# Elliptic solutions of the defocusing NLS equation are stable

## Transcript Of Elliptic solutions of the defocusing NLS equation are stable

IOP PUBLISHING J. Phys. A: Math. Theor. 44 (2011) 285201 (24pp)

JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL doi:10.1088/1751-8113/44/28/285201

Elliptic solutions of the defocusing NLS equation are stable

Nathaniel Bottman1,3, Bernard Deconinck1 and Michael Nivala2

1 Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA 2 Department of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, CA 90095, USA

E-mail: [email protected], [email protected] and [email protected]

Received 20 January 2011, in ﬁnal form 19 May 2011 Published 9 June 2011 Online at stacks.iop.org/JPhysA/44/285201

Abstract The stability of the stationary periodic solutions of the integrable (onedimensional, cubic) defocusing nonlinear Schro¨dinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additional argument using an appropriate Krein signature allows us to conclude the (nonlinear) orbital stability of all stationary solutions of the defocusing NLS equation with respect to so-called subharmonic perturbations: perturbations that have period equal to an integer multiple of the period of the amplitude of the solution. All results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase proﬁles.

1. Introduction

The defocusing one-dimensional nonlinear Schro¨dinger equation with cubic nonlinearity is given by

ı t = − 12 xx + | |2 .

(1)

Here (x, t) is a complex-valued function, describing the slow modulation of a carrier wave in a dispersive medium. Due to both its physical relevance and its mathematical properties, (1) is one of the canonical equations of nonlinear dynamics. The equation has been used extensively to model, among other applications, waves in deep water [2, 39], propagation

3 Present address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA.

1751-8113/11/285201+24$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA

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in nonlinear optics with normal dispersion [23, 28], Bose–Einstein condensates with repulsive self-interaction [21, 34] and electron plasma waves [12]. Equation (1) is completely integrable [1, 40]. This will be used extensively later on.

The equation has a large class of stationary solutions; these are written as

= e−ıωt φ (x),

(2)

where ω is a real constant. Among this class of solutions are the dark and grey solitons, for which φ(x) is expressed in terms of hyperbolic functions. These solutions may be regarded as limit cases of the so-called elliptic solutions studied in this paper. The stationary solutions (2) are either periodic or quasi-periodic as functions in x. The amplitude of φ(x) of the elliptic solutions is expressed in terms of Jacobi elliptic functions. A thorough discussion of the stationary solutions is found in, for instance, [9]. The details relevant to our investigations are presented in section 2.

The stability analysis of the stationary solutions was begun in [39], where the now classical calculation for the modulational stability of the plane-wave solution (φ(x) constant) is given. The literature discussing the stability of the soliton solutions is extensive, see [29], and references therein. Rowlands [35] may have been the ﬁrst to consider the stability of the elliptic solutions directly. He studied the spectral stability problem for these solutions using regular perturbation theory with the Floquet parameter as a small expansion parameter. At the origin in the spectral plane, this parameter is zero, thus Rowlands was able to obtain expressions for the different branches of the continuous spectrum near the origin. For the focusing NLS equation these calculations demonstrate that the spectrum lies partially in the right-half plane, which leads to the conclusion of instability. For the defocusing NLS equation (1), the ﬁrst approximation to these branches lies on the imaginary axis, and Rowlands’ method is inconclusive with regards to stability or instability of the elliptic solutions. More recently, the stability of the elliptic solutions has been examined by Gallay and Ha˘ra˘gus¸ [17, 18]. In [18], they established the spectral stability of small-amplitude solutions of the form (2) of (1), as well as their (nonlinear) orbital stability with respect to perturbations that are of the same period as |φ(x)|. In [17], the restriction on the amplitude for the orbital stability result is removed. Ha˘ra˘gus¸ and Kapitula [22] put some of these results in a more general framework valid for spectral problems with periodic coefﬁcients originating from Hamiltonian systems. They establish that the small-amplitude elliptic solutions investigated in [18] are not only spectrally but also linearly stable. Lastly, we should mention a recent paper by Ivey and Lafortune [26]. They undertake a spectral stability analysis of the cnoidal wave solution of the focusing NLS equation, by exploiting the squared-eigenfunction connection, like we do in [5] for the cnoidal wave solutions of the Korteweg–de Vries equation and here, see below. Their calculations use Floquet theory for the spatial Lax operator to construct an Evans function for the spectral stability problem, whose zeros give the point spectrum corresponding to periodic perturbations. They also obtain a description of the continuous spectrum (which contains this point spectrum) using a Floquet discriminant. Their description of the spectrum is explicit in the sense that no differential equations remain to be solved. By computing level curves of this Floquet discriminant numerically, they obtain a numerical description of the spectrum.

In this paper, we conﬁrm the recent ﬁndings on spectral and orbital stability of the elliptic solutions of the defocusing equation and extend their validity to solutions of arbitrary amplitude. In addition, we extend the stability results to the class of so-called subharmonic perturbations, i.e. perturbations that are periodic with period equal to an integer multiple of the period of the amplitude |φ(x)|. Further, exploiting the integrability of (1), we are able to provide an explicit analytic description of the spectrum and the eigenfunctions associated with the linear stability problem of all elliptic solutions. We follow the same method as in [5],

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using the algebraic connection between the eigenfunctions of the Lax pair of (1) and those of the spectral stability problem. This explicit characterization of the spectrum, as well as the extension of the spectral stability results to arbitrary amplitude, and the results involving subharmonic perturbations are new. It appears that the methods of Ivey and Lafortune [26] allow for an equally explicit description when applied to the defocusing case. They rely on the general theory of hyperelliptic Riemann surfaces and theta functions, which are restricted to the elliptic case, through a nontrivial reduction process. We never leave the realm of elliptic functions, resulting in a signiﬁcantly more straightforward approach. The explicit characterization of the spectrum is an obvious starting point for the stability analysis of more general solutions to non-integrable generalizations of the NLS equations, such as the twodimensional NLS equation [10, 11] or one-dimensional perturbations of the NLS equation which might include such effects as dissipation or external potentials, see e.g., [7, 26]. As in [17, 18, 22], we prove the spectral stability of the elliptic solutions of (1), without imposing a restriction on the amplitude. The results of [22] allow us to prove the completeness of the eigenfunctions of the linear stability problem, resulting in a conclusion of linear stability. Similarly to the last section of [13], we employ an appropriate Krein signature calculation to allow us to invoke the classical results of Grillakis, Shatah and Strauss [20], from which (nonlinear) orbital stability follows.

