Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation
Patrick Gerard, Zhifei Zhang
本文利用Zakharov-Shabat逆散射方法证明了一维Gross-Pitaevskii方程定常行波解的非线性轨道稳定性。
In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation by using Zakharov-Shabat's inverse scattering method.
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We consider a Gross-Pitaevskii equation with a nonlocal interaction potential. We provide sufficient conditions on the potential such that there exists a range of speeds in which nontrivial traveling waves do not exist
Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces
我们考虑所谓的Gross-Pitaevskii方程补充非标准边界条件。在Zhidkov空间中证明了与这类方程的初值问题有关的两个数学结果。
The energy space for the Gross-Pitaevskii equation with magnetic field
研究了具有磁场和无穷远处非零条件的Gross-Pitaevskii方程的能量空间。给出了磁场能量空间非空的充要条件。
Extensions to higher-dimensions are given for a convolution estimate used by Klainerman and Machedon in their study of uniqueness of solutions for the Gross-Pitaevskii hierarchy. Such estimates determine more general forms of Stein-Weiss integrals involving restriction to smooth submanifolds.
We prove existence and uniqueness of a positive solution to a system of two coupled Gross-Pitaevskii equations. We give a full asymptotic expansion of this solution into powers of the semi classical parameter $\varepsilon$ in the Thomas--Fermi limit $\varepsilon\to 0$.
Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term
本文在空间$\Real^2$中建立了带转动角动量项的Gross-Pitaevskii方程Cauchy问题的整体适定性.
On the linear wave regime of the Gross-Pitaevskii equation
本文研究了Gross-Pitaevskii方程对应于模为1的定常态扰动的长波渐近性。我们表现出下界的第一次出现可能的零（涡），并比较相应的解决方案的线性波动方程或变量的解决方案。结果依赖于使用的Madelung变换，产生的流体动力学形式的格罗斯-Pitaevskii方程，以及一个增强系统。
We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.
Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains
我们证明了3维非俘获域外部散焦三次非线性Schroedinger和Gross-Pitaevskii方程在能量空间中的全局适定性。主要成分是一个半经典的Eschhartz估计与外部域的平滑效果相结合，得到Eschhartz估计。
We prove the global-in-time well-posedness of the one dimensional Gross-Pitaevskii equation in the energy space, which is a complete metric space equipped with a newly introduced metric and with the energy norm describing the $H^s$ regularities of the solutions. We establish a family of conserved energies for the one dimensional Gross-Pitaevskii equation, such that the energy norms of the solutions are conserved globally in time. This family of energies is also conserved by the complex modified Korteweg-de Vries flow.
We study dynamics of vortices in solutions of the Gross-Pitaevskii equation $i \partial_t u = Δu + \varepsilon^{-2} u (1 - |u|^2)$ on $\mathbb{R}^2$ with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter $\varepsilon$. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [21].
In the nineties, Klein, Majda and Damodaran have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross-Pitaevskii equation.
Orbital stability of the black soliton to the Gross-Pitaevskii equation
我们建立了一维Gross-Pitaevskii方程的黑孤子或扭结解$\v_0(x) = þ\big(\frac{x}{\sqrt{2}} \big)$的轨道稳定性。
We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.
Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions
本文在空间$\Real^3$中建立了具有角动量转动项的Gross-Pitaevskii方程Cauchy问题的整体适定性，其中角速度等于各向同性捕获频率.
We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument.
Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.
We study the Gross-Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with non-zero boundary condition at infinity, in any dimension. We focus on even potentials that are positive definite or positive tempered distributions.
On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves
我们提供了一个严格的数学推导收敛在长波跨音速极限的最小化行波的二维Gross-Pitaevskii方程向基态的Kadomtsev-Petviashvili方程（KP I）。
In this paper we consider traveling waves for the Gross-Pitaevskii equation which are T-periodic in each variable. We prove that if T is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small T the problem admits only constant solutions.
We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center-unstable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stab…
We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP). Under some assumptions on the scattering data, these well-posedness results provide regular solutions for (GP). We also construct particular solutions, called $N-$soliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when $t\rightarrow\pm \infty$ is also made.
