Asymptotic properties of excited states in the Thomas–Fermi limit

# Asymptotic properties of excited states in the Thomas--Fermi limit

## Abstract

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.

## 1 Introduction

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 [9]. 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

 iεut+ε2uxx+(1−x2−|u|2)u=0, (1)

where is a small parameter to model the Thomas–Fermi asymptotic regime. Let be the real positive solution of the stationary equation

 ε2η′′ε(x)+(1−x2−η2ε(x))ηε(x)=0,x∈R. (2)

Main results of Ignat & Millot [6, 7] and Gallo & Pelinovsky [3] 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

 η0(x):=limε→0ηε(x)={[]cl(1−x2)1/2,for|x|<1,0,for|x|>1. (3)

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

 ∥ηε−η0∥C1(K)⩽CKε2. (4)
• For any small , there is such that

 ∥ηε−η0∥L∞⩽Cε1/3,∥η′ε∥L∞⩽Cε−1/3. (5)
• There is such that for any .

Properties [P1] and [P2] follow from Proposition 2.1 in [6]. Properties [P3] and [P4] follow from Theorem 1 in [3]. To clarify the proof of bound (5), we represent the ground state in the equivalent form

 ηε(x)=ε1/3νε(y),y=1−x2ε2/3, (6)

where solves

 4(1−ε2/3y)ν′′ε(y)−2ε2/3ν′ε(y)+yνε(y)−ν3ε(y)=0,y∈(−∞,ε−2/3).

Let be the unique solution of the Painlevé–II equation

 4ν′′0(y)+yν0(y)−ν30(y)=0,y∈R,

such that as and decays to zero as faster than any exponential function. By Theorem 1 in [3], is a function on , which is expanded into the asymptotic series for any fixed :

 νε(y)=N∑n=0ε2n/3νn(y)+ε(2N+1)/3RN,ε(y), (7)

where are uniquely defined -independent functions on and is the remainder term on . It was proved in [3] that is uniformly bounded for small in -norm. If we denote , then the above arguments shows that there is such that

 ∥uN,ε∥L∞⩽CN,∥u′N,ε∥L∞⩽CNε−2/3.

For any fixed , it follows from the above bounds that the remainder term is smaller in norm than the leading-order term . The error estimate (5) follows from (6), (7), and the fact that .

We shall consider excited states of the Gross–Pitaevskii equation (1), which are real non-positive solutions of the stationary equation

 ε2u′′ε(x)+(1−x2−u2ε(x))uε(x)=0,x∈R. (8)

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 [8], 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 [1], 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 [1] 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

 uε(0)=0,uε(x)>0\rm for allx>0,\rm andlimx→∞uε(x)=0. (9)

Variational theory can be used to prove existence of this solution, similar to the analysis of Ignat & Millot in [7]. 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.

###### Theorem 1

For sufficiently small , there exists a unique solution with properties (9) and there is such that

 ∥∥ ∥∥uε−ηεtanh(⋅√2ε)∥∥ ∥∥L∞⩽Cε2/3. (10)

In particular, the solution converges pointwise as to

 u0(x):=limε→0uε(x)=η0(x)sign(x),x∈R.
###### Remark 1

Function is termed as the dark soliton. It is a solution of the second-order equation

 ε2v′′ε(x)+(1−v2ε(x))vε(x)=0,x∈R,

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

 Lεwε=Hε+Nε(wε), (11)

where

 Lε:=−ε2∂2x+x2−1+3η2ε(x)tanh2(x√2ε),
 Hε(x):=ηε(x)(η2ε(x)−1)sech2(x√2ε)tanh(x√2ε)+√2εη′ε(x)sech2(x√2ε),

and

 Nε(wε)(x)=−3ηε(x)tanh(x√2ε)w2ε(x)−w3ε(x).

Let , where is a new variable, and denote

 ^ηε(z):=ηε(√2εz),^wε(z):=wε(√2εz),^Hε(z):=Hε(√2εz),^Nε(^wε)(z):=Nε(wε)(√2εz).

