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.     

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.     

An adaptive algorithm for the Thomas–Fermi equation

A free boundary value problem is introduced to approximate the original Thomas–Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.     

The Sinc-collocation method for solving the Thomas–Fermi equation

A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results.     

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.     

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).     

Solution of the Thomas–Fermi equation with a convergent approach

The explicit analytic solution of the Thomas–Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao [1]. However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. Optimum convergence control parameter concept is used together with a mathematical proof of the convergence.     

Inverse scattering transformation and soliton stability for a nonlinear Gross–Pitaevskii equation with external potentials

We consider the Gross–Pitaevskii(GP) equation with the combination of periodic and harmonic external potentials. In particular, the method of inverse scattering transformation is applied to the GP equation with external potentials. Furthermore, some exact soliton solutions are obtained for the GP equation by using inverse scattering transformation, in which some physically relevant bright solutions are described. The stabilities of the obtained matter-wave solutions are addressed numerically such that some stable solutions are found, and some solitons can be stable in a wide region. These results may raise the possibility of relative experiments and potential applications.     

Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $\tau h^{-2}\le C$ where $\tau $ and $h$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $\eta $ then the associated numerical solution remains close to the orbit of $\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$ , for very long times.     

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.     

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.     

Multi-symplectic variational integrators for the Gross–Pitaevskii equations in BEC

In this paper, a variational integrator is constructed for Gross–Pitaevskii equations in Bose–Einstein condensate. The discrete multi-symplectic geometric structure is derived. The discrete mass and energy conservation laws are proved. The numerical tests show the effectiveness of the variational integrator, and the performance of the proved discrete conservation law.     

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.     

On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system

We study the semi-classical ground states of the nonlinear Maxwell–Dirac system: $$\begin{aligned} \left\{ \begin{array}{l} \alpha \cdot \left( i\hbar \nabla + q(x)\mathbf{A }(x)\right) w-a\beta w -\omega w - q(x)\phi (x) w = P(x)g(\left| w\right| ) w\\ -\Delta \phi =q(x)\left| w\right| ^2\\ -\Delta {A_k}=q(x)(\alpha _k w)\cdot \bar{w}\ \ \ \ k=1,2,3 \end{array} \right. \end{aligned}$$ for \(x\in \mathbb{R }^3\) , where \(\mathbf{A }\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hbar \) small. We also describe the concentration behavior of the solutions as \(\hbar \rightarrow 0\) .