Stable pulse solutions for the nonlinear Schrödinger equation with higher order dispersion management

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1 Stable pulse solutions for the nonlinear Schrödinger equation with higher order dispersion management Jamison T. Moeser, Christopher K..T. Jones, Vadim Zharnitsky September 11, 3 Abstract The evolution of optical pulses in fiber optic communication systems with strong, higher order dispersion management is modeled by a cubic nonlinear Schrödinger equation with periodically varying linear dispersion at second and third order. Through an averaging procedure, we derive an approximate model for the slow evolution of such pulses, and show that this system possesses a stable ground state solution. Furthermore, we characterize the ground state numerically. The results explain the experimental observation of higher order DM solitons, providing theoretical justification for modern communication systems design. 1 Introduction 1.1 Conventional dispersion management The technique of dispersion management (DM, introduced in the early eighties [16] and refined during the past decade [7], has emerged as the dominant technology for high bandwidth data transmission through optical fibers. In a dispersion managed fiber link, short sections of fiber with opposite linear dispersion characteristics are joined together in a periodically repeated structure, forming a fiber whose linear dispersion is effectively canceled out over each period of dispersion management. In such a system, the characteristic length of local dispersion is much shorter than that of nonlinearity or average dispersion, so that on the scale of a typical dispersion management segment, the effects of nonlinearity and average dispersion can be made small relative to those of the local dispersion. In this regime, destabilizing effects such as four-wave mixing [, 5, ] and Gordon-Haus jitter [14, 36] are minimized. In the case of dispersion management at second order, the propagation equation for the wave envelope can be written in the dimensionless form iu z + d (zu tt + ɛ u u = (1 1

2 with the dispersion coefficient d (z decomposed into its varying and average components d (z = d (z + ɛα, where 1 d (z dz =. Typically d (z is piecewise constant and periodic, and the parameter ɛ corresponds to the ratio of the characteristic length scales of local dispersion to that of nonlinearity and average dispersion [1, 1]. The system performance of dispersion managed fiber links is truly remarkable [8, 5]. Not only are stable pulse structures observed for α >, the focusing (anomalous regime for NLS [9], but also for the case when α [14, 8]. These DM solitons are characterized by nearly Gaussian central peaks and rapidly decaying secondary peaks which comprise the tails [, 8]. Also, in contrast to the soliton solutions for the NLS equation, the DM solitons possess a nontrivial quadratic phase component, namely chirp [1]. The energy of the DM soliton is higher than that of the corresponding NLS soliton with the same full width at half maximum (FWHM and average dispersion [9, 37]. This makes DM solitons more resistant to the effects of spontaneously emitted amplifier noise, giving dispersion managed systems a higher signal to noise ratio than traditional soliton based systems [34]. Also, this energy enhancement can be exploited to reduce energy variation per channel in wavelength division multiplexing (WDM systems operating near zero average dispersion [9]. The first analytical results for DM solitons were obtained in the late nineties, when an averaged equation describing the slow evolution of solutions to (1 was derived in [1] through path averaging and in [1] through multiple scales expansion. A rigorous justification for this averaged equation was given in [38], where, moreover, it was shown that for α >, the Hamiltonian corresponding to the averaged equation possesses a ground state solution in the class of functions A λ = {u : u = λ, u t < }. These results indicate the existence of a stable, stationary solution to the averaged equation that propagates nearly periodically for (1 on time scales up to O( 1 ɛ. The existence of a standing wave solution for the averaged equation in this regime was also established in [1] by means of a general theorem on bifurcation of solutions from the essential spectrum. Furthermore, the existence of a ground state for the case α = was recently demonstrated [13]. 1. Higher order dispersion management Waves of the form exp(id(ωz iωt traveling through an optical fiber satisfy the dispersion relation D(ω = n(ωω, c where D(ω is termed the propagation constant, n(ω is the index of refraction, ω is the frequency, and c is the speed of light in a vacuum [3]. Thus, in general, the propagation constant is a complicated function of frequency. Explicit

3 forumlas for D(ω are generally unknown, and in the derivation of the evolution equation for the electric field s slowly varying amplitude, D(ω is approximated by its Taylor polynomial in a neighborhood of the carrier frequency. D(ω d + d 1 (ω ω + d (ω ω +... where d n = D(n (ω n!. In the derivation of the conventional dispersion management model (1, one assumes that pulses are sufficiently narrow in the frequency domain, ω ω << 1, so that D(ω is accurately approximated by its quadratic Taylor polynomial, and the effects of changes in the values of d in a neighborhood of ω are neglected. However, for the propagation of pulses which are broader in the frequency domain, the inclusion of the cubic term in the Taylor approximation is necessary, with the resulting model taking into account variations in d. This additional term in the Taylor series gives rise to a third order linear dispersive term in the governing NLS-type equation for the electric field s slowly varying envelope [3]. In single channel systems, third order dispersion generally causes an asymmetric broadening of pulses. Moreover, in WDM systems, which utilize many optical channels separated in the frequency domain, third order dispersion can prevent conventional dispersion compensation across neighboring channels. A natural way to surmount these difficulties is to manage dispersion at both second and third order. By utilizing this technique of higher order dispersion management (HODM, the asymmetric broadening that takes place for ultrashort optical pulses in single channel systems with conventional DM is almost exactly compensated for. Furthermore, in WDM systems, HODM makes it possible to compensate for dispersion over many neighboring frequency channels simultaneously. In fact, advances in fiber manufacturing techniques [18] have made it possible to incorporate this idea into new optical fibers, termed dispersion slope compensating fibers, and recent experiments have yielded impressive results [7, 11, 19, 3, 6]. The evolution of optical pulses in a fiber with dispersion management at second and third order is governed by the following dimensionless nonlinear Schrödinger type equation [4] with iu z + d (zu tt + id 3 (zu ttt + ɛ nl u u =, d j (z = d j (z + ɛ j α j and 1 d j (z dz =. 3

