Abstract. We consider the Dispersion Managed Nonlinear Schrödinger Equation in the case of zero residual dispersion. Using dispersive properties of the ...
REGULARITY OF GROUND STATE SOLUTIONS OF DISPERSION ¨ MANAGED NONLINEAR SCHRODINGER EQUATIONS M. STANISLAVOVA Abstract. We consider the Dispersion Managed Nonlinear Schr¨ odinger Equation in the case of zero residual dispersion. Using dispersive properties of the equation and estimates in Bourgain spaces we show that the ground state solutions of DMNLS are smooth. The existence of smooth solutions in this case matches the well-known smoothness of the solutions in the case of nonzero residual dispersion. In the case x ∈ R2 we prove that the corresponding minimization problem with zero residual dispersion has no solution.
1. Introduction and Main Result Our work is motivated by the study of parametrically excited NLS with periodically varying dispersion coefficient iut + D(t)uxx + C(t)|u|2 u = 0, which arises as an envelope equation in the problem of an electromagnetic wave propagating in an optical waveguide. The balance between the dispersion and the nonlinearity in this equation is the key factor that determines the existence of stable pulses. In the last decade, a technique that uses fibers with alternating sections having oposite dispersion was introduced. This technology, called dispersion management, proved to be incredibly successful in producing stable, soliton-like pulses. The idea is to use rapidly varying dispersion with approximately zero mean and small nonlinearity in hope that the balance between the small residual dispersion and the small nonlinearity will produce a soliton-like solutions. There have been an enourmous amount of technological advances in this direction with an array of numerical and phenomenological explanations and a recent theoretical understanding of the strong stability properties of the dispersion managed (DM) systems. The envelope equation that describes the propagation of electromagnetic pulses in optical fibers in the regime of strong dispersion management, derived by Gabitov and Turitsyn in 1996 ([5] [6]) is a nonlinear Schr¨odinger equation with periodically varying coefficients. After rescaling the equation takes the form (1)
iut + d(t)uxx + ε(|u|2 u + αuxx ) = 0,
where t is the propagation distance, x is the retarded time and d(t) is the mean-zero component of the dispersion, see [15]. Note that the average dispersion and nonlinearity are Key words and phrases. Dispersion Managed Nonlinear Schr¨ odinger equation, regularity, ground states. 1
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small compared to the local dispersion, which is a characteristic feature of the strong dispersion management. Performing Van der Pol transformation in (1) and averaging in the Hamiltonian we obtain the averaged variational principle hHi = ε
(2)
Z+∞Z1
1 (α|vx |2 − |T (t)v|4 )dxdt 2
−∞ 0
with the corresponding Euler-Lagrange equation (averaged), see [1], [6] (3)
ivt + εαvxx + εhQi(v, v, v) = 0,
where hQi(v1 , v2 , v3 ) =
Z1
Q(v1 , v2 , v3 , t)dt.
0
Here T (t) is the fundamental solution of iut + d(t)uxx = 0 and Q(v1 , v2 , v3 , t) = T −1 (t)(T (t)v1 T (t)v2 T (t)v3 ). In [15] the existence of ground state solution for the averaged equations is proved, as well as an averaging result, which guarantees the existence of nearly periodic stable pulses. The ground state of the averaged equation exists as a solution of the constrained minimization problem (4)
Z+∞ Pλ = inf{E(v) = hHi(v), v ∈ H 1 , |v|2 dx = λ}. −∞
This result is for the case of positive average dispersion α and using bootstraping procedure it is shown that the minimizer is smooth in this case. The variational problem in the case of zero-average dispersion is more subtle due to the absence of a priori bounds in spaces different from L2 . In this case the functional is formally the singular perturabation limit α → 0 of (4), see [10] and [15]. In [9] the corresponding minimization problem Z+∞ Pλ = inf{ϕ(u), u ∈ L , |u|2 dx = λ}, 2
(5)
−∞
where ϕ(u) = −
R1 +∞ R
2
2
|eit∂x u(x)|4 dxdt has been studied. By eit∂x we denote the semigroup
0 −∞
2
generated by the free Schr¨odinger equation in one dimension, i.e. u(t, x) = (eit∂x u0 )(x) solves iut + uxx = 0, u(0, x) = u0 (x). Exploring the dispersive properties of the Schr¨odinger evolution and using Lion’s concentration compactness in L2 , the existence of a minimizer u ∈ L2 ∩ L∞ has been derived.
REGULARITY OF GROUND STATE SOLUTIONS OF DMNLS
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In the current paper we follow the same idea as in [9], but make use of Bourgain spaces Xs,b to simplify the proof and show that the existing minimizer u is smooth. More presisely, we prove the following theorem. Theorem 1. The minimization problem (5) has a solution u ∈ C ∞ ∩ L2 . It is interesting to study the two-dimensional case x ∈ R2 , which is physically relevant since x is the coordinate of the sections orthogonal to the fiber and t is the distance along the fiber. In this case the corresponding model is the variable coefficients nonlinear Sch¨odinger equation in two-space dimensions iut + d(t)∆u + c(t)|u|2 u = 0.
