We consider the initial-value problem for the cubic-quintic NLS \[ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ\] in three spatial dimensions in the class of solutions with $|ψ(x)|\to c >0$ as $|x|\to\infty$. Here $α_1$, $α_3$, $α_5$ and $c$ are such that $ψ(x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $ψ(x)\to 1$ as $|x|\to\infty$, we study the associated initial-value problem for $u=ψ-1$ and prove a scattering result for small initial data in a weighted Sobolev space.
We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.
We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
We study the quintic Non Linear Schrödinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.
We consider the defocusing energy-critical nonlinear Schrödinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.
We consider the cubic-quintic nonlinear Schrödinger equation in two space dimensions. For this model, X. Cheng established scattering for $H^1$ data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R. Carles and C. Sparber utilized the pseudoconformal energy estimate to obtain scattering at the sharp threshold for data belonging to a weighted Sobolev space. In this work, we remove the weighted assumption and establish scattering at the threshold for radial data in $H^1$.
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
We consider the nonlinear Schrödinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper we prove scattering in a larger region where the virial quantity is no longer guaranteed to be sign definite.
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…
This paper studies the quintic nonlinear Schrödinger equation on $\mathbb{R}^d$ with randomized initial data below the critical regularity $H^{\frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s \in (\frac{d-2}{2}, \frac{d-1}{2})$. The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.
Consider the defocusing quintic nonlinear Schrödinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.
In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d = 3$: $$ iu_t = -Δu + K_1(x)|u|^2u + K_2(x)|u|^4u. $$ We study local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $K_1, K_2 \in C^4(\mathbb R^3 \setminus \{0\})$ satisfy growth condition $|\partial^j K_i(x)| \lesssim |x|^{b_i-j}\, (j = 0, 1, 2, 3, 4)$ for some $b_i \ge 0$ and for $x \neq 0$. To this end we use the Sobolev inequality for the functions $f \in H^n \,(n = 1, 2)$ such that $\||\mathbf L|^\ell f\|_{H^n} < \infty \,(\ell = 1, 2)$, where $\mathbf L$ i…
We consider the periodic cubic-quintic nonlinear Schrödinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +Δ)u=μ_1 |u|^2 u+μ_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$ with $μ_1,μ_2\in \mathbb{R} \setminus\{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given $μ_1$ and $μ_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in the case $μ_2>0…
We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ$$ in three spatial dimensions in the class of solutions with $|ψ(x)|\to c >0$ as $|x|\to\infty$. This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state --- the limiting modulus $c$ corresponds to a local minimum in the energy density. Our arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross--Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geom…
We study the influence of Szeg{ö} projector $Π$ on the L 2 --critical one-dimensional non linear focusing Schr{ö}dinger equation, leading to the quintic focusing NLS-Szeg{ö} equation i$\partial$ t u + $\partial$ 2 x u + $Π$(|u| 4 u) = 0, (t, x) $\in$ R x R, u(0, $\times$) = u 0. This equation is globally well-posed in H 1 + = $Π$(H 1 (R)), for every initial datum u 0. The solution L 2-scatters both forward and backward in time if u 0 has sufficiently small mass. We prove the orbital stability with scaling of the traveling wave : u $ω$,c (t, x) = e i$ω$t Q(x + ct), for some $ω$, c > 0, where Q …
We consider the quintic nonlinear Schrödinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that thi…
We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +Δu + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(\mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(\mathbb{R}^3)$ with corresponding solutions that bl…
The aim of this paper is to show the small data scattering for 2D ICQNLS: $$iu_t=-Δu + K_1(x)|u|^2u+K_2(x)|u|^4u.$$ Under the assumption that $\left| \partial^j K_l \right| \lesssim |x|^{b_l -j}$ for $j=0, 1, 2, l=1, 2$ and $0 \le b_l \le l - \frac23$, we prove the small data scattering in an angularly regular Sobolev space $H_θ^{1,1}$. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space $H_θ^{1, 1} \subset H^1$ with angular regularity such that $\|\partial_θf\|_{H^1} < \infty$, and also use the recently developed angularly averaged Stri…
Asymmetric minimal mass blow-up for the quintic NLS on a star graph
Sapienza Università di Roma（罗马大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Università degli Studi di Pisa（比萨大学）；Waseda University（早稻田大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
We consider the nonlinear Schrödinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2π$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.
We consider the $L^{2}$-critical quintic focusing nonlinear Schrödinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$ spherically symmetric blow-up solutions of the $L^{2}$-critical focusing NLS, there is a minimal amount of concentration of the $L^{2}$-norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the one-dimensional case and rougher initial data, $(u_{0}\in H^{s}, s<1)$, without any additional assumption.
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of Duyckaerts and Merle (2009). When we try to obtain the corresponding result, we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag (2012).
For the Schrödinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.
We consider the nonlinear Schrödinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The core of this article is the notion of stability of solitary waves. We recall the two standard notions of orbital stability in the context of nonlinear Schrödinger equations, and show that they must be considered as independent from each other. We investigate numerically the notion of orbital stability of ground states in the radially symmetric case, confirming existing conjectures or leading to new ones.
We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrödinger equation. Our proof is based on, and extends, methods of Erdös-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.
In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schrödinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves with \textit{dnoidal} profile and such construction can be used to prove that the associated period map is strictly increasing in terms of the energy levels. The monotonicity is also useful to obtain the behaviour of the non-positive spectrum for the associated linearized operator around the wave. Concerning the stability, we prove that the dnoidal waves are or…
We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the…
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…
We prove that solutions to non-linear Schrödinger equations in two dimensions and in the exterior of a bounded and smooth star-shaped obstacle scatter in the energy space. The non-linear potential is defocusing and grows at least as the quintic power.
We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schrödinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.
We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to $H^s(\mathbb{T}) \setminus L^2(\mathbb{T})$ for some $s \in (-\frac18, 0)$. The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.
In this short note we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical NLS in four space dimensions obtained previously by Ryckman and Visan. The argument is inspired by the recent work of Dodson on the mass-critical NLS.
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.
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.
We prove global well-posedness for the L^{2}-critical cubic defocusing nonlinear Schrödinger equation on R^{2} with data u_{0} \in H^{s}(R^{2}) for s > {1/3}.
We review some recent results concerning the Initial Value Problem of 1d-cubic non-linear Schrödinger equation (NLS) and other related systems as the Schrödinger Map. For the latter we prove the existence of a cascade of energy. Finally, some new examples of the Talbot effect at the critical level of regularity are given.
We consider the cubic-quintic nonlinear Schrödinger equation: \[ i\partial_t u = -Δu - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter part of the paper we prove that scatt…
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.
Modified scattering for the cubic dispersion-managed NLS
University of Oregon（俄勒冈大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；National Key Laboratory of Computational Physics（计算物理学国家重点实验室）
In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.