4 Here the α j, j =, 3 are order one measures of average dispersion at second and third order, respectively, ɛ nl is a small parameter representing the ratio of characteristic lengths of the local dispersions to the nonlinearity, and the ɛ j are small parameters representing the ratio of characteristic lengths of the local dispersions to the average dispersions. The dispersion coefficients d j (z, j =, 3, are piecewise constant and, due to the manufacture process, periodic with the same period, here normalized to be 1. In the operating regimes we consider, the parameters satisfy ɛ 3 << ɛ nl ɛ, so we set ɛ = ɛ nl = ɛ, neglect the effects of average third order dispersion by setting α 3 =, and consider the equation iu z + d (zu tt + id 3 (zu ttt + ɛ u u = d (z = d (z + ɛα ( d 3 (z = d 3 (z. We develop an averaging theory for (, and show that for the case α >, the corresponding averaged equation possesses a ground state solution which propagates nearly periodically for the full equation. Furthermore, we solve the Euler-Lagrange equation numerically, revealing the structure of this new DM soliton. We also report that analysis for the case α = will appear elsewhere. Averaging Solutions of ( evolve on two distinct spatial scales, which suggests performing an averaging procedure. We note that the analysis in this section does not require the condition α 3 =, but is necessary later when proving the existence of ground states..1 Averaged equation We first perform the transformation u(z, t = L(z{v(z, t}, where L{ } is the unitary semigroup for the linear evolution equation iu z + d (zu tt + i d 3 (zu ttt =. (3 The operator is easily computed via Fourier transform L(z{v(, t} = 1 θ(z, kˆv(, k exp(iktdk (4 π where θ(z, k = exp z i[k d (τ k 3 d3 (τ]dτ. We observe that L(z is an isometry on H s ( for all s. Moreover, due to the periodicity of d (z and d 3 (z, both θ(z, k and L(z are periodic in z. For 4

5 ease of notation, we henceforth suppress the variable dependencies of v. Using u z = ( L(z{v} z = L(z { } v + i z d (z ( L(z{v} ( t d 3 (z 3 L(z{v} t 3 we obtain by direct substitution into (, the evolution equation for v i v ( z + ɛ v α t + C(z{v} =, (5 where C(z{v} = L( z{ L(z{v} L(z{v}}. (6 Formally the averaged equation is i v ( z + ɛ v α t + C {v} where C {v} = 1 In Fourier space, (7 takes the form = (7 L( z { L(z {v} L(z {v}}dz. (8 i ˆv z + ɛ( k α ˆv + Ĉ {v} = (9 with 1 Ĉ {v} = δ(k k 1 + k k 3 Θ(z, k, k 1, k, k 3 ˆv 1 (z ˆv (zˆv 3 (zdk 1 dk dk 3 dz. (1 3 Here ˆv i (z = ˆv(z, k i and z Θ(z, k, k 1, k, k 3 = exp i{ d (τ[k1 k + k3 k ] + d 3 (τ[k1 3 k 3 + k3 3 k 3 ]}dτ. Performing the integration over z in (1 gives Ĉ {v} = 3 δ(k k 1 + k k 3 Θ l (k, k 1, k, k 3 ˆv 1 (z ˆv (zˆv 3 (zdk 1 dk dk 3 (11 where Θ l (k, k 1, k, k 3 = 1 Θ(z, k, k 1, k, k 3 dz. (1 5

6 is a bounded function on 4. The averaged equation (7 corresponds to the variational equation u z = J H where J = i is a skew-symmetric operator, is the Fréchet derivative, and H is the Hamiltonian H (v = α v t dt 1 1 L(z {v} 4 dtdz (13 We note that H (v is a bounded functional on H 1 (, as L{v} 4 L M ( L{v} L{v} 3 4 L t = M L{v t} L L{v} 3 L = M v t L v 3 L L where we have used the Gagliardo-Nirenberg inequality [4] and the fact that L(z is an isometry on any space H s (. We comment briefly on the regularity of the averaged operator C. We will first show that C { } is bounded on H s ( for s > 1. Now Ĉ {v} = δ(k k 1 + k k 3 Θ l (k, k 1, k, k 3 ˆv(k 1 ˆv(k ˆv(k 3 dk 1 dk dk 3 3 Θ l L ( 4 δ(k k 1 + k k 3 ˆv(k 1 ˆv(k ˆv(k 3 dk 1 dk dk 3 3 Θ l L ( 4 ˆv(k 1 ˆv(k ˆv(k k 1 + k dk 1 dk If we denote Î{v} = ˆv(k 1 ˆv(k ˆv(k k 1 + k dk 1 dk, then by the above estimate, it suffices to show that I{v} H s ( v 3 H s ( Now for any u and w H s (, we have that uw H s ( u H s ( w H s (, (14 or equivalently in Fourier domain, û ŵ L w ( û L w ( ŵ L w (, (15 where is the convolution operator and u L w ( = (1 + k s û L (. If we denote ˆF (k + k = ˆv(k 1 ˆv(k k 1 + k dk 1 = ˆv ˆv 6