(6)
The results in [15] transfer to the two-dimentional case. There exists a solution for every α, λ > 0 of the corresponding variational problem Z Z1 Z Z 1 2 4 1 (7) min{α |∇u| dx − |U (t)u| dxdt : u ∈ H , |u|2 dx = λ}, 2 0 R2
R2
R2
More recently Kunze in [11] has shown that again in the case of nonzero residual dispersion the functional Z Z1 Z 1 (8) ϕ(u) = (|∇u|2 + |u|2 )dx − |U (t)u|4 dxdt, u ∈ H1 , 2 0 R2
R2
where U (t)u0 = eit∆ u0 is the evolution operator of the free Schr¨odinger equation admits a sequence (uj ) ⊂ H1 of critical points such that uj are radially symmetric and |uj |H 1 → ∞ as j → ∞. Here α 6= 0 is taken equal to 1 whithout loss of generality and the constraint kukL2 = 1 is included in the functional. In [9] the author posed the problem about the existence of a constrained minimizer for the functional Z1 Z |U (t)u|4 dxdt, u ∈ L2 (9) ϕ(u) = − 0 R2
in the two-dimensional case x ∈ R2 . In the next theorem we give negative answer to this question. Theorem 2. In R2 a solution of the constrained minimization problem Z1 Z P = inf{ϕ(u) = − |U (t)u|4 dxdt, u ∈ L2 , kukL2 = 1} 0 R2
does not exist. Another case of interest is to consider a one-dimensional NLS with quintic nonlinearity iut + uxx + |u|4 u = 0,
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which arises if the electromagnetic field is so strong that higher order nonlinearity can not be neglected. If we inroduce dispersion management with rapidly varying dispersion the corresponding model is given by iut + d(t)uxx + |u|4 u = 0. In [15] the autors follow the averaging procedure to produce the equation ivt + αvxx + bQ5 (v, v, v, v, v) = 0, with the averaged Hamiltonian Z1 Z+∞ 1 hHi = (α|vx |2 − |T (t)v|6 )dxdt. 2 0 −∞ 1
A solution v ∈ H of the constrained minimization problem Z+∞ 1 Pλ = inf{E(v) = hHi(v), v ∈ H , |v|2 dx = λ} −∞
when α 6= 0 was found in [15]. We prove the following: Theorem 3. In R1 a solution for the constrained minimization problem Z1 Z∞ Z+∞ 6 2 P = inf{ϕ(u) = − |T (t)u| dxdt, u ∈ L , |u|2 dx = 1} 0 −∞
−∞
does not exist. 2. Proof of Theorem 1 Introduce the Bourgain spaces Xs,b [2], [3] as the set of all functions u with Z |ˆ u(ξ, τ )|2 < τ − |ξ|2 >2b < ξ >2s dξdτ < ∞, where < ξ >:= (1+|ξ|2 )1/2 and < τ −|ξ|2 >:= (1+|τ −|ξ|2 |2 )1/2 and uˆ(ξ, τ ) is the time-space − Fourier transform. We also introduce the space Xs,b as Z − 2 2 2b 2s Xs,b := u : |ˆ u(ξ, τ )| < τ + |ξ| > < ξ > dξdτ < ∞ . Note that Xs,b spaces are Hilbert spaces with norm Z kukXs,b = |ˆ u(ξ, τ )|2 < τ − |ξ|2 >2b < ξ >2s dξdτ R − if and only if v ∈ Xs,b . We include the and that kukXs,b = sup uvdxdt. Thus v ∈ Xs,b − v∈X−s,−b
following well-known lemma for convinience next.
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Lemma 4. Let ψ ∈ C0∞ (R1 ), suppψ ⊂ (−1, 1). Then
2
(1) ψ(t)eit∂x u0 ≤ Cb ku0 kH s Xs,b
(2) kukL∞ s ≤ Cε kukX s,1/2+ε t Hx
Proof. To prove (1), compute the Fourier transform of the left hand side 2 ˆ − |ξ|2 )uˆ0 (ξ). F(ψ(t)eit∂x u0 )(τ, ξ) = ψ(τ
Thus
it∂x2
ψ(t)e u0
Xs,b
≤
Z
≤
ˆ − |ξ|2 )|2 |uˆ0 (ξ)|2 < τ − |ξ|2 >2b < ξ >2s dξdτ. |ψ(τ
R ˆ − |ξ|2 )|2 < τ − |ξ|2 >2b dτ ≤ Cb . Then (1) follows from |ψ(τ ukL2 L1τ we have For part (2) since kukL∞ 2 ≤ kˆ t Lx ξ Z Z 2 kukL∞ ( |ˆ u(τ, ξ)|dτ )2 < ξ >2s dξ s ≤ t Hx Z Z Z 2 2 1+2ε ≤ ( |ˆ u(τ, ξ| < τ − |ξ| > dτ ).(
dτ ) < ξ >2s dξ ≤ Cε kuk2Xs,1/2+ε . 2 1+2ε < τ − |ξ| >
We will need to use the following lemma [13], p.21 on the smoothing effect of the Duhamel operator on the space Xs,b . Lemma 5. Let ψ be a smooth characteristic function of the interval [−1, 1]. Then for any ε>0
Zs
ψ(s) ei(s−t)∂x2 F (t)dt ≤ kF kX s,−1/2+2ε .
s,1/2+ε 0
X
Next, we introduce the Littlewood-Paley decomposition. Let ϕ ∈ C0∞ (R1 ) and ϕ(ξ) = 1 if |ξ| ≤ 1 and ϕ(ξ) = 0 if |ξ| > 2. Define the function ψ(ζ) = ϕ(ζ) − ϕ(2ζ). Then ϕ(ζ) +
∞ X
ψ(2−k ζ) = 1
k=1
for every ζ ∈ R, ζ 6= 0. Define the Littlewood-Paley operators as and
−k d ˆ P k f (ζ) = ψ(2 ζ)f (ζ)
d ˆ ˆ P 0 f (ζ) = ϕ(ζ)f (ζ) ∼ χ[−1,1] (ζ)f (ζ).
k−1 d Note that P ≤ |ζ| ≤ 2k+1 . k f (ζ) 6= 0 only if 2
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Let Pk−5
