For the focusing mass-critical NLS $iu_t+Δu=-|u|^{\frac 4d}u$, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for $H^1_x$ initial data in dimensions $d=2,3$ with spherical symmetry and $d\ge 4$ with certain splitting-spherically symmetric initial data.
Nonlinear Schrödinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically.
We consider the initial-value problem for the one-dimensional nonlinear Schrödinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as $t\to\infty$. In particular, we establish the asymptotic stability of the family of small solitary waves.
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 prove existence and stability results for a two-parameter family of solitary-wave solutions to a system in which an equation of nonlinear Schrödinger type is coupled to an equation of Korteweg-de Vries type. Such systems model interactions between short and long dispersive waves. The results extend earlier results of Angulo, Albert and Angulo, and Chen. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we establish the sub-a…
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…
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state solitary wave, which is then shown to be the previously constructed solution. Finally, we pres…
We construct in this paper global (for $t \geq 0$) and bounded solutions $u(t)$ for the nonlinear Schrödinger equation \[i \partial_t u + Δu + |u|^{p-1} u = 0, \quad t \in \mathbb{R}, x \in \mathbb{R}^d\] in mass sub-critical cases ($1 < p < 1 + \frac{4}{d}$) and mass super-critical ($1 + \frac{4}{d} < p < \frac{d+2}{d-2}$) such that $u(t)$ decomposes asymptotically into two solitary waves with logarithmic distance \[ \|u(t) - e^{i γ(t)} \sum_{k=1}^2 Q(\cdot - x_k(t))\|_{H^1} \to 0 \] and \[|x_1(t) - x_2(t)| \sim 2 \log t, \quad \mbox{as}t \to + \infty.\] The logarithmic distance is related to…
We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conform…
Local exact controllability of the 1D NLS (subject to zero boundary conditions) with distributed control is shown to hold in a $H^1$--neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to J.-L. Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail.
We consider dispersion generalized nonlinear Schrödinger equations (NLS) of the form $i \partial_t u = P(D) u - |u|^{2 σ} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $σ\in \mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.
We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4ω>c^2$ and the critical parameters $4ω=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation sym…
Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schrödinger equation $iu_t+Δu=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.
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…
In this paper, we continue the study in \cite{MiaoTX:DNLS:Stab}. We use the perturbation argument, modulational analysis and the energy argument in \cite{MartelMT:Stab:gKdV, MartelMT:Stab:NLS} to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $σ>1$ we considered corresponds to the $L^2$-supercritical case.
We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument…
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.
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.
We consider a class of $1D$ NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.
Asymptotic stability of fast solitary waves to the Benjamin Equation
Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS（德尼·普瓦松研究所，图尔大学，奥尔良大学，法国国家科学研究中心）；Laboratory of Mathematics-EDST, Department of Mathematics, Faculty of Sciences I, Lebanese University（黎巴嫩大学理学院数学系数学与EDST实验室）
We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-s…
In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig-Merle's compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.
In this paper, we study the critical norm conjecture for the inter-critical nonlinear Schr{ö}dinger equation with critical index $s_c$ satisfying $\frac{1}{2}<s_c<1$ when $d\geq 5$. Under the assumption of uniform boundedness of the critical norm, we prove the global well-posedness and scattering for the Cauchy problem. We follow the standard `Concentration compactness/Rigidity method' established in \cite{KenigMerle1,KenigMerle2}, and treat three scenarios for the critical element respectively. Moreover, double Duhamel method and interaction Morawetz estimate are applied to exclude the critic…
We consider the energy super critical nonlinear Schrödinger equation $$i\pa_tu+Δu+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding he…
We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $κ>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $κ= 1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $κ\neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Keni…
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones …
Threshold Scattering for the Energy-Critical NLS with a Repulsive Inverse Square Potential
The Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；School of Mathematics, SouthEast University（东南大学数学学院）；Department of Mathematics, University of Iowa（爱荷华大学数学系）
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
On scattering for NLS: rigidity properties and numerical simulations via the lens transform
CNRS & Univ. Rennes IRMAR - UMR 6625（法国国家科学研究中心与雷恩大学）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
We consider the focusing nonlinear Schrödinger equation $i u_t + Δu + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + Δv + (|x|^{-(N-γ)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $γ<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1(\mathbb{R}^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [16] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,14]); for the generalized Hartre…
We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb tψ=-iα\cdot \nabla ψ+m βψ-f(ψ\sp\ast βψ) βψ$, $ψ(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp 2(\R)$, where $α_j$, $j = 1,...,n$, and $β$ are $N \times N$ Hermitian matrices which satisfy $α_j^2=β^2=I_N$, $α_j β+βα_j=0$, $α_j α_k + α_k α_j =2 δ_{jk} I_N$. We study the spectral stability of solitary wave solutions $ϕ(x)e^{-iωt}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $ω\to m$, proving that if $k>2/n$, then one positive…
The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schrödinger equation (NLS) as an effective model for the wavepacket envelope. We provide a detailed proof of this approximation result for the Gross-Pitaevskii equation (GP) and a semilinear wave equation, both with periodic coefficients in $\mathbb{N}\ni d$ spatial dimensions and with cubic nonlinearities. The proof is carried out in Bloch expansion variables with estimates in an $L^1$-type norm, which translates to an estimate…
In this paper, we consider the $L_x^2$ solution $u$ to mass critical NLS $iu_t+Δu=\pm |u|^{\frac 4d} u$. We prove that in dimensions $d\ge 4$, if the solution is spherically symmetric and is \emph{almost periodic modulo scaling}, then it must lie in $H_x^{1+\eps}$ for some $\eps>0$. Moreover, the kinetic energy of the solution is localized uniformly in time. One important application of the theorem is a simplified proof of the scattering conjecture for mass critical NLS without reducing to three enemies(see the work of Killip-Tao-Visan, and Killip-Visan-Zhang). As another important application…
In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\partial_t u+(u_{xx}+u^5-γu|u|^{q-1})_x=0$, where $q>5$ and $0<γ\ll1$. For any initial data $u_0\in H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solitons, and our goal is to classify the dynamics near soliton. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $γ^{-2/(q-1)}$, as $γ\rightarrow0$; (ii) the solution is always in a sma…
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
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…
We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schrödinger equation (DNLS) in dimensions $d=1,2$ and $3$. These states are obtained via a bifurcation analysis about the continuum (NLS) limit. One family consists {\it on-site symmetric} (vertex-centered) states; these are spatially localized solitary standing waves which are symmetric about any fixed lattice site. The other spatially localized states are {\it off-site symmetric}. Depending on the spatial dimension, these may be bond-centered, cell-centered, or face-centered. Finally, we show that t…