7 then (14 and (15 applied to û = ŵ = ˆv L w yields ( ˆv ˆv ˇ H s ( ( ˆv ˇ H s ( = v H s ( whereˇdenotes the inverse Fourier transform. Now Î{v} = ˆv(k ˆF (k + k dk = ˆv ˆF, so we apply the above argument to û = ˆv, ŵ = ˆF to conclude that I{v} H s ( v 3 H s (. A standard extension of this argument shows that C { } is locally Lipschitz on H s ( for s > 1. C {u v} H s ( M u v H s ( where M depends on u Hs ( and v Hs (.. Well posedness The averaged equation (7 is similar in form to the focusing NLS equation, and local well posedness is a straightforward application of semigroup theory. Theorem..1 If v H s (, s > 1, then there exists z max > and a unique solution v(z, t C([, z max, H s ( for (7 with initial data v, with the property that either z max = or z max < and lim z zmax v(z H s =. Proof : The linear part can be solved via Fourier transform, generating a C group of unitary operators S(z on H s ( for z. Since C { } is locally Lipschitz from H s ( H s (, local existence follows [33]. To prove a global existence theorem, a priori estimates on solutions of (7 of the form v(z Hs ( < C(z for any z + are needed. This is possible for initial data in H s (, s 1, using conservation of the L norm, conservation of the Hamiltonian, and regularity of the operator C. Theorem.. If v H s (, s 1, then there exists a unique solution v(z, t C([,, H s ( for (7 with initial data v. Proof : 7

8 Multiplying (9 by ˆv, its conjugate by ˆv, subtracting, and integrating over yields z ˆv dk = Im ˆv Ĉ {v}dk, where ˆv Ĉ {v}dk = with ˆ v δ(k k 1 + k k 3 Θ l (, 3 ˆv 1 ˆvˆv 3 dk 1 dk dk 3 dk = 3 3 δ(k k 1 + k k 3 Θ l (, 3 ˆvˆv 1 ˆvˆv 3 dk 1 dk dk 3 dk = k 1 k + k 3 k 3 = k 3 1 k 3 + k 3 3 k 3. Since Θ l (, 3 = Θ l (, 3, making the change of variables k k 3, k 1 k we see that δ(k k 1 + k k 3 Θ l (, 3 ˆvˆv 1 (z ˆv (zˆv 3 (zdk 1 dk dk 3 dk = 3 so that 3 δ(k k 1 + k k 3 Θ l (, 3 ˆvˆv 1 (z ˆv (zˆv 3 (zdk 1 dk dk 3 dk, z v = z and the L norm is conserved. By conservation of the Hamiltonian, H (v = α v t dt 1 ˆv dk = 1 L(z {v } 4 dtdz = Thus H (v = α v t v t dt 1 dt = H (v α + 1 α H (v α M α v 3/ L(z {v} 4 dtdz. L(z {v} 4 dtdz L ( v t L ( 8

9 by the Sobolev inequality. From standard estimates [35], v t dt M At this point we note that the boundedness of v(z H 1 implies the boundedness of v(z L p for p. We complete the proof by demonstrating that v(z Hs ( < M(z. We start with the integral formulation of (7 v(z, t = S(z{v } so that for s 1, z S(z z { C {v(z }}dz (16 v(z Hs ( S(z{v } Hs ( + z = v H s ( + v Hs ( + S(z z { C {v(z }} Hs (dz z v Hs ( + z C {v(z } H s (dz z v(z 3 H s ( dz v(z L ( v(z Hs (dz z v H s ( + M v(z H s (dz Gronwall s inequality gives that v(z H s ( is bounded on [, z, and this, in combination with the local well posedness result, gives global existence [3]..3 Averaging theorem For this section it is most convenient to rescale z z ɛ in the transformed and averaged equations (5 and (7 so that and i vɛ z + α v ɛ ( z t + C {v ɛ } = (17 ɛ i v z + α v + C {v} = (18 t The validity of the averaging procedure is addressed in the following theorem. 9

