This paper discusses about solutions of the nonlocal nonlinear Schrodinger equation. We prove that the solution remains close to the orbit of the soliton for a large-time, if the initial data is close to the ground state solitons. The proof uses the hyperbolic dynamics near ground state, which exhibits properties of local structural stability of solutions with respect to the flows of the nonlocal nonlinear Schrodinger equation.
We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E^s_σ$ for which the norms are defined by $$ \|f\|_{E^s_σ} = \|\langleξ\rangle^σ2^{s|ξ|}\hat{f}(ξ)\|_{L^2}, \ s<0, \ σ\in \mathbb{R}. $$ Any Sobolev space $H^{r}$ is a subspace of $E^s_σ$, i.e., $H^r \subset E^s_σ$ for any $ r,σ\in \mathbb{R}$ and $s<0$. Let $s<0$ and $σ>-1/2$ ($σ>0$) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to $E^s_σ$ and their Fourier transforms are supported in $(0, \infty)$…
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 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…
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）
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.
We apply the GBDT version of the Bäcklund-Darboux transformation to the nonlocal NLS (focusing and defocusing cases). The matrix case is included and solutions in the form of rectangular $m_1 \times m_2$ matrix functions are dealt with. The wave function is also constructed explicitly. Families of explicit examples are considered in detail. Some initial results and representations for the more complicated algebro-geometric solutions are obtained as well.
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…
We prove the nonexistence of ground states for NLS on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass.
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
We prove orbital stability result for physical ground states of a nonlinear Schrödinger (NLS) equation in the sense that the set of these ground states is contained in the set of prescribed mass solutions which is orbital stable by the Cazenave-Lions theorem. We apply the nonlinear generalized Rayleigh quotients method which allows establishing a one-to-one correspondence between the values of the mass $m$, the frequency $λ$, and the action level $S$ of the physical ground states.
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.
In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.
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 transpose work by T.Mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 1D. As an application we extend to dimension 1D a result on asymptotic stability of ground states of NLS proved by Cuccagna & Mizumachi for dimensions $\ge 3$
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.
In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm assumption. We rigorously show that the spectral problem for the nonlocal NLS equation admits no eigenvalues or resonances, as well as Zhou vanishing lemma is effective under the $L^1(\R)$ small-norm assumption. With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse …
The paper aims at giving a first insight on the existence/nonexistence of ground states for the $L^2$-critical NLS equation on metric graphs with localized nonlinearity. In particular, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations.
We investigate the existence of ground states with fixed mass for the nonlinear Schrödinger equation with a pure power nonlinearity on periodic metric graphs. Within a variational framework, both the $L^2$-subcritical and critical regimes are studied. In the former case, we establish the existence of global minimizers of the NLS energy for every mass and every periodic graph. In the critical regime, a complete topological characterization is derived, providing conditions which allow or prevent ground states of a certain mass from existing. Besides, a rigorous notion of periodic graph is introd…
In this paper we generalize a weak sequential result of \cite{fan20182} to a non-scattering solutions in dimension $d \geq 2$. No symmetry assumptions are required for the initial data. We build on a previous result of \cite{dodson20202} for one dimension.
We study the nature of the Nonlinear Schrödinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.
We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration of the minimal mass blow up solution.
We consider the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schrödinger Energy with power nonlinearity on noncompact metric graphs. We first establish that the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass $μ$, up to an at most countable set of masses. Then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs. Finally we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. Our proofs are based on careful variat…
We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a global minimizer of the action functional on the Nehari manifold and the defect considered is a Fülöp-Tsutsui $δ$ type, namely a $δ$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results from the Grillakis-Shatah-Strauss theory.
We prove that standing-waves solutions to the non-linear Schrödinger equation in dimension one whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term $ G $ satisfies a Euler differential inequality. When the non-linear term $ G $ is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.
We investigate the existence of ground states of prescribed mass, for the nonlinear Schroedinger energy on a noncompact metric graph G. While in some cases the topology of G may rule out or, on the contrary, guarantee the existence of ground states of any given mass, in general also some metric properties of G, and their quantitative relation with the actual value of the prescribed mass, are relevant as concerns the existence of ground states. This may give rise to interesting phase transitions from nonexistence to existence of ground states, when a certain quantity reaches a critical threshol…
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 consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya-Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder.
We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrödinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ν$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.
We extend to a specific class of systems of nonlinear Schrödinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.
We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.
We consider the Kawahara model and two fourth order semi-linear Schrödinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. At the same time, we verify and clarify recent numerical simulations of the stability of these solitons. For the fourth order NLS models, we improve upon recent results on stability of very special, explicit solutions in the one dimensional case. Our multidimen…
In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schrödinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce and Vega for the KdV equa…
We investigate the existence of ground states for the focusing Nonlinear Schrödinger Equation on the infinite three-dimensional cubic grid. We extend the result found for the analogous two-dimensional grid by proving an appropriate Sobolev inequality giving rise to a family of critical Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from $10/3$ and $6$, namely, from the $L^2$-critical power for the same problem in $\mathbb{R}^3$ to the critical power for the same problem in $\mathbb{R}$. Given the Gagliardo-Nirenberg inequality, the problem of the existence of ground st…
We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^αu\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $α\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we…
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.
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_{s}^{\ast}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $μ\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we firs…
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…
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…
We consider a mass-critical system of nonlinear Schödinger equations \begin{align*} \begin{cases} i\partial_t u +Δu =\bar{u}v,\\ i\partial_t v +κΔv =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} where $(u,v)$ is a $\mathbb{C}^2$-valued unknown function and $κ>0$ is a constant. If $κ=1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(ϕ,ψ)$, where $M(u,v)$ denotes the mass and $(ϕ,ψ)$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Do…
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 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…
The uniqueness of the positive ground state solutions of fractional Shrodinger equations with a harmonic potential has not been covered by the breakthrough method developed in [1, 2]. It has remained an open question for years. [3] and [5] were quite recently able to prove the existence of the ground state solutions and to derive important properties but they failed to address the uniqueness. [3] left it as an open question, and [5] run numerical simulations that were in favor of the uniqueness. Very recently, the authors of this note developed a general and unified method to prove the uniquen…
We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend results of Cho, Hong, and Lee from the radial to the non-radial setting.
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.