It should be emphasized that our results are equally valid for elliptic solutions that have trivial phase (φ(x) real) as for solutions with a non-trivial phase proﬁle (φ(x) not purely real). Similar calculations to the ones presented here apply to the focusing NLS equation, without the conclusion of stability, of course. That case is more complicated, due to the Lax operator associated with that integrable equation not being self adjoint. It will be presented separately elsewhere.

Before entering the main body of the paper, we wish to apologize to the reader for the use of no less than three different incarnations of the NLS equation, in addition to (1). One is obtained through a scaling transformation with a time-dependent exponential factor, to allow the stationary solutions to appear as equilibrium solutions. The second one is used to facilitate our proof of spectral stability and involves a time- and space-dependent exponential factor. The last NLS form writes the second one in terms of its real and imaginary parts, and is useful for our proof of orbital stability. All forms are introduced because we beneﬁt greatly from their use. None are new to the literature. It does not appear straighforward to avoid the use of any of them without much added complication.

2. Elliptic solutions of the defocusing NLS equation

The results of this section are presented in more detail in [9]. We restrict our considerations to the bare necessities for what follows.

Stationary solutions (2) of (1) satisfy the ordinary differential equation

ωφ = − 12 φxx + φ|φ|2.

(3)

Substituting an amplitude-phase decomposition

φ (x) = R(x) eıθ(x)

(4)

in (3), we ﬁnd ordinary differential equations satisﬁed by the amplitude R(x) and the phase θ (x) by separating real and imaginary parts, after factoring out the overall exponential factor. Here we explicitly use that both amplitude and phase are real-valued functions. The equation for the phase θ (x) is easily solved in terms of the amplitude. One ﬁnds

θ (x) = c x 1 dy. (5) 0 R2(y)

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Here c is a constant of integration. Using standard methods for elliptic differential equations (see for instance [8, 30]), one shows that the amplitude R(x) is given by

R2(x) = k2sn2 (x, k) + b,

(6)

where sn(x, k) is the Jacobi elliptic sine function, and k ∈ [0, 1) is the elliptic modulus [8, 30]. The amplitude R(x) is periodic with period T (k) = 2K(k), where K(k) is the complete elliptic integral of the ﬁrst kind [8, 30]:

π/2

1

K(k) =

dy.

(7)

0

1 − k2 sin2 y

The form of the solution (6) leads to

ω = 12 (1 + k2) + 32 b, (8) and

c2 = b(b + 1)(b + k2).

(9)

Conditions on the reality of the amplitude and phase lead to the constraint b ∈ R+ (including

zero) on the offset parameter. The class of solutions constructed here is not the most general

class of stationary solutions of (1). We did not specify the full class of parameters allowed

by the Lie point symmetries of (1), which allow for a scaling in x, multiplying by a unitary

constant, etc. The methods introduced in the remainder of this paper apply equally well and

with similar results to the full class of stationary elliptic solutions.

If the constant c is zero, the solution is referred to as a trivial-phase solution. Otherwise

it is called a nontrivial-phase solution. It is clear from the above that the only trivial-phase

solutions are (up to symmetry transformations)

(x,

t)

=

ksn(x,

k)

e−

ı 2

(1+k 2 )t

.

(10)

This one-parameter family of solutions is found from the two-parameter family of stationary

solutions by equating b = 0. The trivial-phase solutions are periodic in x. Their period is

4K(k). In contrast, the nontrivial-phase solutions are typically not periodic in x. The period

of their amplitude is T (k) = 2K(k), whereas the period τ (k) of their phase is determined by

θ (τ (k)) = 2π . Unless τ (k) and T (k) are rationally related, the nontrivial-phase solution is

quasi-periodic instead of periodic.

This quasi-periodicity is more immediately obvious using a different form of the elliptic

solutions (see [17, 18]), which will prove useful in section 6. We split the integrand of (5) as

c = κ(k, b) + K(x; k, b), (11) R2(x)

where κ(k, b) is the average value of c/R2(x) over an interval of length T (k). Thus the average

value of K(x; k, b) is zero. Then the elliptic solutions may be written as

(x, t ) = e−ıωt+ıκx Rˆ (x),

(12)

where Rˆ (x + T (k)) = Rˆ (x) is typically not real. It is clear from this formulation of the elliptic

solutions that they are generically quasiperiodic with two incommensurate spatial periods

T (k) and 2π/κ(k, b).

3. The linear stability problem

Before we study the orbital stability of the elliptic solutions, we examine their spectral and

linear stability. To this end, we transform (1) so that the elliptic solutions are time-independent

solutions of this new equation. Let

(x, t) = e−ıωt ψ(x, t).

(13)

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Then

ıψt = −ωψ − 12 ψxx + ψ|ψ|2.

(14)

As stated, the elliptic solutions are those solutions for which ψt ≡ 0. Next, we consider perturbations of such an elliptic solution. Let

ψ (x, t) = eıθ(x) (R(x) + u(x, t) + ı v(x, t)) + O( 2),

(15)

where is a small parameter and u(x, t) and v(x, t) are real-valued functions. Since their dependence on both x and t is unrestricted, there is no loss of generality from factoring out the temporal and spatial phase factors. Substituting (15) into (1) and separating real and imaginary parts, the terms of zero order in vanish, since R(x) eıθ(x) solves (1). Next, we equate terms of order to zero and separate real and imaginary parts, resulting in

∂ u = L u = J L+ S

u ,

(16)

∂t v

v

−S L− v

where

J= 0 1 ,

(17)

−1 0

and the linear operators L−, L+ and S are deﬁned by L− = − 12 ∂x2 + R2(x) − ω + 2Rc42(x) , (18)

L+ = − 12 ∂x2 + 3R2(x) − ω + 2Rc42(x) , (19)

c

cR (x) c

1

S = R2(x) ∂x − R3(x) = R(x) ∂x R(x) .

(20)

We wish to show that perturbations u and v that are initially bounded remain so for all times. By ignoring terms of order 2 and higher we are restricting ourselves to linear stability. The elliptic solution φ(x) = R(x) eıθ(x) is by deﬁnition linearly stable if for all ε > 0 there is a δ > 0 such that if ||u(x, 0) + ıv(x, 0)|| < δ then ||u(x, t) + ıv(x, t)|| < ε for all t > 0. It should be noted that this deﬁnition depends on the choice of the norm || · || of the perturbations. In the next section this norm will be speciﬁed. The linear stability problem (16) is written in its standard form to allow for a straightforward comparison with the results of other authors, see for instance [17, 18, 22, 35], and many references where only the soliton case is considered. Some of our calculations are more conveniently done using a different form of the linear stability problem (16) or the spectral stability problem (22, below). These forms will be introduced as necessary.

