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437
WKB analysis for nonlinear Schrödinger equations with a potential
 Comm. Math. Phys
"... Abstract. We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. ..."
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Cited by 53 (11 self)
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Abstract. We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity. (1.1) (1.2)
SECONDORDER CORRECTIONS TO MEAN FIELD EVOLUTION OF WEAKLY INTERACTING BOSONS. II.
"... Abstract. We study the evolution of a Nbody weakly interacting system of Bosons. Our work forms an extension of our previous paper I [13], in which we derived a secondorder correction to a meanfield evolution law for coherent states in the presence of small interaction potential. Here, we remove ..."
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Cited by 33 (2 self)
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Abstract. We study the evolution of a Nbody weakly interacting system of Bosons. Our work forms an extension of our previous paper I [13], in which we derived a secondorder correction to a meanfield evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the secondorder correction. This implies an improved Fockspace estimate for our approximation of the Nbody state. 1.
Mathematical theory and numerical methods for BoseEinstein condensation
 Kinet. Relat. Models
"... Abstract. In this paper, we mainly review recent results on mathematical theory and numerical methods for BoseEinstein condensation (BEC), based on the GrossPitaevskii equation (GPE). Starting from the simplest case with onecomponent BEC of the weakly interacting bosons, we study the reduction of ..."
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Cited by 30 (14 self)
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Abstract. In this paper, we mainly review recent results on mathematical theory and numerical methods for BoseEinstein condensation (BEC), based on the GrossPitaevskii equation (GPE). Starting from the simplest case with onecomponent BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, wellposedness of the Cauchy problem as well as the finite time blowup. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finitedifference methods andtimesplitting spectralmethods arereviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (ThomasFermi
Optimal error estimates of finite difference methods for the GrossPitaevskii equation with angular momentum rotation
 Math. Comp
"... Abstract. We analyze finite difference methods for the GrossPitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative CrankNicolson finite difference (CNFD) method and semiimplicit finite difference (SIFD) ..."
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Cited by 21 (13 self)
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Abstract. We analyze finite difference methods for the GrossPitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative CrankNicolson finite difference (CNFD) method and semiimplicit finite difference (SIFD) method, at the order of O(h 2 + τ 2)inthel 2norm and discrete H 1norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain apriori bound of the numerical solution in the l ∞norm by using the inverse inequality and the l 2norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods. 1.
Splitting and composition methods in the numerical integration of differential equations
, 2008
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On PhaseSeparation Model: Asymptotics and Qualitative Properties
"... In this paper we study bound state solutions of a class of twocomponent nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain th ..."
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Cited by 19 (2 self)
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In this paper we study bound state solutions of a class of twocomponent nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful to derive rigorously the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of [23]. When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blowup nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for this type of systems in higher dimensions.
Nonlinear waves in BoseEinstein condensates: physical relevance and mathematical techniques, preprint available at: http://wwwrohan.sdsu.edu/∼rcarrete
"... This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more infor ..."
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Cited by 18 (6 self)
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This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact
Dynamics of rotating Bose–Einstein condensates and its efficient and accurate numerical computation
 SIAM J. Appl. Math
"... Abstract. In this paper, we study the dynamics of rotating Bose–Einstein condensates (BEC) based on the Gross–Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentu ..."
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Cited by 17 (8 self)
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Abstract. In this paper, we study the dynamics of rotating Bose–Einstein condensates (BEC) based on the Gross–Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. By formulating the equation in either the twodimensional polar coordinate system or the threedimensional cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient timesplitting method which is time reversible, unconditionally stable, efficient, and accurate for the problem. Moreover, it conserves the position density. We also apply the numerical method to study issues such as the stability of central vortex states and the quantized vortex lattice dynamics in rotating BEC.
Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/GrossPitaevskii equations, Discrete Contin
 Dyn. Syst
"... Abstract. We consider a class nonlinear Schrödinger / GrossPitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and meanfield quantum manybody phenomena. It is kn ..."
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Cited by 16 (2 self)
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Abstract. We consider a class nonlinear Schrödinger / GrossPitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and meanfield quantum manybody phenomena. It is known that there is a critical L 2 norm (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking point. The source of this behavior in the full Hamiltonian PDE is related to the dynamics of a finitedimensional Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a shadowing theorem on the persistence of solutions, with oscillating masstransport between wells, on very long, but finite, time scales within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / radiation modes. (1.1)
Numerical analysis of nonlinear eigenvalue problems, Preprint arXiv:0905.1645
"... We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A∇u) + V u + f(u 2)u = λu, ‖u ‖ L 2 = 1. We focus in particular on the Fourier spectral approximation (for periodic pro ..."
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Cited by 16 (3 self)
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We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A∇u) + V u + f(u 2)u = λu, ‖u ‖ L 2 = 1. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P1 and P2 finiteelement discretizations. Denoting by (uδ, λδ) a variational approximation of the ground state eigenpair (u, λ), we are interested in the convergence rates of ‖uδ − u ‖ H 1, ‖uδ − u ‖ L 2, λδ − λ, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A, V and f satisfy certain conditions, λδ − λ  goes to zero as ‖uδ − u ‖ 2 H1 + ‖uδ − u‖L2. We also show that under more restrictive assumptions on A, V and f, λδ − λ  converges to zero as ‖uδ − u ‖ 2 H1, thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error uδ − u in negative Sobolev norms.