Asymptotic properties of excited states in the Thomas--Fermi limit
Excited states are stationary localized solutions of the Gross–Pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis. Existence and asymptotic properties of excited states are considered in the semi-classical (Thomas-Fermi) limit. Using the method of Lyapunov–Schmidt reductions and the known properties of the ground state in the Thomas–Fermi limit, we show that excited states can be approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger equation with nonzero boundary conditions) and the ground state. The dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail-to-tail interaction potential is achieved.
The defocusing nonlinear Schrödinger equation is derived in the mean-field approximation to model Bose–Einstein condensates with repulsive inter-atomic interactions between atoms. This equation is referred in this context to as the Gross–Pitaevskii equation . When the Bose–Einstein condensate is trapped by a magnetic field, the Gross–Pitaevskii equation has a harmonic potential. In the strongly nonlinear limit, referred to as the Thomas–Fermi limit [4, 11], the Bose–Einstein condensate is a nearly compact cloud, which may contain localized dips of the atomic density. The nearly compact cloud is modeled by the ground state of the Gross–Pitaevskii equation, whereas the localized dips are modeled by the excited states. Asymptotic properties of the stationary excited states in the Thomas–Fermi limit are analyzed in this article.
The Gross–Pitaevskii equation with a harmonic potential and a repulsive nonlinear term can be rewritten in the form
where is a small parameter to model the Thomas–Fermi asymptotic regime. Let be the real positive solution of the stationary equation
Main results of Ignat & Millot [6, 7] and Gallo & Pelinovsky  state that for any sufficiently small there exists a unique smooth positive solution that decays to zero as faster than any exponential function. The ground state converges pointwise as to the compact Thomas–Fermi cloud
The ground state and the convergence of to is characterized by the following properties:
for any .
For any small and any compact subset , there is such that
For any small , there is such that
There is such that for any .
Properties [P1] and [P2] follow from Proposition 2.1 in . Properties [P3] and [P4] follow from Theorem 1 in . To clarify the proof of bound (5), we represent the ground state in the equivalent form
Let be the unique solution of the Painlevé–II equation
such that as and decays to zero as faster than any exponential function. By Theorem 1 in , is a function on , which is expanded into the asymptotic series for any fixed :
where are uniquely defined -independent functions on and is the remainder term on . It was proved in  that is uniformly bounded for small in -norm. If we denote , then the above arguments shows that there is such that
We shall consider excited states of the Gross–Pitaevskii equation (1), which are real non-positive solutions of the stationary equation
We classify the excited states by the number of zeros of on . A unique solution with zeros exists near for by the local bifurcation theory , where is computed from the linear theory as , . Because of the symmetry of the harmonic potential, the -th excited state is even on for even and odd on for odd .
This paper continues the previous research on the ground state in the Thomas–Fermi limit that was developed by Gallo & Pelinovsky in [2, 3]. We focus now on the existence and asymptotic properties of the excited states as . Using the method of Lyapunov–Schmidt reductions, we show that the -th excited state is approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger equation with nonzero boundary conditions) and the ground state . The dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail-to-tail interaction potential is achieved.
Note that this paper gives a rigorous justification of the variational approximations found by Coles et al. in , where the -th excited states was approximated by a variational ansatz in the form of a product of dark solitons with time-dependent parameters and the ground state. Time-evolution equations for the parameters of the variational ansatz were found from the Euler-Lagrange equations. Critical points of these equations give approximations of the equilibrium positions of the dark solitons relative to the center of the harmonic potential and to each others, whereas the linearization around the critical points give the frequencies of oscillations of dark solitons near such equilibrium positions. Variational approximations were found in  to be in excellent agreement with numerical solutions of the stationary equation (8).
This article is organized as follows. The first excited state centered at is considered in Section 2. Although existence of this solution can be established from the calculus of variations, we develop the fixed-point iteration scheme to study this solution as . The second excited state is approximated in Section 3. We will work with the method of Lyapunov–Schmidt reductions to find the equilibrium position of two dark solitons as . Section 4 discusses the existence results for the general -th excited state with .
Before we proceed with main results, let us discuss some notations. If and are two quantities depending on a parameter in a set , the notation as indicates that remains bounded as . If depends on and , the notation as indicates that remains bounded as . Different constants are denoted with the same symbol if they can be chosen independently of the small parameter .
2 First excited state
The first excited state is an odd solution of the stationary equation (8) such that
Variational theory can be used to prove existence of this solution, similar to the analysis of Ignat & Millot in . Since we are interested in asymptotic properties of the first excited state as , we will obtain both existence and convergence results from the fixed-point arguments. Our main result is the following theorem.
For sufficiently small , there exists a unique solution with properties (9) and there is such that
In particular, the solution converges pointwise as to
Function is termed as the dark soliton. It is a solution of the second-order equation
which arises in the context of the defocusing nonlinear Schrödinger equation.
The proof of Theorem 1 consists of six steps.