Since (16) is autonomous in t, we can separate variables and consider solutions of the form

u(x, t) = eλt U (x, λ) ,

(21)

v(x, t)

V (x, λ)

so that the eigenfunction vector (U (x, λ), V (x, λ))T satisﬁes the spectral problem

U λ

=L U

=J

L+

S

U .

(22)

V

V

−S L− V

Since −S is the Hermitian conjugate of S, this latter form of the spectral problem emphasizes the Hamiltonian structure of the problem. In what follows, we suppress the λ

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dependence of U and V. In order to show that the solution φ(x) = R(x) eıθ(x) is spectrally stable, we need to verify that the spectrum σ (L) does not intersect the open right-half of the complex λ plane. To avoid confusion with other spectra deﬁned below, we refer to σ (L) as the stability spectrum of the elliptic solution φ(x). Since the nonlinear Schro¨dinger equation (1) is Hamiltonian [2], the spectrum of its linearization is symmetric with respect to both the real and the imaginary axis [38], so proving the spectral stability of an elliptic solution is equivalent to proving the inclusion σ (L) ⊂ ıR.

Spectral stability of an elliptic solution implies its linear stability if the eigenfunctions corresponding to the stability spectrum σ (L) are complete in the space deﬁned by the norm || · ||. In that case all solutions of (16) may be obtained as linear combinations of solutions of (22).

The ﬁrst goal of this paper is to prove the spectral and linear stability of all solutions (2) by analytically determining the stability spectrum σ (L), as well as its associated eigenfunctions. It is already known from [18] and [22] that the inclusion σ (L) ⊂ ıR holds for solutions of small amplitude, or, equivalently, solutions with small elliptic modulus, leading to spectral stability. We strengthen these results by providing a completely explicit description of σ (L) and its eigenfunctions, without requiring any restriction on the elliptic modulus. To conclude the completeness of the eigenfunctions associated with σ (L), and thus the linear stability of the elliptic solutions, we rely on the SCS lemma, see Ha˘ra˘gus¸ and Kapitula [22].

4. Numerical results

In the next few sections, we determine the spectrum of (22) analytically. Before we do so, we compute it numerically, using Hill’s method [15]. Hill’s method is ideally suited to a periodiccoefﬁcient problem such as (22). It should be emphasized that almost none of the elliptic solutions are periodic in x, as discussed in section 2. Nevertheless, since we have factored out the exponential phase factor eıθ(x) and the remaining coefﬁcients are all expressed in terms of R(x), the spectral problem (22) is a problem with periodic coefﬁcients, even for elliptic solutions that are quasi-periodic.

Using Hill’s method, we compute all eigenfunctions using the Floquet–Bloch decomposition

U (x) = eiμx Uˆ (x) , Uˆ (x + T (k)) = Uˆ (x), Vˆ (x + T (k)) = Vˆ (x),

(23)

V (x)

Vˆ (x)

with μ ∈ [−π/2T (k), π/2T (k)). It follows from Floquet’s theorem [3] that all bounded solutions of (22) are of this form. Here bounded means that maxx∈R{|U (x)|, |V (x)|} is ﬁnite. Thus

U, V ∈ Cb0(R).

(24)

By a similar argument as that given at the end of section 2, the typical eigenfunction (23) obtained this way is quasi-periodic, with periodic eigenfunctions ensuing when the two periods T (k) and 2π/μ are commensurate. Speciﬁcally, our investigations include perturbations of an arbitrary period that is an integer multiple of T (k), i.e., subharmonic perturbations.

Figure 1 shows discrete approximations to the spectrum of (22), computed using SpectrUW 2.0 [14]. The solution parameters for the top two panels (a) and (b) are b = 0 (thus corresponding to a trivial-phase solution (10)) and k = 0.8. The numerical parameters (see [14, 15]) are N = 20 (41 Fourier modes) and D = 40 (39 different Floquet exponents). The right panel (b) is a blow-up of the left panel (a) around the origin. First, it appears

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(a)

(b)

(c)

(d )

Figure 1. Numerically computed spectra (imaginary part of λ vs. real part of λ) of (22) for different solutions (2), with parameter values given below, using Hill’s method with N = 20 (41 Fourier modes) and D = 40 (39 different Floquet exponents), see [14, 15]. (a) A trivial-phase sn-solution with k = 0.5. (b) A blow-up of (a) around the origin, showing a band of higher spectral density. (c) A nontrivial-phase solution with b = 0.2 and k = 0.5. (d) A blow-up of (c) around the origin,

similarly showing a band of higher spectral density.

that the spectrum is on the imaginary axis4, indicating spectral stability of the snoidal solution (10). Second, the numerics show that a symmetric band around the origin has a higher spectral density than does the rest of the imaginary axis. This is indeed the case, as shown in more detail in ﬁgure 2(a), where the imaginary parts in [−1, 1] of the computed eigenvalues are displayed as a function of the Floquet parameter μ. This shows that λ values with imaginary parts in [−0.37, 0.37] (approximately) are attained for four different μ values in [−π/2T (k), π/2T (k)). The rest of the imaginary axis is only attained for two different μ values. This picture persists if a larger portion of the imaginary λ axis is examined. These numerical results are in perfect agreement with the theoretical results below.

The bottom two panels (c) and (d) correspond to a nontrivial-phase solution with b = 0.2 and k = 0.5. The numerical parameters are identical to those for panels (a) and (b). Again, the spectrum appears to lie on the imaginary axis, with a higher spectral density around the origin. The clumping of the eigenvalues outside of the higher-density band is a consequence of aliasing. This is an artifact of the numerics and the graphics. A plot of the imaginary parts of the computed eigenvalues as a function of μ is shown in ﬁgure 2(b). As for the trivial-phase case this shows the quadruple covering of the spectrum of a band around the origin of the imaginary axis, and the double covering of the rest of the imaginary axis. Due to the nontrivial-phase proﬁle, the curves in ﬁgure 2(b) have lost some symmetry compared to those in ﬁgure 2(a).

4 The order of magnitude of the largest real part computed is 10−10.

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(a)

(b)

Figure 2. The imaginary part of λ as a function of μ, demonstrating the higher spectral density (four vs. two) corresponding to ﬁgure 1(b) (left panel) and to ﬁgure 1(d) (right panel). The parameter values are identical to those of ﬁgure 1.