10 Theorem.3.1 Let v(z, t C ([, z ], H s ( be a solution of (18 on the time interval [, z ] for any z > and s > 7. Then for ɛ sufficiently small, there exists v ɛ (z, t a solution of (17 with initial data v(, t such that v ɛ v L ([, z ɛ ],Hs 3 ( < Cɛ. emark : We note that since u = L{v ɛ }, the standard averaging result u L{v} L ([, z ɛ ],Hs 3 ( < ɛ follows immediately by isometry. Proof : The proof is similar in spirit to classical averaging results in finite dimensions [3], and follows closely the method of [38]. We first split C( z ɛ {v} into its average and varying components ( z ( z C {v} = C {v} + {v} ɛ ɛ where ( z ( z ˆ {v} = δ(k k 1 + k k 3 A ɛ ɛ, k, k 1, k, k 3 ˆv(z, k 1 ˆv(z, k ˆv(z, k 3 dk 1 dk dk 3 3 and ( z ( z A ɛ, k, k 1, k, k 3 = Θ ɛ, k, k 1, k, k 3 Θ l (k, k 1, k, k 3. The function A is bounded in its spatial variables, uniformly in z and ɛ. Now consider z ( τ z B ɛ (z, k, k 1, k, k 3 = A ɛ, k, k ɛ 1, k, k 3 dτ = ɛ A(τ, k, k 1, k, k 3 dτ where τ = τ ɛ. Since the integrand is 1 - periodic with zero mean, we may write z ɛ z ɛ A(τ, k, k 1, k, k 3 dτ = ɛ A(τ, k, k 1, k, k 3 dτ where z = z ɛ [z ɛ ] [, 1, with [ ] denoting the greatest integer function. Thus z B ɛ L ( 5 ɛ A L ( 5 ɛz A L ( 5 Mɛ where M is independent of z. We define the local average ṽ = v + v 1, where ˆv 1 (z, k = i δ(k k 1 + k k 3 B ɛ (z, k, k 1, k, k 3 ˆv(z, k 1 ˆv(z, k ˆv(z, k 3 dk 1 dk dk

11 By direct estimation ˆv 1 (z, k = i δ(k k 1 + k k 3 B ɛ (z, k, k 1, k, k 3 ˆv(z, k 1 ˆv(z, k ˆv(z, k 3 dk 1 dk dk 3 3 B ɛ L ( 5 δ(k k 1 + k k 3 ˆv(z, k 1 ˆv(z, k ˆv(z, k 3 dk 1 dk dk 3. 3 epeating the arguments for regularity of C in Section.1, we have that for every σ > 1, v 1 (z H σ Mɛ v(z 3 H σ Moreover, by the energy estimate in the well posedness Theorem.., the bound is uniform in z for s 1, We directly compute sup z z v 1 (z H σ Mɛ sup z z v(z 3 H σ M ɛ. i ˆv 1 z = ˆ ˆ, where ˆ = i δ(k k 1 +k k 3 B ɛ (z, k, k 1, k, k 3 z {ˆv(z, k 1 ˆv(z, k ˆv(z, k 3 }dk 1 dk dk 3. 3 Using equation (7, we estimate The local average ṽ satisfies where By continuity of C( z ɛ { } for all s > 1, and so Finally we consider H s 3 Mɛ. i ṽ z + α ṽ ( z t + C {ṽ} =, ɛ ( z ( z = C {ṽ} C {v} + v 1 + α ɛ ɛ t. ( z ( z C {ṽ} C {v} H s Mɛ ɛ ɛ H s 3 Mɛ. f ɛ = v ɛ ṽ 11

12 which satisfies i f ɛ z + α f ( ɛ z ( z t + C {ṽ + f ɛ } C {ṽ} =. ɛ ɛ Again by continuity of C( z ɛ { } ( z ( z C {ṽ + f ɛ } C {ṽ} M f ɛ ɛ ɛ H s 3 H s 3. Writing (19 in Fourier space, multiplying the equation by (1 + k s 3 ˆf ɛ, its conjugate by (1 + k s 3 f ɛ, subtracting and integrating over k, one obtains z f ɛ H s 3 Mɛ f ɛ H s 3 + M f ɛ 4 H s 3. Since f ɛ H < M, we can write the estimate s 3 z f ɛ H M ɛ + M f s 3 ɛ H s 3. Now using the fact that f ɛ ( H s 3 = and applying Gronwall s inequality we have f ɛ (z H s 3 e M ɛz M ɛ e K M ɛ where K is a time-independent constant for z O( 1 ɛ, so sup z z ɛ f ɛ H s 3 M ɛ. Overall, we have sup z z ɛ v ɛ v H s 3 sup z z ɛ = sup z z ɛ v ɛ ṽ H s 3 + f ɛ H s 3 + sup z z ɛ sup z z ɛ ṽ v H s 3 v 1 (z H s 3 Mɛ 3 Existence of a ground state Here we show that for the case α > and α 3 =, the averaged Hamiltonian possesses a minimizer in the class of admissible functions A λ = {v : v = λ, v t < }. We adapt an argument first established for the NLS equation [6] and later adapted for the case of second order dispersion management [38]. We first present properties of the Hamiltonian that are essential to the minimization argument. 1

13 3.1 Properties of H inf Aλ H {v} < Proof : We first assume that the 1 - periodic dispersion maps d i (z, i =, 3, are piecewise constant and of the following form, which is standard in optical communications d i (z = { Di if z [, θ or z [1 θ, 1 D i if z [θ, 1 θ. For these dispersion profiles we define the map strength parameters s i by s i = θ D i. The Hamiltonian can be written H {v} = α v t dt e i(k1 k+k3 k4t e i z d(z +i 3 z d3(z dz ˆv(k 1 ˆv(k ˆv(k 3 ˆv(k 4 dk 1 dk dk 3 dk 4 dtdz where = k 1 k + k 3 k 4 and 3 = k 3 1 k 3 + k 3 3 k 3 4. Performing the integration in z yields 1 H {v} = α 4 e i(k1 k+k3 k4t v t dt ( θ sin(s + s 3 3 s + s 3 3 ˆv(k 1 ˆv(k ˆv(k 3 ˆv(k 4 dk 1 dk dk 3 dk 4 dt Let v be an arbitrary element of A λ and consider the rescaled function v (t = 1 v ( t, which is also an element of A λ. A scaling property of the Fourier transform gives that and the chain rule yields ˆv (k = 1 ˆv( k v (t t = 3 v(t t 13

