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Modulated amplitude waves with nonzero phases in Bose-Einstein condensates

In this paper we give a frame for application of the averaging method to Bose-Einstein condensates (BECs) and obtain an abstract result upon the dynamics of BECs. Using aver- aging method, we determine the location where the modulated amplitude waves (periodic or quasi-periodic) exist and we also study the stability and instability of modulated amplitude waves (periodic or quasi-periodic). Compared with the previous work, modulated amplitude waves studied in this paper have nontrivial phases and this makes the problem become more diffcult, since it involves some singularities.

In collisionally inhomogeneous BECs, the scattering length is subjected to a spatial periodic variation: a(x) = a(x + L 0 ) for some period L 0 leading to nonlinear potentials, which has been realized experimentally 21,22 and studied theoretically 23,24 . Also, we can refer to a recent comprehensive review 25 .
Spatially periodic potentials V (x), created as optical lattices (OLs), which arise as interference patterns produced by coherent counterpropagating laser beams illuminating the condensate, are of interest in the context of BECs and have been employed in both experimental and theoretical studies [26][27][28][29][30] .
In this study, we investigate spatially extended solutions of BECs in periodic OLs. We apply a coherent structure ansatz to (1.1), yielding a parametrically forced Duffing equation with singularities, which describes the spatial evolution of the field. We employ the averaging method to study the periodic orbits (MAWs) including their hyperbolic spatial structures, and illustrate their dynamical behaviors with numerical simulation of the GP equation.
Compared with the previous work 23, [31][32][33][34][35][36] , the phases of MAWs considered in this paper are nontrivial, not constant, which continuously depend on the spatial variation. Nontrivial phase solutions have more complicated dynamics 9,37 and imply nonzero current of the matter -it is proportional to R 2 (x)θ(x) = c, for amplitude R(x) of MAW and nonzero constant c -along x-axis, and hence seem to have no direct relation to present experimental setting for BECs 38,39 (remember that the condensate in this paper is confined to be a parabolic trap). Of course, dealing with the MAWs with nonzero phases will take more difficulties than before. One reason is that the forced Duffing oscillator derived from GP equation has singularities at the origin and the directly application of the usual perturbation theory is unavailable because of the loss of smoothness and the existence of strong nonlinear term (singular term).
The averaging method 40-42 at its heart is a transformation procedure leading to a systematic perturbation expansion and completes with error bounds on the difference between exact and approximation solutions. Also, it is a important tool for proving properties of the exact problem based on properties of the approximation problem 43 . For example, the existence of periodic orbits can be proved using averaging together with the implicit function theorem, and the existence of invariant tori can be proved using averaging together with the Moser twist theorem. However, applying the averaging method for singular system, the first problem we must deal with is how to put the system into the standard form of averaging.
The paper is organized as follows. In Section II, we introduce modulated amplitude waves involving periodic and quasi-periodic, and in Section III, we apply a transformation to transform the GP equation to a standard form of averaging. An abstract result upon the dynamics to BECs is obtained in Section IV, and in Section V we analyze and demonstrate some of the spatial dynamical features of BECs with a positive chemical potential. Finally, we summarize our results in Section VI.

