for a sufficiently large ma…
We prove that a system of coupled nonlinear Schr{ö}dinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6]. This is the first example of unstable tori for a 1d PDE.
We study the nonlinear Schrödinger equation with an arbitrary real potential $V(x)\in (L^1+L^\infty)(Γ)$ on a star graph $Γ$. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $-γ<0$. We show the existence of ground states $φ_ω(x)$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on $V(x)$. Moreover, for $V(x)=-\dfracβ{x^α}$, in the supercritical case, we prove that the standing waves $e^{iωt}φ_ω(x)$ are orbitally unstable in $H^{1}(Γ)$ when $ω$ is large enough. Analogous result holds…
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.
We consider a nonlinear Schroedinger equation with a finite bands periodic potential in R . We assume the existence of an orbitally stable family of ground states. We prove that under appropriate hypotheses the ground states are asymptotically stable.
We study standing waves for a nonlinear Schrödinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $ i \frac{d}{dt}Ψ_t = H Ψ_t - | Ψ_t |^{2μ} Ψ_t $, where $H$ is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $ω> \fra…
We consider a nonlinear Schrödinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of $δ$-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass ($L^2$-norm). When existing, the minimizer is called ground state and it is…
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.
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 study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobo…
Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the period…
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.
We consider the nonlinear Schrödinger (NLS) equation with the subcritical power nonlinearity on a star graph consisting of $N$ edges and a single vertex under generalized Kirchhoff boundary conditions. The stationary NLS equation may admit a family of solitary waves parameterized by a translational parameter, which we call the shifted states. The two main examples include (i) the star graph with even $N$ under the classical Kirchhoff boundary conditions and (ii) the star graph with one incoming edge and $N-1$ outgoing edges under a single constraint on coefficients of the generalized Kirchhoff…
We consider a nonlinear Schrödinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $ i \partial_t Ψ(t) = H Ψ(t) - |Ψ(t)|^{2μ}Ψ(t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $δ$ interaction, with strength $α< 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0<μ<2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^*$ it attains its minimum value at a certain $\hat Ψ_m \in H^1(\GG) $, while for…
For the stationary nonlinear Schrödinger equation $-Δu+ V(x)u- f(u) = λu$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow to match the $L^2$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $L^2$-…
In this paper, we consider the following nonlinear Klein-Gordon equation \begin{align*} \partial_{tt}u-Δu+u=|u|^{p-1}u,\qquad t\in \mathbb{R},\ x\in \mathbb{R}^d, \end{align*} with $1<p< 1+\frac{4}{d}$. The equation has the standing wave solutions $u_ω=e^{iωt}ϕ_ω$ with the frequency $ω\in(-1,1)$, where $ϕ_ω$ obeys \begin{align*} -Δϕ+(1-ω^2)ϕ-ϕ^p=0. \end{align*} It was proved by Shatah (1983), and Shatah, Strauss (1985) that there exists a critical frequency $ω_c\in (0,1)$ such that the standing waves solution $u_ω$ is orbitally stable when $ω_c<|ω|<1$, and orbitally unstable when $|ω|<ω_c$. Fu…
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。
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…
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…
Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^2$ perturbatoins, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial…
Classification and stability of positive solutions to the NLS equation on the $\mathcal{T}$-metric graph
本文对 $\mathcal{T}$-度量图上NLS方程的正 $H^1$-解进行了完整分类，证明了作用基态的唯一性及特定条件下的轨道不稳定性，并提供了能量基态唯一性与稳定性的数值证据。
In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of the local cubic focusing NLS on $\mathbb{R}$. First, finite time blow-up solutions exist with arbitrarily small initial data in $H^s(\mathbb{R})$, for any $s\geqslant0$. On the other hand, the solitons of the local NLS, which are also solutions of the nonlocal equation, are unstable by blow-up for the latter.
Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$.…
Stability transitions of NLS action ground-states on metric graphs
研究了$\mathcal{T}$图和蝌蚪图上非线性Schrödinger方程作用基态的轨道稳定性，发现频率$λ$增大时基态在$L^2$-临界指数附近发生稳定与不稳定的交替转变，并通过渐近分析与数值模拟验证了该理论结果。
Dynamics of threshold solutions for energy critical NLS with inverse square potential
本文研究了能量临界NLS在逆平方势场下的阈值解动力学，证明了动能小于基态的解会散射或收敛到基态，而动能更大的解则在有限时间内发散，仅在五维中存在例外情况。
We study the periodic cubic derivative non-linear Schrödinger equation (dNLS) and the (focussing) quintic non-linear Schrödinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on the line ${\mathbb R}$. We describe the (three parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide a criteria for stabi…
In this note, we give a rigorous proof that the NLS periodic Akhmediev breather is unstable. The proof follows the ideas in [17], in the sense that a suitable modification of the Stokes wave is the global attractor of the local Akhmediev dynamics for sufficiently large time, and therefore the latter cannot be stable in any suitable finite energy periodic Sobolev space.
We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). …
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。
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。
Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation
该研究针对二次非线性薛定谔方程实初值全局适定的猜想，找到会引发爆破的反例初值，提出爆破初值集为对应非线性热方程零平衡点稳定流形的猜想，并用参数化方法给出启发性论证。
Nonlinear modulational instability of two-dimensional deep hydroelastic Stokes waves
本文研究二维深水弹性斯托克斯波的非线性模调不稳定性，通过推导立方非线性薛定谔方程的聚焦近似结果，证明长波扰动下斯托克斯波的非线性不稳定性。
In this paper we consider the existence of multi-soliton structures for the nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently of the unstable character of (NLKG) solitons, it is possible to construct a N-soliton family of solutions to (NLKG), of dimension 2N, globally well-defined in the energy space H^1 \times L^2 for all large positive times. The method of proof involves the generalization of previous works on supercritical NLS and gKdV equations by Martel, Merle and the first author to the wave case, where we replace the unstable mode associated to the linear N…
We consider the nonlinear Schrödinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable.
We extend our previous result on the nonlinear Klein-Gordon equation to the nonlinear Schrodinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which c…
We consider the quadratic and cubic KP - I and NLS models in $1+2$ dimensions with periodic boundary conditions. We show that the spatially periodic travelling waves (with period $K$) in the form $u(t,x,y)=\vp(x-c t)$ are spectrally and linearly unstable, when the perturbations are taken to be with the same period. This strong instability implies other instabilities considered recently - for example with respect to perturbations with periods $nK, n=2, 3, ...$ or bounded perturbations.
On stability and instability of the ground states for the focusing inhomogeneous NLS with inverse-square potential
本文研究含逆平方势的聚焦非齐次NLS方程，在质量次临界、质量临界及临界间三种情形下分别证明基态集合稳定，以及基态驻波因爆破或直接呈不稳定。
The long-time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. First, the IST is formulated over a single sheet of the complex plane without introducing a uniformization variable. The solution of the focusing NLS equation with nonzero boundary conditions is thus associated with a suitable ma…
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