14 where t = t. Substituting v into the Hamiltonian, we have where and F ( = H {v } = α 3 v(t t dt 1 ( θ e i(k1 k+k3 k4t sin(s + s 3 3 s 4 + s 3 3 ˆv ( k1 ( k ˆv ˆv ( k3 = α v(t t dt ( k4 ˆv dk 1 dk dk 3 dk 4 dt 1 3 e i( k 1 ( k + k 3 k 4 t θ sin(s + s s + s ˆv ( k1 ( k ˆv ˆv ( k1 = ( 3 k1 3 = ( k3 ( k4 ˆv dk 1 dk dk 3 dk 4 dt, ( k + ( 3 k + Making the change of variable k j = kj, we have F ( = C ( k3 ( 3 k3 ( k4 ( 3 k4. ( e i(k 1 k +k 3 k 4 θ sin(s t + s s 4 + s ˆv(k 1 ˆv(k ˆv(k 3 ˆv(k 4dk 1dk dk 3dk 4dt = C G(v;, s, s 3, θ 14

15 where = (k 1 (k + (k 3 (k 4, 3 = (k 1 3 (k 3 + (k 3 3 (k 4 3, and G is a functional of v also depending on, s, s 3 and θ. At this stage, we can see that as s j and θ 1, we recover the exact scaling of the integrable NLS Hamiltonian evaluated at v. To show that this Hamiltonian can be made negative, we first note that by continuity of the kernel ( θ K(, θ, s j, j = sin(s + s s + s in, F ( =. Moreover, differentiating the functional F in yields F ( = C G (v;, s, s 3, θ G(v;, s, s 3, θ To compute G, we differentiate under the integral sign and apply the chain rule. This gives G (v;, s, s 3, θ =, so that F ( = G(v;, s, s 3, θ = 1 1 k +k 3 k 4 t 4 e i(k ˆv(k 1 ˆv(k ˆv(k 3 ˆv(k 4 dk 1 dk dk 3 dk 4 dt = 1 v(t 4 dt <. Thus for small enough, the Hamiltonian is negative H {v} is subadditive : If I λ = inf v Aλ H {v}, then I λ1+λ < I λ1 + I λ Claim : For θ > 1, I θλ < θi λ Proof of claim : since I θλ = inf H {v} v A θλ = inf w A λ H { θw} w L = λ θw L = θλ. But H ( θw = α ( θw t dt 1 1 L(z{ θw} 4 dtdz = θα w t dt θ 1 15 L(z{w} 4 dtdz

16 for θ > 1. So < θ ( α I θλ = w t dt 1 1 w 4 dtdz inf w A λ H { θw} < θ inf w A λ H {w} = θi λ. Proof of subadditivity: If we set λ 1 = αλ with α < 1, we have I λ1+λ = I αλ+λ < (α + 1I λ = αi (α 1 λ 1 + I λ < I λ1 + I λ Localization of minimizing sequences In the minimization proof, we used the fact that for a minimizing sequence v k (t H 1 (, there exists a subsequence v km (t which remains localized. That is, for any ɛ > there exists an > such that + w m (t dt > λ ɛ where w m (t = v km (t t m and λ = w m(t dt. To prove this result, we apply a version of Lions concentration-compactness lemma [17, 38]. Lemma If u m H 1 ( is a bounded sequence with u m L = λ, then there exists a subsequence u mk for which one of the following properties hold 1. (localization There exists a sequence t k such that for any ɛ > there exists > and tk + t k. (vanishing For any > lim sup k y u mk dx λ ɛ. y+ y u mk dx. 3. (splitting There exists < < λ such that for any ɛ >, there exist k and two sequences v k, w k with compact support so that for k k v k H 1 + w k H 1 4 sup u mk H 1 (19 k N u mk (v k + w k L ɛ ( v k L ɛ v k L (λ ɛ (1 v k x + w k L x u mk L x + ɛ ( L and dist(supp(v k, supp(w k > ɛ 1. 16

17 Thus, for the minimization problem there exists a localized subsequence of the minimizing sequence if vanishing and splitting can be ruled out. We first rule out vanishing. Let v k be a minimizing sequence for H {v} and assume that a subsequence v mk vanishes. Since v k is a minimizing sequence, for some k we have α v k t dt 1 1 L(z {v k } 4 dtdz < so that 1 Thus for some z, we have L(z {v k } 4 dtdz >. L(z {v k } 4 dt >. Applying a lemma of Cazenave [6] for arbitrary H 1 ( functions gives that u 4 dt C u H 1 sup y y+1 sup y y 1 y+1 y 1 u dt L(z {v k } dt >. (3 Now we relate sup y y+1 y 1 L(z {v k } dt to sup y y+1 y 1 v k dt with the following localization lemma, which is similar to the lemma of [38]. Lemma 3.1. Consider the following linear dispersive equation iu z + d (zu tt + i d 3 (zu ttt = (4 with u H 1 (, u L ( = 1, and d i (z piecewise constant. Let u n (t, z be a sequence of solutions of (4 and define ɛ n (z = sup y y+1 y 1 u n (t, z If u n (t, is vanishing initial data (lim n ɛ n ( = with the constraint u n L ( = 1, then the sequence of the solutions u n (t, z is also vanishing (lim n ɛ n (z =. 17