Step 2: Linear estimates. In new variables, operator becomes

 ^Lε=−12∂2z+2ε2z2−1+3^η2ε(z)tanh2(z)=^L0+^Uε(z),

where

 ^L0:=−12∂2z+2−3sech2(z)

and

 ^Uε(z):=2ε2z2+3(^η2ε(z)−1)tanh2(z).

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

 ∃C>0:∀^f∈L2odd(R):∥^L−10^f∥H2⩽C∥^f∥L2. (12)

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

 ∥^wε∥L∞α:=∥eα|⋅|^wε∥L∞.

The unique solution for any is expressed explicitly by the integral formula

 ^L−10^f(z)=−2sech2(z)∫z0cosh4(z′)(∫z′−∞^f(z′′)sech2(z′′)dz′′)dz′.

For any fixed , it follows from the integral representation that the solution decays exponentially with the same rate as so that

 ∃C>0:∀^f∈L2odd(R)∩L∞α(R):∥^L−10^f∥L∞α⩽C∥^f∥L∞α. (13)

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 [5]). 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

 ~Lε=−ε2∂2x+x2−1+3η2ε(x)

was studied by Gallo & Pelinovsky [3], 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 [3]).

Thanks to the proximity of to near with an exponential accuracy in , the potential is similar to near and satisfies for any fixed :

 ∃C>0:V(x)⩾Cε2/3,|x|⩾x0.

On the other hand, for any fixed , property (P2) implies that

 ∃C>0:sup|z|⩽z0|^Uε(z)|⩽Cε2.

Thanks to the positivity of near and the proximity of the central well near in the potentials and , the quantum tunneling theory [5] 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

 ∃C>0:∀^f∈L2odd(R):∥^L−1ε^f∥H2⩽Cε−2/3∥^f∥L2. (14)

Note that the function has peaks near points and .

Step 3: Bounds on the inhomogeneous and nonlinear terms. By symmetries, we note that

 ^Hε∈L2odd(R)\rm and^Nε(^wε):H2odd(R)↦L2odd(R).

We will show that for small and fixed there is such that

 ∥^Hε∥L2∩L∞α⩽Cε2/3. (15)

Using the triangle inequality, we obtain

 ∥^Hε∥L2⩽∥ηε∥L∞∥(1−^η2ε)sech2(⋅)∥L2+√2ε∥η′ε∥L∞∥sech2(⋅)∥L2.

By properties (P1) and (P2), for small and fixed the first term is estimated by

 ∥(1−^η2ε)sech2(⋅)∥L2 ⩽ ∥(1−^η2ε)sech2(⋅)∥L2(|z|⩽ε−1/3)+∥(1−^η2ε)sech2(⋅)∥L2(|z|⩾ε−1/3) ⩽ ∥1−η2ε∥L∞(|x|<√2ε2/3)∥sech2(⋅)∥L2+α−1/2e−αε−1/3∥sech2(⋅)∥L∞α ⩽ Cε4/3.

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

 ∀^u,^v∈H2(R):∥^u^v∥H2⩽∥^u∥H2∥^v∥H2

For any , we have

 ∥^Nε(^wε)∥L2⩽3∥ηε∥L∞∥^w2ε∥H2+∥^w3ε∥H2⩽3∥^wε∥2H2+∥^wε∥3H2. (16)

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.

Let

 ^wε=^w1+^w2+^φε,^w1=^L−10^Hε,^w2=−3^L−10^ηεtanh(z)^w21.

The remainder term solves the new problem

 Lε^φε=Hε+Nε(^φε), (17)

where the new linear operator is

 Lε:=^Lε+Δ^Uε(z),Δ^Uε(z):=6^ηεtanh(z)(^w1+^w2)+3(^w1+^w2)2,

the new source term is

 Hε:=−^Uε(^w1+^w2)−3^ηεtanh(z)(2^w1^w2+^w22)−(^w1+^w2)3,

and the new nonlinear function is

 Nε(^φε):=−3^ηεtanh(z)^φ2ε−3(^w1+^w2)^φ2ε−^φ3ε.

