Adiabatic theorem for the Gross–Pitaevskii equation

We prove an adiabatic theorem for the nonautonomous semilinear Gross–Pitaevskii equation. More precisely, we assume that the external potential decays suitably at infinity and the linear Schrödinger operator ?Δ+V_s admits exactly one bound state, which is ground state, for any s∈[0,1]. In the nonlinear setting, the ground state bifurcates into a manifold of (small) ground state solutions. We show that, if the initial condition is at the ground state manifold, bifurcated from the ground state of ?Δ+V₀, then, for any fixed s∈[0,1], as 𝜀→0, the solution will converge to the ground state manifold bifurcated from the ground state of ?Δ+V_s. Moreover, the limit is of the same mass to the initial condition.     

Vortex helices for the Gross–Pitaevskii equation

We prove the existence of travelling vortex helices to the Gross–Pitaevskii equation in ${\mathbb{R}}^3$. These solutions have an infinite energy, are periodic in the direction of the axis of the helix and have a degree one at infinity in the orthogonal direction.     

Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.     

Effective Dynamics for N-Solitons of the Gross–Pitaevskii Equation

We consider several solitons moving in a slowly varying external field. We present results of numerical computations which indicate that the effective dynamics obtained by restricting the full Hamiltonian to the finite-dimensional manifold of N-solitons (constructed when no external field is present) provides a remarkably good approximation to the actual soliton dynamics. This is quantified as an error of size h ² where h is the parameter describing the slowly varying nature of the potential. This also indicates that previous mathematical results of Holmer and Zworski (Int. Math. Res. Not. 2008: Art. ID runn026, 2008) for one soliton are optimal. For potentials with unstable equilibria, the Ehrenfest time, log(1/h)/h, appears to be the natural limiting time for these effective dynamics. We also show that the results of Holmer et?al. (arXiv:0912.5122, 2009) for two mKdV solitons apply numerically to a larger number of interacting solitons. We illustrate the results by applying the method with the external potentials used in the Bose?CEinstein soliton train experiments of Strecker et?al. (Nature 417:150?C153, 2002).     

Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the $(x,y)$ -variables as well as the Hermite spectral method in the $z$ -direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.     

The limit equation for the Gross–Pitaevskii equations and S. Terraciniʼs conjecture

We establish the limit system for the Gross–Pitaevskii equations when the segregation phenomenon appears, and shows this limit is the one arising from the competing systems in population dynamics. This covers and verifies a conjecture of S. Terracini et al., both in the parabolic case and the elliptic case.     

Orbital stability of traveling waves for the one-dimensional Gross–Pitaevskii equation

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.     

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross–Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.     

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

We prove a Liouville-type theorem for semilinear parabolic systems of the form

$$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$

in the whole space \({\mathbb R}^N\times {\mathbb R}\). Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for \(m=2\) in dimension \(N\le 2\), and partial results in higher dimensions in the range \(p< N/(N-2)\). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions \(N\ge 3\). In particular, our results solve the important case of the parabolic Gross–Pitaevskii system—i.e. the cubic case \(r=1\)—in space dimension \(N=3\), for any symmetric (m, m)-matrix \((\beta _{ij})\) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space \({\mathbb R}^N_+\times {\mathbb R}\). As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

     

Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies

We consider solutions of the focusing cubic and quintic Gross–Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the L²$L^2$-critical or L²$L^2$-supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.     

Error Analysis of the Strang Time-Splitting Laguerre–Hermite/Hermite Collocation Methods for the Gross–Pitaevskii Equation

The aim of this paper is to carry out a rigorous error analysis for the Strang splitting Laguerre–Hermite/Hermite collocation methods for the time-dependent Gross–Pitaevskii equation (GPE). We derive error estimates for full discretizations of the three-dimensional GPE with cylindrical symmetry by the Strang splitting Laguerre–Hermite collocation method, and for the d-dimensional GPE by the Strang splitting Hermite collocation method.     

Bright soliton solution of a Gross–Pitaevskii equation

We theoretically and numerically study the bright soliton solutions of a Gross–Pitaevskii equation governing one-dimensional (1D)(cigar-shaped) Bose–Einstein condensates (BEC) trapped in an optical lattice of 1D structure. The analytical expression of bright soliton is derived by using the variational approximation, which completely matches the numerical results with a range of potential’s parameters. Moreover, we determined the parameter domains for the persistence and non-persistence of bright soliton solutions.     

On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy

In this paper, we present a uniqueness result for solutions to the Gross–Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³${\mathbb{R}}^3$ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H^α$H^{\alpha }$ for α>1$\alpha >1$. By constructing a specific counterexample, we show that, on T³${\mathbb{T}}^3$, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1$\alpha =1$. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α?1$\alpha ?1$. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.     

Persistence of the Thomas–Fermi approximation for ground states of the Gross–Pitaevskii equation supported by the nonlinear confinement

We justify the Thomas–Fermi approximation for the stationary Gross–Pitaevskii equation with the repulsive nonlinear confinement, which was recently introduced in physics literature. The method is based on the resolvent estimates and the fixed-point iterations. The results cover the case of the algebraically growing nonlinear confinement.