In this paper, we justify the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition on the 3-D upper space under the assumption that the leading order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in \cite{C…
We study the dynamics of vortices in an inhomogeneous Gross-Pitaevskii equation $i \partial_t u = Δu + {1\over \varepsilon^2} (p_\varepsilon^2(x) - |u|^2)$. For a unique scaling regime $|p_\varepsilon(x) - 1 | = O(|\log \varepsilon|^{-1})$, it is shown that vortices can interact both with the background perturbation and with each other. Results for associated parabolic and elliptic problems are discussed.
The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof uses the I-method which is complicated by the fact that no L^2 -conservation law holds. This improves former results of Bethuel-Saut and Gerard.
We prove existence and qualitative properties of ground state solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation. Using a mountain pass argument on spheres and constructing appropriately localized Palais-Smale sequences we are able to prove existence of real positive ground states as saddle points. The analysis is deployed in the set of possible states, thus overcoming the problem that the energy is unbounded below. We also prove a corresponding nonlocal Pohozaev identity with no rest term, a crucial part of the analysis.
Gray and black solitons of nonlocal Gross-Pitaevskii equations: existence, monotonicity and nonlocal-to-local limit
本文研究非局部Gross-Pitaevskii方程暗孤子的存在性、单调性及非局部到局部极限，证明了对称孤子存在并建立了收敛性。
We study the well-posedness of two systems modeling the non-equilibrium dynamics of pumped decaying Bose-Einstein condensates. In particular, we present the local theory for rough initial data using the Fourier restricted norm method introduced by Bourgain. We extend the result globally for initial data in $L^2$.
Global-in-time Well-posedness of the One-dimensional Hydrodynamic Gross-Pitaevskii Equations without Vacuum
本文通过马德卢格变换，建立了无真空情况下一维流体格罗斯-皮塔耶夫斯基方程在特定函数空间中的全局时间well-posed性，核心贡献是函数空间间的局部bilipschitz等价性。
Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems
我们考虑一个R^2中的Gross-Pitaevskii方程组，它模拟了两个具有排斥作用的玻色-爱因斯坦凝聚体的混合物。我们的目标是研究基态和激发态溶液的定性行为。我们允许两个不同的谐波和偏心的捕获势和研究的空间图案的解决方案内的Crachas-Fermi近似，以及大的相互作用制度内的相分离现象。
We study the Gross-Pitaevskii equation in dimension two with periodic conditions in one direction, or equivalently on the product space $\mathbb{R} \times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}.$ We focus on the variational problem consisting in minimizing the Ginzburg-Landau energy under a fixed momentum constraint. We prove that there exists a threshold value for $L$ below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.
In the previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In particular, our result proves the non degeneracy of these travelling waves, which is a key step in the classification of these waves and for the construction of multi-travelling waves.
We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite. The proof uses a suitable modification of the I-method.
We consider the final-data problem for systems of nonlinear Schrödinger equations with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in $L^2(\mathbb{R}^d)$, extending the result of J. Murphy to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross-Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight.
Travelling waves for the Gross-Pitaevskii equation II
本文的目的是提供一个严格的数学证明的存在性行波解的Gross-Pitaevskii方程在二维和三维。我们的论点，基于约束下的最小化，产生一个完整的分支的解决方案，并延长早期的结果，只有一部分的分支。在三维空间中，我们也证明了不存在小能量的行波解。
We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the e…
Logarithmic Gross-Pitaevskii equation
本文研究带对数非线性及无穷远非平凡边界条件的薛定谔方程，证明其柯西问题在能量空间中全局适定，并刻画了一维孤立波与行波。
Fractional Gross-Pitaevskii equations in non-Gaussian attractive Bose-Einstein condensates
本文研究非高斯吸引玻色-爱因斯坦凝聚相关的分数阶Gross-Pitaevskii方程的归一化解，证明了粒子数阈值N*的存在性及不同参数下解的存在性情况，还分析了解在α趋近于2时的渐近行为。
We construct a family of conserved energies for the one dimensional Gross-Pitaevskii equation, but in the low regularity case (in \cite{KL} we have constructed conserved energies in the high regularity situation). This can be done thanks to regularization procedures and a study of the topological structure of the finite-energy space. The asymptotic (regularised conserved) phase change on the real line with values in $ \R/2π\Z$ is studied. We also construct a conserved quantity, the renormalized momentum $H_1$ (see Theorem \ref{thm:E1}), on the universal covering space of the finite-energy spac…
We pursue our work on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schrödi…
In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds $\pm \sqrt{2}$, each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimat…
We consider the nonlocal Gross-Pitaevskii equation that models a Bose gas with general nonlocal interactions between particles in one spatial dimension, with constant density far away. We address the problem of the existence of traveling waves with nonvanishing conditions at infinity, i.e. dark solitons. Under general conditions on the interactions, we prove existence of dark solitons for almost every subsonic speed. Moreover, we show existence in the whole subsonic regime for a family of potentials. The proofs rely on a Mountain Pass argument combined with the so-called "monotonicity trick", …
A new proof of a Liouville theorem for the one dimensional Gross-Pitaevskii equation
本文通过线性化算子的分解恒等式简化谱分析，为一维Gross-Pitaevskii方程黑孤子和暗孤子渐近稳定性所依赖的Liouville定理给出了一个新证明。
In this short report we give a proof of the existence of a stationary solution to the Gross-Pitaevskii equation in $2d$ driven by a space-time white noise.