II. COHERENT STRUCTURE AND MODULATED AMPLITUDE WAVE
We consider uniformly propagating coherent structures with the ansatz where R(x) ∈ R gives the amplitude dynamics of the condensate wave function, θ(x) determines the phase dynamics, and the "chemical potential" µ, defined as the energy which takes to add one more particle to the system, is proportional to the number of atoms trapped in the condensate. When the (temporally periodic) coherent structure (2.1) is also spatially periodic, it is called a modulated amplitude wave (MAW) 44,45 . Similarly, a solution of the equation (1.1) with the (temporally periodic) coherent structure (2.1) is called a quasi-periodic modulated amplitude wave (QMAW) if it is also spatially quasi-periodic.
Inserting (2.1) into (1.1), we obtain the following two couple nonlinear ordinary differential equations and the integration constant c, determined by the velocity and number density, plays the role of "angular momentum" 28 .
Inspecting (2.2) we know that in case of c = 0, i.e., the phase of the condensate wave function is trivial, it is the parametrically driven Duffing equation with the time variable replaced by the spatial coordinate, and MAWs (standing waves) in this system with V (x) = V 0 cos κx or V (x) = V 1 cos κ 1 x + V 2 cos κ 2 x have been widely studied 23,32,36 .
In general, c = 0, the system (2.2) becomes more complicated and the phase is no longer constant 9 . Even the amplitude R(x), a solution of (2.2), is L-periodic, the corresponding condensate wave function ψ(x, t) may be not periodic with respect to the spatial variable x.
In fact, andΘ(x) = Θ(x) − ν is a L-periodic function with zero mean value. If 2π/ν and L are rationally related, then ψ(x, t) is a MAW; if 2π/ν and L are rationally irrelevant, then ψ(x, t) is not periodic but quasi-periodic, which is corresponding to a QMAW with the frequency There also exists an interesting and surprising result. Note that ν = 0 when c = 0 and ν = 0 when c = 0. If we vary c on the interval (−∞, +∞), by continuous dependence of solutions with respect to the parameters, ν can continuously take the value on some interval, which implies that (1.1) has infinitely many (positive measure set) MAWs and QMAWs by adjusting the integration constant c. Thus, all these MAWs and QMAWs form a foliation.
In this paper, we consider the case δ > 0 corresponding to a positive chemical potential.
Also, in order that the mathematical results obtained in this paper do apply to more general periodic functions, we assume that the external potential V (x) is an analytic and L-periodic function (OLs). Note that (2.2) defines on two half-planes, and we only consider the case of the right half-plane since there are no distinct technicalities.

III. TRANSFORMATION TO STANDARD FORM OF AVERAGING
Rewrite equation (2.2) in the planar equivalent form Generally, averaging method involves two steps: transforming to standard form; solving the averaging equation. In order to proceed we need to transform (3.1) to a standard form for the method of averaging. So we have the following result.
Proof. First, it is easy to verify that, for each ρ ∈ 4 c 2 δ , +∞ and θ ∈ T, is a periodic solution with the same period τ = 2π/ √ δ of the unperturbed equation or the equivalent Hamiltonian system with the initial value That is to say, (3.3) is an isochronous system. The total energy of system (3.3) is given by and as we know, all periodic solutions lie on curves of constant energy. We will use these facts below. Now using the variation of constants, the functions ρ(·), θ(·) can be defined such that From the conservation of the Hamiltonian, it follows that so the differentiating with respect to x along with system (3.1) yields Then, we have So, using the definition of Ψ together with equation (3.5), we have the first desired expression of (3.2).
Using the formula for R ′ given in system (3.1) and the definition of ρ(·) and θ(·), we obtain that ∂ϕ ∂x Together with equation (3.5), after some simple algebraic manipulations, it follows that Finally, using the definition of Ψ together with equation (3.5), we have the second desired expression of (3.2).
In general, finding a explicit expression of the transformation is not easy. The transformation Ψ given in Lemma 3.1 has a more delicate information, such as protecting two-form which is not used in this paper. However, we believe that it will be helpful for further study.