Making the opposite choice for the sign on c in (9) results in the ﬁgure being slanted in the

other direction. The above considerations remain true for different values of the offset b ∈ R+ and the

elliptic modulus k ∈ [0, 1), although the spectrum does depend on both, as we will prove in the following sections. Thus, for all values of (b, k) ∈ R+ × [0, 1), the spectrum of the elliptic solutions appears to be conﬁned to the imaginary axis, indicating the spectral stability of these solutions. Similarly, for all these parameter values, the spectrum σ (L) covers a symmetric interval around the origin four times, whereas the rest of the imaginary axis is double covered.

The edge point on the imaginary axis where the transition from spectral density four to two occurs depends on both b and k and is denoted λc(b, k). The k-dependence of λc(b = 0.2, k) is shown in ﬁgure 3. Again, both numerical and analytical results (see section 6) are displayed. For these numerical results, Hill’s method with N = 50 was used.

5. Lax pair representation

Since our analytical stability results originate from the squared-eigenfunction connection

between the defocusing NLS linear stability problem (16) and its Lax pair, in this section we

examine this Lax pair, restricted to the elliptic solutions of the defocusing NLS.

As for the stability problem, we consider the generalized defocusing NLS (14). This

equation is integrable, thus it has a Lax pair representation. Speciﬁcally, (14) is equivalent to

the compatibility condition χxt = χtx of the two ﬁrst-order linear differential equations

χx = −ψı∗ζ ıψζ χ ,

χ = −ıζ 2 − 2ı |ψ |2 + 2ı ω

ζ ψ + 2ı ψx

χ.

(25)

t

ζ ψ ∗ − 2ı ψx∗

ıζ 2 + 2ı |ψ |2 − 2ı ω

Thus (14) is satisﬁed if and only if both equations for χ of (25) are satisﬁable. Written as

a spectral problem with parameter ζ , the ﬁrst equation is seen to be formally self adjoint [27],

thus the spectral parameter ζ is conﬁned to the real axis. Restricting to the elliptic solutions

gives

χ = −ıζ

φ χ,

χ = −ıζ 2 − 2ı |φ|2 + 2ı ω

ζ φ + 2ı φx

χ.

(26)

x

φ∗ ıζ

t

ζ φ∗ − 2ı φx∗

ıζ 2 + 2ı |φ|2 − 2ı ω

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Figure 3. Numerical and analytical results for the imaginary part of the edge point λc(b, k) of the quadruple-covered region as a function of the elliptic modulus k for b = 0.2. The solid curve displays the analytical result, the small circles are obtained numerically.

(This ﬁgure is in colour only in the electronic version)

We refer to the spectrum of the ﬁrst equation of (26) as σL. It is the set of all ζ values for which this equation has a solution bounded in x (as in section (4)). As discussed above, σL ⊂ R. The main goal of this section is the complete analytic determination of σL. For ease of notation, we rewrite the second equation of (26) as

χt = AC −BA χ . (27)

Since A, B and C are independent of t, we may separate variables. Consider the ansatz

χ (x, t) = e t ϕ(x),

(28)

where is independent of t. We refer to the set of all such that χ is a bounded function of x as the t-spectrum σt . Substituting (28) into (27) and canceling the exponential, we ﬁnd

A−

B ϕ = 0.

(29)

C −A −

This implies that the existence of nontrivial solutions requires

2 = A2 + BC = −ζ 4 + ωζ 2 − cζ + 116 (4ωb − 3b2 − k 4), (30)

where k 2 = 1 − k2. We have used the explicit form of φ(x), given in section 2. This demonstrates that is not only independent of t, but also of x. Such a conclusion could also be arrived at by expressing the derivatives of the operators of (26) as matrix commutators, and applying the fact that the trace of a matrix commutator is identically zero [4, 16].

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Having determined as a function of ζ for any given elliptic solution of defocusing NLS (i.e., in terms of the parameters b and k), we now wish to do the same for the eigenvector ϕ(x), determined by (29). Immediately,

ϕ = γ (x) −B(x) ,

(31)

A(x) −

where γ (x) is a scalar function. Indeed, the vector part of (31) ensures that χ (x, t) satisﬁes the second equation of (26). Next, we determine γ (x) so that χ (x, t) also satisﬁes the ﬁrst equation. Substituting (31) in this ﬁrst equation results in two homogeneous linear scalar differential equations for γ (x) which are linearly dependent. Solving gives

γ (x) = γ0 exp − (A − )φ + Bx + ıζ B dx . (32) B

For almost all ζ ∈ C, we have explicitly determined two linearly independent solutions

of the ﬁrst equation of (26). Indeed, for all ζ , there should be two such solutions, and two

have been constructed for all ζ ∈ C for which = 0: the combination of (31) and (32) gives

two solutions, corresponding to the different signs for in (30). These solutions are clearly

linearly independent. For those values of ζ for which = 0, only one solution is generated.

A second one may be found using the method of reduction of order.

To determine the spectrum σL, we need to determine the set of all ζ ⊂ R such that (31)

is bounded for all x. Clearly, the vector part of (31) is bounded as a function of x. Thus, we

need to determine for which ζ the scalar function γ (x) is bounded. For this, it is necessary

and sufﬁcient that

(A − )φ + Bx + ıζ B = 0.

(33)

B

Here · = T (1k) 0T (k) · dx is the average over a period and denotes the real part. The investigation of (33) is signiﬁcantly simpler for the trivial-phase case b = 0 than for the general nontrivial-phase case. We treat these cases separately.

5.1. The trivial-phase case: b = 0

With b = 0, (30) becomes

2 = −ζ 4 + ωζ 2 − k164 = −(ζ − ζ1)(ζ − ζ2)(ζ − ζ3)(ζ − ζ4), (34) with

ζ1 = − 12 (1 + k),

ζ2 = − 12 (1 − k),

ζ3 = 12 (1 − k),

ζ4 = 12 (1 + k). (35)

The graph for 2 as a function of ζ is shown in ﬁgure 4(a).

The explicit form of (33) is different depending on whether is real or imaginary. It

should be noted that since ζ ∈ R, it follows from (34) that these are the only possibilities.

First, we consider being imaginary or zero, requiring |ζ | (k + 1)/2 or |ζ |

(1 − k)/2. It follows from the deﬁnitions of A and B that the integrand in (33) may be

written as a rational function of the periodic function sn2(x, k), multiplied by its derivative

2sn(x, k)cn(x, k)dn(x, k). As a consequence the average of this integrand is zero. Thus, all

these values of ζ belong to the Lax spectrum. Extra care should be taken when ζ = 0, in

which case the denominator in (33) is singular, and not integrable. This case may be dealt

with separately. One ﬁnds that the vector part of (31) cancels the singularity in γ (x). In

fact, the two eigenfunctions of the ﬁrst equation of (26) are (−dn(x, k), kcn(x, k))T and

(−kcn(x, k), dn(x, k))T .