We study the generalized point-vortex problem and the Gross-Pitaevskii equation on surfaces of revolution. We find rotating periodic solutions to the generalized point-vortex problem, which have two two rings of $n$ equally spaced vortices with degrees $\pm 1$. In particular we prove the existence of such solutions when the surface is longitudinally symmetric. Then we seek a rotating solution to the Gross-Pitaevskii equation having vortices that follow those of the point-vortex flow for $\varepsilon$ sufficiently small.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gérard \cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
我们给出了一个新的证明的唯一性的解决方案的Gross-Pitaevskii的层次，首先建立了由Erdos，Schlein和丘，在不同的空间，基于时空估计。
The Gross-Pitaevskii equation with white noise in time perturbations of the harmonic potential is considered. In this article we define a Crank-Nicolson scheme based on a spectral discretization and we show the convergence of this scheme in the case of cubic non-linearity and when the exact solution is uniquely defined and global in time. We prove that the strong order of convergence in probability is at least one.
In this paper we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic potential and an additive space-time white noise. We study the well-posedness of the model using an inhomogeneous Wick renormalization due to the potential, and prove the existence of an invariant measure and of stationary martingale solutions.
We prove an adiabatic theorem for the non-autonomous Gross-Pitaevskii equation in the case of a weak trap. More precisely, we assume that the external potential decays suitably at infinity and admits exactly one bound state.
Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions
我们研究了两个和三个空间维度的Gross-Pitaevskii方程在时间无穷远处接近非零常数平衡点的解的渐近行为。我们构造了一类具有指定色散渐近行为的整体解，该解是根据线性化发展给出的。
In 1982, Ball, Marsden, and Slemrod proved an obstruction to the controllability of linear dynamics with a bounded bilinear control term. This note presents an example of nonlinear dynamics with respect to the state for which this obstruction still holds while the control potential is not bounded.
We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.
Bäcklund transformations are applied to study the Gross-Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwartzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross-Pitaevskii equation are obtained.
We consider systems of $N$ bosons in $\mathbb{R}^3$, trapped by an external potential. The interaction is repulsive and has a scattering length of the order $N^{-1}$ (Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit $N\to \infty$.
Scattering for the Gross-Pitaevskii equation
本文研究了Gross-Pitaevskii（或Ginzburg-Landau Schroedinger）方程在时间无穷远时非零常数平衡点附近解的渐近行为。证明了在大于3维的空间中，小扰动在无穷远时刻可以用线性化演化来逼近，并且在某些索伯列夫空间中，波算子在0附近是同胚的。
We review some mathematical work on the Bose gas in the Gross-Pitaevskii regime. We start with the classical results by Lieb, Seiringer and Yngvason on the ground state energy and by Lieb and Seiringer on the existence of Bose-Einstein condensation. Afterwards, we discuss some more recent progress, based on a rigorous version of Bogoliubov theory.
We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to …
We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy ident…
In this paper we study the bilinear-control problem for the linear and non-linear Schr{ö}dinger equation with harmonic potential. By the means of different examples, we show how space-time smoothing effects (Strichartz estimates, Kato smoothing effect) enjoyed by the linear flow, can help to prove obstructions to controllability.
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