18 Proof: Let χ m (t be a smooth approximation to the characteristic function on the interval [ m, m], with the property that t χ m < C m, χ m(t = 1 if t 1, and χ(t = if t m. Multiplying (4 by ūχ m (t, its conjugate by uχ m (t, subtracting and integrating over t yields Now and e d dz χ m u dt = d (zim χ m ūu tt dt d 3 (ze χ m ūu ttt dt Im χ m ūu tt dt = Im (χ m ū t + χ mūu t dt = Im χ mūu t dt χ m ūu ttt dt = e (χ m ū t + χ mūu tt dt = e χ m ū t u tt dt e χ mūu tt dt = 1 d u t dt χ m + e (χ mū + χ mū t u t dt = 3 u t χ mdt + e χ mūu t dt Overall, d χ m u dt = d (zim dz and integrating from to z gives z χ m u(z dt = χ m u( dt + χ mūu t dt d 3 (z ( 3 u t χ mdt + e χ mūu t dt ( d (z Im χ mūu tdt + d 3 (z ( 3 u t χ m dt + e χ mūu tdt dz χ m u( dt + C m ( u H 1, χ m L χ m L, d j L where C m as m. Let u n (t, z denote a sequence of solutions of (4 with vanishing initial data i.e. ɛ n ( = sup y y+1 y 1 u n (t, If ɛ n (z < ɛ n ( then we are done, so let ɛ n (z > ɛ n (. Choosing χ mn ( t n such that it is centered with respect to u n (z, t, we have χ mn u n (t, z dt ɛ n (z. 18

19 and also χ mn u n (t, dt m n ɛ n (. and taking the limit m n with m n 1 ɛ n( gives the result. eturning to (3 and applying the contrapositive of the localization lemma, we have y+1 sup y y 1 v k dt >, contradicting the assumption that a subsequence v mk vanishes. To rule out splitting, it is enough to show that H {v mk } > H {w k } + H {u k } + α(ɛ where α(ɛ is independent of k and goes to as ɛ, as this causes H {v mk } to violate subadditivity. We directly evaluate H {v mk } H {v mk } = α (u k + w k + h k dt t where 1 1 L{u k + w k + h k } 4 dtdz h k L < ɛ and we have suppressed the notation L(z. Expanding the terms, this can be rewritten H {v mk } = H {u k } + H {w k } +α e ( t u k t w k + t u k t h k + t w k t h k + t h k dt e 1 ( L{uk + w k } L{h k } + 1 L{h k} 4 + L{u k + w k } (L{u k + w k }(L{h k } + (L{u k + w k } (L{h k } +L{u k + w k } L{h k } Lh k dtdz 19

20 + 1 1 ( L{uk } L{w k } + L{u k } L{u k }L{w k } +(L{u k } (L{w k } + L{w k } L{u k }L{w k } dtdz. We proceed exactly as in [38]. The terms α e ( t u k t w k + t u k t h k + t w k t h k + t h k dt can be estimated from below by C 1 ɛ, with C 1 depending only on λ and α. The terms 1 ( e L{uk + w k } L{h k } + 1 L{h k} 4 + L{u k + w k } (L{u k + w k }(L{h k } + (L{u k + w k } (L{h k } +L{u k + w k } L{h k } Lh k dtdz are all estimated by Holder s inequality and the Sobolev inequality 1 L(z{v} 4 dtdz M v(t 3 L ( v t(t L (, yielding a lower bound of the form C (ɛ. The remaining terms 1 ( L{uk } L{w k } + L{u k } L{u k }L{w k } +(L{u k } (L{w k } + L{w k } L{u k }L{w k } dtdz are estimated using the boundedness of H 1 ( solutions of linear Schrödinger equations in L (, and the following lemma, which is a straightforward consequence of the localization lemma : In the notation of the Concentration-Compactness Lemma the following estimates hold t t k t c L{w k } dt Cɛ t t k t c L{u k } dt Cɛ, where t c = t1+t. Overall, we have H {v mk } > H {w k } + H {u k } + α(ɛ where α(ɛ is independent of k and goes to as ɛ. Thus splitting causes H to violate subadditivity, a contradiction.