Thanks to bounds (12), (13), and (15), we have for fixed and

 ∃C>0:∥^w1∥H2∩L∞α⩽Cε2/3,∥^w2∥H2∩L∞α⩽Cε4/3. (18)

As a result, for any small , there is such that

 ∥^ηεtanh(z)(2^w1^w2+^w22)∥L2⩽Cε2,∥(^w1+^w2)3∥L2⩽Cε2.

Let us now estimate the term in . By properties (P1) and (P2), for small and fixed there are constants such that

 ∥^Uε^wj∥L2 ⩽ 2ε2∥z2^wj∥L2+3∥(^η2ε−1)^wj∥L2 ⩽ ε2C(α)∥^wj∥L∞α+3∥(1−^η2ε)^wj∥L2(|z|⩽ε−1/3)+3∥(1−^η2ε)^wj∥L2(|z|⩾ε−1/3) ⩽ ε2C(α)∥^wj∥L∞α+3∥1−η2ε∥L∞(|x|<√2ε2/3)∥^wj∥L2+3α−1/2e−αε−1/3∥^wj∥L∞α ⩽ ~C(α)ε4/3∥^wj∥L2∩L∞α,j=1,2.

In view of bound (18), for any small there is such that

 ∥^Uε(^w1+^w2)∥L2⩽Cε2. (19)

Combining all together, we have established that and for any small , there is such that

 ∥Hε∥L2⩽Cε2. (20)

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

 ∥Nε(^φε)∥L2⩽C(δ)∥^φε∥2H2. (21)

Similarly, we obtain that is Lipschitz continuous in the ball and for any small there is such that

 ∀^φε,^φε∈Bδ(H2odd):∥Nε(^φε)−Nε(^φε)∥L2⩽C(δ)(∥^φε∥H2+∥^φε∥H2)∥^φε−^φ∥H2. (22)

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

 ∃C>0:∀^f∈L2odd(R):∥L−1ε^f∥H2⩽Cε−2/3∥^f∥L2. (23)

Let us rewrite equation (17) as the fixed-point problem

 ^φε∈H2odd(R):^φε=L−1εHε+L−1εNε(^φε). (24)

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

 ∃C>0:∥^φε∥H2⩽Cε4/3.

By Sobolev’s embedding of to , for any small there is such that

 ∥wε∥L∞=∥^wε∥L∞⩽C∥^w1+^w2+^φε∥H2⩽Cε2/3,

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

 u′′ε(x)=1ε2(x2−1+u2ε(x))uε(x)<0,x∈(x0,~x0).

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 .

Combining results in Steps (5) and (6), we conclude that is the first excited state of the stationary equation (8) that satisfies properties (9).

## 3 Second excited state

The second excited state is an even solution of the stationary equation (8) such that

 uε(x)>0\rm for all|x|>x0,uε(x)<0\rm for all|x|

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 .

###### Theorem 2

For sufficiently small , there exists a unique solution with properties (25) and there exist and such that

 Missing or unrecognized delimiter for \right (26)

and

 a=−ε√2(log(ε)+12log|log(ε)|−32log(2)+o(1))\rm asε→0. (27)
###### Remark 2

Since as while near , we have

 x0=a+O(ε5/3)\rm asε→0.
###### Remark 3

Exactly the same asymptotic expansion (27) has been obtained with the use of the averaged Lagrangian approximation and has been confirmed numerically [1].

The proof of Theorem 2 follows the same steps as the proof of Theorem 1 with an additional step on the Lyapunov–Schmidt bifurcation equation.

Step 1: Decomposition. Let and substitute

 uε(x)=ηε(x)tanh(x−a√2ε)