Step 1: Decomposition. Let us substitute to the stationary equation (8) and obtain an equivalent problem for written in the operator form
Let , where is a new variable, and denote
Step 2: Linear estimates. In new variables, operator becomes
Operator is well known in the linearization of the defocusing NLS equation at the dark soliton. The spectrum of in consists of two eigenvalues at and with eigenfunctions and and the continuous spectrum on . For any , there exists a unique such that
Let us consider functions that decay to zero as with a fixed exponential decay rate . Let be the exponentially weighted space with the supremum norm
The unique solution for any is expressed explicitly by the integral formula
For any fixed , it follows from the integral representation that the solution decays exponentially with the same rate as so that
Figure 1 shows the confining potential of operator (solid line) and the bounded potential of operator (dots) versus . The confining potential has two wells near and a deeper central well near . The two wells near are absent in the potential .
Because of the confining potential, the spectrum of is purely discrete (Theorem 10.7 in ). It contains small eigenvalues that correspond to eigenfunctions localized in the central well near and in the two smaller wells near .
We note that a similar operator at the ground state
was studied by Gallo & Pelinovsky , where it was shown that for all . By property (P4), is bounded away from zero near by the constant of the order of . As a consequence, the purely discrete spectrum of in includes small positive eigenvalues of the order with the eigenfunctions localized in the two wells near (see Theorem 2 in ).
Thanks to the proximity of to near with an exponential accuracy in , the potential is similar to near and satisfies for any fixed :
On the other hand, for any fixed , property (P2) implies that
Thanks to the positivity of near and the proximity of the central well near in the potentials and , the quantum tunneling theory  implies that the simple zero eigenvalue of persists as a small eigenvalue of . This eigenvalue of corresponds to an even eigenfunction. The other eigenvalue of corresponding to an odd eigenfunction is bounded away from zero.
All other eigenvalues of are small positive of the size . As a result, operator is still invertible on but bound (12) is now replaced by
Note that the function has peaks near points and .
Step 3: Bounds on the inhomogeneous and nonlinear terms. By symmetries, we note that
We will show that for small and fixed there is such that
Using the triangle inequality, we obtain
By properties (P1) and (P2), for small and fixed the first term is estimated by
By property (P3), the second term is estimated by . As a result, for any small there is such that . By similar arguments, for any and there is such that .
To deal with the nonlinear terms, we recall that is Banach algebra with respect to multiplication in the sense that
For any , we have
Similarly, is a Banach algebra with respect to multiplication for any .
Step 4: Normal-form transformations. Because we are going to lose as a result of bound (14), we need to perform transformations of solution , usually referred to as the normal-form transformations. We need two normal-form transformations to ensure that the resulting operator of a fixed-point equation is a contraction.
The remainder term solves the new problem
where the new linear operator is
the new source term is
and the new nonlinear function is
As a result, for any small , there is such that
Let us now estimate the term in . By properties (P1) and (P2), for small and fixed there are constants such that
In view of bound (18), for any small there is such that
Combining all together, we have established that and for any small , there is such that
For the nonlinear term, we still have . Thanks to bound (18), for any in the ball of radius , for any small there is such that
Similarly, we obtain that is Lipschitz continuous in the ball and for any small there is such that
Step 5: Fixed-point arguments. Thanks to bound (18) and Sobolev embedding of to , is as small as in the central well near and is exponentially small in in the two wells near . As a result, small positive eigenvalues of of the size persist in the spectrum of and have the same size, so that bound (14) extends to operator in the form
Let us rewrite equation (17) as the fixed-point problem
The map is Lipschitz continuous in the neighborhood of . Thanks to bounds (21) and (23), the map is a contraction in the ball if . On the other hand, thanks to bounds (20) and (23), the source term is as small as in norm. By Banach’s Fixed-Point Theorem in the ball with , there exists a unique of the fixed-point problem (24) such that
By Sobolev’s embedding of to , for any small there is such that
which completes the proof of bound (10).
Step 6: Properties (9). Solution constructed in Step (5) is a odd continuously differentiable function of on vanishing at infinity, so that and . By bootstrapping arguments for the stationary equation (8), we have . It remains to prove that is positive for all .
Recall that for all . By property (P4) and bound (10), there is such that for all . We shall prove that for all . Assume by contradiction that there is such that and . (If , then is the only solution of the second-order equation (8).) The continuity of implies that for every for some . Using the differential equation (8), we obtain
Then, , so that is a negative, decreasing function of for all with . This fact is a contradiction with the decay of to zero as . Therefore, for all .
3 Second excited state
The second excited state is an even solution of the stationary equation (8) such that
Here determines a location of two symmetric zeros of at . The second excited state is approximated as by a product of two copies of dark solitons (Remark 1) placed at with as . Our analysis is based on the method of Lyapunov–Schmidt reductions, which gives existence and convergence properties for the second excited state, as well as an analytical expansion of for small .
For sufficiently small , there exists a unique solution with properties (25) and there exist and such that
Since as while near , we have
Step 1: Decomposition. Let and substitute