21 3. Minimization theorem Theorem 3..1 Let α > and α 3 =. Then there exists a solution to the following constrained minimization problem : Minimize H {v} = α v t dt 1 1 L(z {v} 4 dtdz over the set of admissible functions A λ = {v H 1 (, v = λ} Moreover, every minimizing sequence has a subsequence which converges strongly in H 1 (. emark : We note that the constraint v = λ is quite natural, as the L norm of the initial data is preserved by solutions of the Euler-Lagrange equation (7. Posing the problem in this way is also critical to proof of the stability of the ground state which is given in a later section. Proof : We follow the arguments of [6, 38]. The idea is to first show strong convergence in L ( by using Lions concentration-compactness principle. This involves using structural properties of the Hamiltonian to rule out possible loss of compactness. Strong convergence in L (, along with an appropriate Sobolev inequality, implies convergence of the quartic term in the Hamiltonian. These results, in combination with lower semicontinuity of the H 1 ( norm, give the existence of a minimizer. We show a posteriori that all minimizing sequences have a subsequence which converges strongly in H 1 (. We first argue that I λ >. To prove the lower bound, we use the Sobolev inequality [4] Lv 4 L 4 C Lv t L Lv 3 L = C v t L v 3 L = Cλ3/ v t L. Integrating the inequality over z gives Thus 1 + L(zv 4 dtdz Cλ 3/ v t L. H (v v t L Cλ3/ v t L = Cλ3/ L ( v t C λ 3 4 >, for all v H 1 (. Taking the infimum over v A λ gives the desired result. Let v k be a minimizing sequence for H (v. By the previous inequality, 1

22 v k H 1 must be bounded. By Alaoglu s theorem, there exists a weakly converging subsequence in H 1 (, v km. We will prove strong convergence of v km to a minimizer in H 1 (, and first establish strong convergence in L (. From previous analysis, we conclude that the minimizing sequence remains localized as m. That is, for any ɛ > there exists an > such that + w m (t dt > λ ɛ. (5 where w m (t = v km (t t m. Now w m w for some w H 1 (. For any >, the embedding H 1 ( L ([, ] is compact and we have Together with (5, this implies and therefore + w dt = lim m + w m dt. w dt > λ ɛ for any ɛ >, + w dt = λ. This norm convergence, along with weak convergence in L (, gives strong convergence in L (. Since w m converges weakly to w and the Sobolev norm H1 ( is weakly lower semi-continuous, we have w H1 ( lim inf m w m H1 (, which together with w m w L (, implies that t w m L ( lim inf m tw m L ( (6 Now for any u, u m H 1 ( the Sobolev inequality gives + u m u 4 dt C + It follows that if u m u in L (, + ( + t u m t u dt C u m u 4 dt. ( + 3/ u m u dt 3/ u m u dt

23 Applying the same argument to L(zw m and L(zw, we establish that and so Combing (6 and (7, which can only happen if L(zw m L(zw in L 4 (. L(zw L4 ( = lim m L(zw m L4 (. (7 H (w lim inf m H (w m, H (w = lim m H (w m, (8 so the weak limit w is a minimizer. Furthermore, by (8 t w L ( = lim m tw m L (. Together with weak convergence, this implies strong convergence of t w m in L (, so w m w strongly in H 1 ( 4 Properties of the ground state 4.1 egularity The minimizer for the constrained minimization problem is also a weak solution to the Euler-Lagrange equation If we rewrite (9 in the form ωv + α v tt + C {v} = (9 v tt = 1 α (ωv C {v} = f(v, where f(v H 1 ( by continuity of C {v}, we may use standard elliptic regularity theory [15] to conclude that v H 3 (. Again, by continuity of C {v}, ωv α v tt = C {v} H 3 (, forcing v H 5 (. We repeat this procedure indefinitely, obtaining v H s ( for every s 1, or v C (. We note however that the H s ( norm of v may depend on α. 3

24 4. Stability It is clear that the minimizer is not unique, as any translation v( + τ, τ, or rotation e iθ v, θ, of the minimizer is also a solution of the constrained minimization problem. Also, it is not known that translations and rotations give all possible minimizers. From now on we consider the class of ground state solutions S λ = {v g A λ, H (v g = I λ }. Using the strong convergence of minimizing sequences and conservation laws for (7, one can show that the minimizer is stable in the following orbital sense. Theorem 4..1 Let S λ be the set of ground states S λ = {v g A λ, H (v g = I λ }. For any ɛ >, there exists a δ > such that if inf Sλ v v g H 1 δ, then the solutions of (7 corresponding to initial data v and v g, denoted v(z and v g (z, satisfy sup z inf Sλ v(z v g (z H 1 ɛ. Proof: We argue by contradiction. Let v k ( be a sequence of initial conditions such that inf Sλ v k ( v g H 1, and assume that v k (z and v g (z satisfy sup z inf Sλ v k (z v g (z H 1 ɛ for some ɛ >. For definiteness, let z n to be the first time that inf Sλ v k (z v g (z H 1 = ɛ. By conservation of the L norm and of the Hamiltonian, we have v k (z n dt = v k ( dt H {v k (z n } = H {v k (}. By the assumption on v k ( and continuity of H, we have v k (z n dt = v k ( dt λ By choosing for example w k = functions such that H {v k (z n } = H {v k (} H {v g }. λ 1 v k (z n ( v k(z n dt 1 w k v k (z n H 1, let w k be a sequence of H 1 and w k(z dt = λ. By continuity of H, w k is a minimizing sequence and must have a subsequence w mk which converges to a ground state. But v k (z n v g (z H 1 v k (z n w mk H 1 + w mk v g (z H 1, and taking the infimum over S λ gives ɛ = inf v k (z n v g (z H 1 v k (z n w mk H 1 + inf w mk v g (z H 1, S λ S λ a contradiction. Thus the class of ground states must be orbitally stable. 4