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JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL doi:10.1088/1751-8113/44/28/285201

Elliptic solutions of the defocusing NLS equation are stable

Nathaniel Bottman1,3, Bernard Deconinck1 and Michael Nivala2

1 Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA 2 Department of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, CA 90095, USA

E-mail: [email protected], [email protected] and [email protected]

Received 20 January 2011, in ﬁnal form 19 May 2011 Published 9 June 2011 Online at stacks.iop.org/JPhysA/44/285201

Abstract The stability of the stationary periodic solutions of the integrable (onedimensional, cubic) defocusing nonlinear Schro¨dinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additional argument using an appropriate Krein signature allows us to conclude the (nonlinear) orbital stability of all stationary solutions of the defocusing NLS equation with respect to so-called subharmonic perturbations: perturbations that have period equal to an integer multiple of the period of the amplitude of the solution. All results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase proﬁles.

1. Introduction

The defocusing one-dimensional nonlinear Schro¨dinger equation with cubic nonlinearity is given by

ı t = − 12 xx + | |2 .

(1)

Here (x, t) is a complex-valued function, describing the slow modulation of a carrier wave in a dispersive medium. Due to both its physical relevance and its mathematical properties, (1) is one of the canonical equations of nonlinear dynamics. The equation has been used extensively to model, among other applications, waves in deep water [2, 39], propagation

3 Present address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA.

1751-8113/11/285201+24$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA

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in nonlinear optics with normal dispersion [23, 28], Bose–Einstein condensates with repulsive self-interaction [21, 34] and electron plasma waves [12]. Equation (1) is completely integrable [1, 40]. This will be used extensively later on.

The equation has a large class of stationary solutions; these are written as

= e−ıωt φ (x),

(2)

where ω is a real constant. Among this class of solutions are the dark and grey solitons, for which φ(x) is expressed in terms of hyperbolic functions. These solutions may be regarded as limit cases of the so-called elliptic solutions studied in this paper. The stationary solutions (2) are either periodic or quasi-periodic as functions in x. The amplitude of φ(x) of the elliptic solutions is expressed in terms of Jacobi elliptic functions. A thorough discussion of the stationary solutions is found in, for instance, [9]. The details relevant to our investigations are presented in section 2.

The stability analysis of the stationary solutions was begun in [39], where the now classical calculation for the modulational stability of the plane-wave solution (φ(x) constant) is given. The literature discussing the stability of the soliton solutions is extensive, see [29], and references therein. Rowlands [35] may have been the ﬁrst to consider the stability of the elliptic solutions directly. He studied the spectral stability problem for these solutions using regular perturbation theory with the Floquet parameter as a small expansion parameter. At the origin in the spectral plane, this parameter is zero, thus Rowlands was able to obtain expressions for the different branches of the continuous spectrum near the origin. For the focusing NLS equation these calculations demonstrate that the spectrum lies partially in the right-half plane, which leads to the conclusion of instability. For the defocusing NLS equation (1), the ﬁrst approximation to these branches lies on the imaginary axis, and Rowlands’ method is inconclusive with regards to stability or instability of the elliptic solutions. More recently, the stability of the elliptic solutions has been examined by Gallay and Ha˘ra˘gus¸ [17, 18]. In [18], they established the spectral stability of small-amplitude solutions of the form (2) of (1), as well as their (nonlinear) orbital stability with respect to perturbations that are of the same period as |φ(x)|. In [17], the restriction on the amplitude for the orbital stability result is removed. Ha˘ra˘gus¸ and Kapitula [22] put some of these results in a more general framework valid for spectral problems with periodic coefﬁcients originating from Hamiltonian systems. They establish that the small-amplitude elliptic solutions investigated in [18] are not only spectrally but also linearly stable. Lastly, we should mention a recent paper by Ivey and Lafortune [26]. They undertake a spectral stability analysis of the cnoidal wave solution of the focusing NLS equation, by exploiting the squared-eigenfunction connection, like we do in [5] for the cnoidal wave solutions of the Korteweg–de Vries equation and here, see below. Their calculations use Floquet theory for the spatial Lax operator to construct an Evans function for the spectral stability problem, whose zeros give the point spectrum corresponding to periodic perturbations. They also obtain a description of the continuous spectrum (which contains this point spectrum) using a Floquet discriminant. Their description of the spectrum is explicit in the sense that no differential equations remain to be solved. By computing level curves of this Floquet discriminant numerically, they obtain a numerical description of the spectrum.

In this paper, we conﬁrm the recent ﬁndings on spectral and orbital stability of the elliptic solutions of the defocusing equation and extend their validity to solutions of arbitrary amplitude. In addition, we extend the stability results to the class of so-called subharmonic perturbations, i.e. perturbations that are periodic with period equal to an integer multiple of the period of the amplitude |φ(x)|. Further, exploiting the integrability of (1), we are able to provide an explicit analytic description of the spectrum and the eigenfunctions associated with the linear stability problem of all elliptic solutions. We follow the same method as in [5],

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using the algebraic connection between the eigenfunctions of the Lax pair of (1) and those of the spectral stability problem. This explicit characterization of the spectrum, as well as the extension of the spectral stability results to arbitrary amplitude, and the results involving subharmonic perturbations are new. It appears that the methods of Ivey and Lafortune [26] allow for an equally explicit description when applied to the defocusing case. They rely on the general theory of hyperelliptic Riemann surfaces and theta functions, which are restricted to the elliptic case, through a nontrivial reduction process. We never leave the realm of elliptic functions, resulting in a signiﬁcantly more straightforward approach. The explicit characterization of the spectrum is an obvious starting point for the stability analysis of more general solutions to non-integrable generalizations of the NLS equations, such as the twodimensional NLS equation [10, 11] or one-dimensional perturbations of the NLS equation which might include such effects as dissipation or external potentials, see e.g., [7, 26]. As in [17, 18, 22], we prove the spectral stability of the elliptic solutions of (1), without imposing a restriction on the amplitude. The results of [22] allow us to prove the completeness of the eigenfunctions of the linear stability problem, resulting in a conclusion of linear stability. Similarly to the last section of [13], we employ an appropriate Krein signature calculation to allow us to invoke the classical results of Grillakis, Shatah and Strauss [20], from which (nonlinear) orbital stability follows.

It should be emphasized that our results are equally valid for elliptic solutions that have trivial phase (φ(x) real) as for solutions with a non-trivial phase proﬁle (φ(x) not purely real). Similar calculations to the ones presented here apply to the focusing NLS equation, without the conclusion of stability, of course. That case is more complicated, due to the Lax operator associated with that integrable equation not being self adjoint. It will be presented separately elsewhere.