IV. AN ABSTRACT RESULT OF AVERAGING TO BECS
In this section, we will give an abstract result to BECs by the method of averaging. We also assume that the period L of the external potential V (x) satisfies that where τ is the least period of the unperturbed system. The basic idea that leads to the application of the method of averaging arises from an inspection of system (3.2). The derivatives with respect to the spatial variable x are all proportional to ε. Hence, if ε is small, the variables would be expected to remain near their constant unperturbed values over a long spatial scale.
A good approximation of (3.2) up to the spatial domains of order 1/ε is given by the averaged system We have the following theorem.
with g 1 , g 2L -periodic functions of x. Moreover, , ρ ε (x)) and (θ 0 (x), ρ 0 (x)) are solutions of the original system (3.2) and the averaged system (4.1) respectively, with the initial values such that for the spatial domains x of order 1/ε.
Proof. Conditions (4.2)-(4.6) imply that (θ 0 , ρ 0 ) is a instable and hyperbolic fixed point 2), which is some artificial "instability" in terms of the evolution in x. There is a set of methods for the study of modulational instability in time t, e.g., see [47][48][49][50] . Recently, based on spectrum theory and Hamiltonian floquet theory, the method of studying modulational (temporal) instability of standing waves (with trivial phases) has been developed for NLS equation with constant nonlinearity coefficients and periodic potentials by Bronski and Rapti 51 , later applied by Porter et al. 23 . However, here we can not provide any information upon it with our methods.

V. EQUILIBRIUMS AND THE AVERAGED EQUATION
In this section, we will analyze and demonstrate some of the spatial dynamical features of BECs with a positive chemical potential. We also assume that V (x) is an analytic and L-periodic function with the least positive period L = τ = π/ √ δ, i.e., p = q = 1.
First, expanding V in a Fourier series, we have where all coefficients are real. Let us substitute the expansion for V (x) given by (5.1) into (4.1), and by an easy (perhaps lengthy) computation, we obtain the averaging system If α < 0, corresponding to the repulsive nonlinearity, we take θ 0 = kπ − φ/2. Let us define the function f : 4 c 2 δ , +∞ → R by It is easy to verify that Using the mean-value theorem, there is at least one root ρ 0 of f on the interval 4 c 2 δ , +∞ . If α > 0, corresponding to the attractive case, we take θ 0 = kπ + (π − φ)/2. Similarly, we can induce that the function f also has at least one root ρ 0 on the interval 4 c 2 δ , +∞ . Thus, in any case, the averaged system (5.2) has at least one equilibrium (θ 0 , ρ 0 ).
After a simple computation, the Jacobian matrix of the averaged system (5.2) at the equilibrium (θ 0 , ρ 0 ) is given by Notice that if M has no zero eigenvalue, then the equilibrium (θ 0 , ρ 0 ) can be continuable. If M has no imaginary eigenvalue, the equilibrium (θ 0 , ρ 0 ) is hyperbolic since the two eigenvalues of M have opposite signs. So it follows that the original system (3.2) has a hyperbolic L-periodic solution (θ ε (x), ρ ε (x)) with respect to the spatial variable x near (θ 0 , ρ 0 ) such To demonstrate the process of averaging to BECs, a specific example of numerical computation is given in the following. We take the integration constant c = 1 and the parameters (kπ + π 2 , 3.00), (kπ + π 2 , 2.00), (kπ, 10.00), k = 0, ±1, ±2, · · · for the averaged system (5.2).

VI. DISCUSSION AND CONCLUSION
We have given an abstract result to BECs, see Theorem 4.1. Using averaging method, we determine the location where the MAWs or QMAWs exist. Comparing the previous work, we do not restrict our discussion near the origin since the equation we have dealt with has some singularities. On the other hand, the MAWs or QMAWs studied in this paper, which have nontrivial phases, form a foliation. This is a new result in this context.
However, we can not determine the spatial dynamics of some periodic orbits corresponding to MAWs or QMAWs since the equilibriums are not hyperbolic. Maybe second or even higher-order averaging is required, and this question is left for our further study. In addition, we can not provide any information upon modulational instability in time t of MAWs or QMAWs. One reason is that MAWs or QMAWs obtained in this paper are not standing waves, but the waves with nontrivial phases, which does not allow us to apply the method developed by Bronski and Rapti 51 directly. This problem may be not easy but a good topic for further study.
From the view point of a physical application, it might be reasonable to use the averaging principle to replace a mathematical model by the corresponding averaged system, to use the averaged system to make a prediction, and then to test the prediction against the results of a physical experiment. The study of this paper exactly gives a frame for application of the averaging method to BECs.