25 5 Numerical studies 5.1 Solution of the eigenvalue problem In Fourier space, the Euler-Lagrange equation (9 becomes ωˆv α k ˆv + Ĉ {v} =. (3 We propose the following explicit iteration scheme so that ω nˆv n+1 α k ˆv n+1 + Ĉ {v n} =, ˆv n+1 = Ĉ {v n} ω n + α k. If we multiply (3 by ˆv(k and integrate over k, we can derive a formula for ω n+1. ˆv n+1 ω n+1 = Ĉ {v n+1}dk α k ˆv n+1 dk ˆv. n+1 dk This idea also suggests a definition for a relaxation factor to hasten convergence [1, 31], s n+1 = ω n+1 ˆv n+1 dk ˆv n+1 Ĉ {v n+1}dk α k ˆv n+1 dk. The factor s n can be used to compensate the nonlinearity, and as the scheme converges, s n 1. We also note that convergence depends strongly on the initial guess, which is typically taken to be Gaussian. Incorporating the relaxation factor, the overall scheme becomes ˆv n+1 = (s n p Ĉ {v n} ω n + α k, ˆv n+1 ω n+1 = Ĉ {v n+1}dk α k ˆv n+1 dk ˆv, n+1 dk s n+1 = ω n+1 ˆv n+1 dk ˆv n+1 Ĉ {v n+1}dk α k ˆv n+1 dk, where we use p = 1.5. To confine our search for minimizers to a fixed level set of L (, we normalize the result of each iteration so that its energy is that of the initial guess, In practice the algorithm is v n L ( = v L( = λ. 5

26 1. Choose an initial profile ˆv, typically Gaussian.. Compute ω with ˆv. 3. Compute s with ˆv and ω. 4. Compute ˆv 1 using the scheme. 5. escale ˆv 1 so that ˆv 1 L ( = ˆv L (. 6. Compute ω 1 and s epeat 3, 4 and 5 until desired accuracy reached. 5. Solution of evolution equations Both the full evolution equation ( and averaged equation (7 can easily be solved with a version of the well known Fourier split-step scheme, which applies to a wide class of NLS-type equations. Given an evolution equation of the form iu z + L{u} + N {u} =, where L is a self-adjoint operator on a Hilbert space and N is a continuous nonlinear operator, the solution may be written formally as u(z, t = u(, te i z (L(s{u}+N (s{u}ds. (31 We consider the evolution for a small propagation step z so that we may approximate (31 using a formal Taylor expansion u( z, t = u(, te i z (L(s{u}+N (s{u}ds u(, te i z/ L(s{u} e i z N (s{u}ds e i z/ L(s{u}ds, with a local error on the order of ( z 3. Performing the approximation for O( 1 z time steps gives a global error on the order of ( z. Formally, e i z is the semigroup for the linear evolution iu z + L{u} = and can be computed explicitly in Fourier domain. Also, e i z N (s{u}ds is the solution operator for the evolution equation iu z + N {u} =, L(s{u}ds and can be computed either by a standard ODE method such as fourth-order unge-kutta or, in special cases, by using conservation laws for the equation. The overall scheme becomes 6

27 1. Choose an initial profile u(, t and compute its Fourier transform with the FFT.. In Fourier space, evolve the linear dispersive operator for z/. 3. Evolve the nonlinear operator for z. 4. Evolve the linear dispersive operator for z/. 5. epeat 3 and 4 until the final propagation step. 6. Evolve the linear dispersive operator for z/ and compute the inverse Fourier transform with the IFFT. 5.3 Existence of nearly periodic solutions Combining the averaging and minimization results, we see that there exist stationary solutions for (7 that evolve nearly periodically for (. The ground state v g (t for the variational problem corresponds to a standing wave solution for (7, v(z, t = exp(iωzv g (t. The averaging theorem gives that ( z u L {v(z, t} ɛ ɛ L ([, z ɛ ],Hs 3 ( so that u(z, t is nearly periodic on the scale of validity for the averaging theorem. Moreover, by well-posedness of the averaged equation, the same result is true for initial data chosen close to the class of ground states inf Sλ v v g H 1 ɛ. We define a higher order dispersion managed soliton to be an element from the class of ground states S λ and demonstrate the existence of such solutions numerically. Figure 1 shows the shape of the ground state solution for the parameters d (z = d 3 (z = { 5. if z [,.5 or z [.75, if z [.5,.75 (3 and α = 1., ɛ =.1. The solution is computed on the time domain [-3,3] with 48 Fourier modes. The logarithm of the amplitude v(t is plotted vs. time on the interval [-,]. One observes a nearly Gaussian central peak, along with many secondary peaks which decay rapidly. This is similar to the structure of the ground states observed for dispersion management at second order [1, 8]. From the averaging theorem, one would expect the ground state to evolve nearly periodic for z O(1. Figure depicts the evolution of the maximum amplitude of the ground state for the corresponding full equation. The individual oscillations are due to the linear compensation of dispersion, and we observe that the evolution of the amplitude is, in fact, nearly periodic on z scales much longer than those predicted by the averaging theorem. 7

28 .5 Ground state for the averaged Hamiltonian.5 1 log( u t Figure 1: High order dispersion managed soliton 8

29 3.85 Amplitude vs. distance u z Figure : Nearly periodic evolution of the ground state 9

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