Before entering the main body of the paper, we wish to apologize to the reader for the use of no less than three different incarnations of the NLS equation, in addition to (1). One is obtained through a scaling transformation with a time-dependent exponential factor, to allow the stationary solutions to appear as equilibrium solutions. The second one is used to facilitate our proof of spectral stability and involves a time- and space-dependent exponential factor. The last NLS form writes the second one in terms of its real and imaginary parts, and is useful for our proof of orbital stability. All forms are introduced because we beneﬁt greatly from their use. None are new to the literature. It does not appear straighforward to avoid the use of any of them without much added complication.

2. Elliptic solutions of the defocusing NLS equation

The results of this section are presented in more detail in [9]. We restrict our considerations to the bare necessities for what follows.

Stationary solutions (2) of (1) satisfy the ordinary differential equation

ωφ = − 12 φxx + φ|φ|2.

(3)

Substituting an amplitude-phase decomposition

φ (x) = R(x) eıθ(x)

(4)

in (3), we ﬁnd ordinary differential equations satisﬁed by the amplitude R(x) and the phase θ (x) by separating real and imaginary parts, after factoring out the overall exponential factor. Here we explicitly use that both amplitude and phase are real-valued functions. The equation for the phase θ (x) is easily solved in terms of the amplitude. One ﬁnds

θ (x) = c x 1 dy. (5) 0 R2(y)

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Here c is a constant of integration. Using standard methods for elliptic differential equations (see for instance [8, 30]), one shows that the amplitude R(x) is given by

R2(x) = k2sn2 (x, k) + b,

(6)

where sn(x, k) is the Jacobi elliptic sine function, and k ∈ [0, 1) is the elliptic modulus [8, 30]. The amplitude R(x) is periodic with period T (k) = 2K(k), where K(k) is the complete elliptic integral of the ﬁrst kind [8, 30]:

π/2

1

K(k) =

dy.

(7)

0

1 − k2 sin2 y

The form of the solution (6) leads to

ω = 12 (1 + k2) + 32 b, (8) and

c2 = b(b + 1)(b + k2).

(9)

Conditions on the reality of the amplitude and phase lead to the constraint b ∈ R+ (including

zero) on the offset parameter. The class of solutions constructed here is not the most general

class of stationary solutions of (1). We did not specify the full class of parameters allowed

by the Lie point symmetries of (1), which allow for a scaling in x, multiplying by a unitary

constant, etc. The methods introduced in the remainder of this paper apply equally well and

with similar results to the full class of stationary elliptic solutions.

If the constant c is zero, the solution is referred to as a trivial-phase solution. Otherwise

it is called a nontrivial-phase solution. It is clear from the above that the only trivial-phase

solutions are (up to symmetry transformations)

(x,

t)

=

ksn(x,

k)

e−

ı 2

(1+k 2 )t

.

(10)

This one-parameter family of solutions is found from the two-parameter family of stationary

solutions by equating b = 0. The trivial-phase solutions are periodic in x. Their period is

4K(k). In contrast, the nontrivial-phase solutions are typically not periodic in x. The period

of their amplitude is T (k) = 2K(k), whereas the period τ (k) of their phase is determined by

θ (τ (k)) = 2π . Unless τ (k) and T (k) are rationally related, the nontrivial-phase solution is

quasi-periodic instead of periodic.

This quasi-periodicity is more immediately obvious using a different form of the elliptic

solutions (see [17, 18]), which will prove useful in section 6. We split the integrand of (5) as

c = κ(k, b) + K(x; k, b), (11) R2(x)

where κ(k, b) is the average value of c/R2(x) over an interval of length T (k). Thus the average

value of K(x; k, b) is zero. Then the elliptic solutions may be written as

(x, t ) = e−ıωt+ıκx Rˆ (x),

(12)

where Rˆ (x + T (k)) = Rˆ (x) is typically not real. It is clear from this formulation of the elliptic

solutions that they are generically quasiperiodic with two incommensurate spatial periods

T (k) and 2π/κ(k, b).

3. The linear stability problem

Before we study the orbital stability of the elliptic solutions, we examine their spectral and

linear stability. To this end, we transform (1) so that the elliptic solutions are time-independent

solutions of this new equation. Let

(x, t) = e−ıωt ψ(x, t).

(13)

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Then

ıψt = −ωψ − 12 ψxx + ψ|ψ|2.

(14)

As stated, the elliptic solutions are those solutions for which ψt ≡ 0. Next, we consider perturbations of such an elliptic solution. Let

ψ (x, t) = eıθ(x) (R(x) + u(x, t) + ı v(x, t)) + O( 2),

(15)

where is a small parameter and u(x, t) and v(x, t) are real-valued functions. Since their dependence on both x and t is unrestricted, there is no loss of generality from factoring out the temporal and spatial phase factors. Substituting (15) into (1) and separating real and imaginary parts, the terms of zero order in vanish, since R(x) eıθ(x) solves (1). Next, we equate terms of order to zero and separate real and imaginary parts, resulting in

∂ u = L u = J L+ S

u ,

(16)

∂t v

v

−S L− v

where

J= 0 1 ,

(17)

−1 0

and the linear operators L−, L+ and S are deﬁned by L− = − 12 ∂x2 + R2(x) − ω + 2Rc42(x) , (18)

L+ = − 12 ∂x2 + 3R2(x) − ω + 2Rc42(x) , (19)

c

cR (x) c

1

S = R2(x) ∂x − R3(x) = R(x) ∂x R(x) .

(20)

We wish to show that perturbations u and v that are initially bounded remain so for all times. By ignoring terms of order 2 and higher we are restricting ourselves to linear stability. The elliptic solution φ(x) = R(x) eıθ(x) is by deﬁnition linearly stable if for all ε > 0 there is a δ > 0 such that if ||u(x, 0) + ıv(x, 0)|| < δ then ||u(x, t) + ıv(x, t)|| < ε for all t > 0. It should be noted that this deﬁnition depends on the choice of the norm || · || of the perturbations. In the next section this norm will be speciﬁed. The linear stability problem (16) is written in its standard form to allow for a straightforward comparison with the results of other authors, see for instance [17, 18, 22, 35], and many references where only the soliton case is considered. Some of our calculations are more conveniently done using a different form of the linear stability problem (16) or the spectral stability problem (22, below). These forms will be introduced as necessary.

Since (16) is autonomous in t, we can separate variables and consider solutions of the form

u(x, t) = eλt U (x, λ) ,

(21)

v(x, t)

V (x, λ)

so that the eigenfunction vector (U (x, λ), V (x, λ))T satisﬁes the spectral problem

U λ

=L U

=J

L+

S

U .

(22)

V

V

−S L− V

Since −S is the Hermitian conjugate of S, this latter form of the spectral problem emphasizes the Hamiltonian structure of the problem. In what follows, we suppress the λ

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dependence of U and V. In order to show that the solution φ(x) = R(x) eıθ(x) is spectrally stable, we need to verify that the spectrum σ (L) does not intersect the open right-half of the complex λ plane. To avoid confusion with other spectra deﬁned below, we refer to σ (L) as the stability spectrum of the elliptic solution φ(x). Since the nonlinear Schro¨dinger equation (1) is Hamiltonian [2], the spectrum of its linearization is symmetric with respect to both the real and the imaginary axis [38], so proving the spectral stability of an elliptic solution is equivalent to proving the inclusion σ (L) ⊂ ıR.

Spectral stability of an elliptic solution implies its linear stability if the eigenfunctions corresponding to the stability spectrum σ (L) are complete in the space deﬁned by the norm || · ||. In that case all solutions of (16) may be obtained as linear combinations of solutions of (22).

The ﬁrst goal of this paper is to prove the spectral and linear stability of all solutions (2) by analytically determining the stability spectrum σ (L), as well as its associated eigenfunctions. It is already known from [18] and [22] that the inclusion σ (L) ⊂ ıR holds for solutions of small amplitude, or, equivalently, solutions with small elliptic modulus, leading to spectral stability. We strengthen these results by providing a completely explicit description of σ (L) and its eigenfunctions, without requiring any restriction on the elliptic modulus. To conclude the completeness of the eigenfunctions associated with σ (L), and thus the linear stability of the elliptic solutions, we rely on the SCS lemma, see Ha˘ra˘gus¸ and Kapitula [22].

4. Numerical results

In the next few sections, we determine the spectrum of (22) analytically. Before we do so, we compute it numerically, using Hill’s method [15]. Hill’s method is ideally suited to a periodiccoefﬁcient problem such as (22). It should be emphasized that almost none of the elliptic solutions are periodic in x, as discussed in section 2. Nevertheless, since we have factored out the exponential phase factor eıθ(x) and the remaining coefﬁcients are all expressed in terms of R(x), the spectral problem (22) is a problem with periodic coefﬁcients, even for elliptic solutions that are quasi-periodic.

Using Hill’s method, we compute all eigenfunctions using the Floquet–Bloch decomposition

U (x) = eiμx Uˆ (x) , Uˆ (x + T (k)) = Uˆ (x), Vˆ (x + T (k)) = Vˆ (x),

(23)

V (x)

Vˆ (x)

with μ ∈ [−π/2T (k), π/2T (k)). It follows from Floquet’s theorem [3] that all bounded solutions of (22) are of this form. Here bounded means that maxx∈R{|U (x)|, |V (x)|} is ﬁnite. Thus

U, V ∈ Cb0(R).

(24)

By a similar argument as that given at the end of section 2, the typical eigenfunction (23) obtained this way is quasi-periodic, with periodic eigenfunctions ensuing when the two periods T (k) and 2π/μ are commensurate. Speciﬁcally, our investigations include perturbations of an arbitrary period that is an integer multiple of T (k), i.e., subharmonic perturbations.

Figure 1 shows discrete approximations to the spectrum of (22), computed using SpectrUW 2.0 [14]. The solution parameters for the top two panels (a) and (b) are b = 0 (thus corresponding to a trivial-phase solution (10)) and k = 0.8. The numerical parameters (see [14, 15]) are N = 20 (41 Fourier modes) and D = 40 (39 different Floquet exponents). The right panel (b) is a blow-up of the left panel (a) around the origin. First, it appears

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(a)

(b)

(c)

(d )

Figure 1. Numerically computed spectra (imaginary part of λ vs. real part of λ) of (22) for different solutions (2), with parameter values given below, using Hill’s method with N = 20 (41 Fourier modes) and D = 40 (39 different Floquet exponents), see [14, 15]. (a) A trivial-phase sn-solution with k = 0.5. (b) A blow-up of (a) around the origin, showing a band of higher spectral density. (c) A nontrivial-phase solution with b = 0.2 and k = 0.5. (d) A blow-up of (c) around the origin,

similarly showing a band of higher spectral density.

that the spectrum is on the imaginary axis4, indicating spectral stability of the snoidal solution (10). Second, the numerics show that a symmetric band around the origin has a higher spectral density than does the rest of the imaginary axis. This is indeed the case, as shown in more detail in ﬁgure 2(a), where the imaginary parts in [−1, 1] of the computed eigenvalues are displayed as a function of the Floquet parameter μ. This shows that λ values with imaginary parts in [−0.37, 0.37] (approximately) are attained for four different μ values in [−π/2T (k), π/2T (k)). The rest of the imaginary axis is only attained for two different μ values. This picture persists if a larger portion of the imaginary λ axis is examined. These numerical results are in perfect agreement with the theoretical results below.

The bottom two panels (c) and (d) correspond to a nontrivial-phase solution with b = 0.2 and k = 0.5. The numerical parameters are identical to those for panels (a) and (b). Again, the spectrum appears to lie on the imaginary axis, with a higher spectral density around the origin. The clumping of the eigenvalues outside of the higher-density band is a consequence of aliasing. This is an artifact of the numerics and the graphics. A plot of the imaginary parts of the computed eigenvalues as a function of μ is shown in ﬁgure 2(b). As for the trivial-phase case this shows the quadruple covering of the spectrum of a band around the origin of the imaginary axis, and the double covering of the rest of the imaginary axis. Due to the nontrivial-phase proﬁle, the curves in ﬁgure 2(b) have lost some symmetry compared to those in ﬁgure 2(a).

4 The order of magnitude of the largest real part computed is 10−10.

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(a)

(b)

Figure 2. The imaginary part of λ as a function of μ, demonstrating the higher spectral density (four vs. two) corresponding to ﬁgure 1(b) (left panel) and to ﬁgure 1(d) (right panel). The parameter values are identical to those of ﬁgure 1.

Making the opposite choice for the sign on c in (9) results in the ﬁgure being slanted in the

other direction. The above considerations remain true for different values of the offset b ∈ R+ and the

elliptic modulus k ∈ [0, 1), although the spectrum does depend on both, as we will prove in the following sections. Thus, for all values of (b, k) ∈ R+ × [0, 1), the spectrum of the elliptic solutions appears to be conﬁned to the imaginary axis, indicating the spectral stability of these solutions. Similarly, for all these parameter values, the spectrum σ (L) covers a symmetric interval around the origin four times, whereas the rest of the imaginary axis is double covered.

The edge point on the imaginary axis where the transition from spectral density four to two occurs depends on both b and k and is denoted λc(b, k). The k-dependence of λc(b = 0.2, k) is shown in ﬁgure 3. Again, both numerical and analytical results (see section 6) are displayed. For these numerical results, Hill’s method with N = 50 was used.

5. Lax pair representation

Since our analytical stability results originate from the squared-eigenfunction connection

between the defocusing NLS linear stability problem (16) and its Lax pair, in this section we

examine this Lax pair, restricted to the elliptic solutions of the defocusing NLS.

As for the stability problem, we consider the generalized defocusing NLS (14). This

equation is integrable, thus it has a Lax pair representation. Speciﬁcally, (14) is equivalent to

the compatibility condition χxt = χtx of the two ﬁrst-order linear differential equations

χx = −ψı∗ζ ıψζ χ ,

χ = −ıζ 2 − 2ı |ψ |2 + 2ı ω

ζ ψ + 2ı ψx

χ.

(25)

t

ζ ψ ∗ − 2ı ψx∗

ıζ 2 + 2ı |ψ |2 − 2ı ω

Thus (14) is satisﬁed if and only if both equations for χ of (25) are satisﬁable. Written as

a spectral problem with parameter ζ , the ﬁrst equation is seen to be formally self adjoint [27],

thus the spectral parameter ζ is conﬁned to the real axis. Restricting to the elliptic solutions

gives

χ = −ıζ

φ χ,

χ = −ıζ 2 − 2ı |φ|2 + 2ı ω

ζ φ + 2ı φx

χ.

(26)

x

φ∗ ıζ

t

ζ φ∗ − 2ı φx∗

ıζ 2 + 2ı |φ|2 − 2ı ω

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Figure 3. Numerical and analytical results for the imaginary part of the edge point λc(b, k) of the quadruple-covered region as a function of the elliptic modulus k for b = 0.2. The solid curve displays the analytical result, the small circles are obtained numerically.

(This ﬁgure is in colour only in the electronic version)

We refer to the spectrum of the ﬁrst equation of (26) as σL. It is the set of all ζ values for which this equation has a solution bounded in x (as in section (4)). As discussed above, σL ⊂ R. The main goal of this section is the complete analytic determination of σL. For ease of notation, we rewrite the second equation of (26) as

χt = AC −BA χ . (27)

Since A, B and C are independent of t, we may separate variables. Consider the ansatz

χ (x, t) = e t ϕ(x),

(28)

where is independent of t. We refer to the set of all such that χ is a bounded function of x as the t-spectrum σt . Substituting (28) into (27) and canceling the exponential, we ﬁnd

A−

B ϕ = 0.

(29)

C −A −

This implies that the existence of nontrivial solutions requires

2 = A2 + BC = −ζ 4 + ωζ 2 − cζ + 116 (4ωb − 3b2 − k 4), (30)

where k 2 = 1 − k2. We have used the explicit form of φ(x), given in section 2. This demonstrates that is not only independent of t, but also of x. Such a conclusion could also be arrived at by expressing the derivatives of the operators of (26) as matrix commutators, and applying the fact that the trace of a matrix commutator is identically zero [4, 16].

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Having determined as a function of ζ for any given elliptic solution of defocusing NLS (i.e., in terms of the parameters b and k), we now wish to do the same for the eigenvector ϕ(x), determined by (29). Immediately,

ϕ = γ (x) −B(x) ,

(31)

A(x) −

where γ (x) is a scalar function. Indeed, the vector part of (31) ensures that χ (x, t) satisﬁes the second equation of (26). Next, we determine γ (x) so that χ (x, t) also satisﬁes the ﬁrst equation. Substituting (31) in this ﬁrst equation results in two homogeneous linear scalar differential equations for γ (x) which are linearly dependent. Solving gives

γ (x) = γ0 exp − (A − )φ + Bx + ıζ B dx . (32) B

For almost all ζ ∈ C, we have explicitly determined two linearly independent solutions

of the ﬁrst equation of (26). Indeed, for all ζ , there should be two such solutions, and two

have been constructed for all ζ ∈ C for which = 0: the combination of (31) and (32) gives

two solutions, corresponding to the different signs for in (30). These solutions are clearly

linearly independent. For those values of ζ for which = 0, only one solution is generated.

A second one may be found using the method of reduction of order.

To determine the spectrum σL, we need to determine the set of all ζ ⊂ R such that (31)

is bounded for all x. Clearly, the vector part of (31) is bounded as a function of x. Thus, we

need to determine for which ζ the scalar function γ (x) is bounded. For this, it is necessary

and sufﬁcient that

(A − )φ + Bx + ıζ B = 0.

(33)

B

Here · = T (1k) 0T (k) · dx is the average over a period and denotes the real part. The investigation of (33) is signiﬁcantly simpler for the trivial-phase case b = 0 than for the general nontrivial-phase case. We treat these cases separately.

5.1. The trivial-phase case: b = 0

With b = 0, (30) becomes

2 = −ζ 4 + ωζ 2 − k164 = −(ζ − ζ1)(ζ − ζ2)(ζ − ζ3)(ζ − ζ4), (34) with

ζ1 = − 12 (1 + k),

ζ2 = − 12 (1 − k),

ζ3 = 12 (1 − k),

ζ4 = 12 (1 + k). (35)

The graph for 2 as a function of ζ is shown in ﬁgure 4(a).

The explicit form of (33) is different depending on whether is real or imaginary. It

should be noted that since ζ ∈ R, it follows from (34) that these are the only possibilities.

First, we consider being imaginary or zero, requiring |ζ | (k + 1)/2 or |ζ |

(1 − k)/2. It follows from the deﬁnitions of A and B that the integrand in (33) may be

written as a rational function of the periodic function sn2(x, k), multiplied by its derivative

2sn(x, k)cn(x, k)dn(x, k). As a consequence the average of this integrand is zero. Thus, all

these values of ζ belong to the Lax spectrum. Extra care should be taken when ζ = 0, in

which case the denominator in (33) is singular, and not integrable. This case may be dealt

with separately. One ﬁnds that the vector part of (31) cancels the singularity in γ (x). In

fact, the two eigenfunctions of the ﬁrst equation of (26) are (−dn(x, k), kcn(x, k))T and

(−kcn(x, k), dn(x, k))T .

10