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Solutions to the nonlinear Schrödinger equation with sequences of initial data converging to a Dirac mass
Physica D 239 (2010) 2050–2056
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Solutions to the nonlinear Schrödinger equation with sequences of initial data converging to a Dirac mass J.P. Newport a,c,∗ , K.D.T.-R. McLaughlin b,c a
Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, NC 27599, United States
b
Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721-0089, United States
c
Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-900, Brazil
article
info
Article history: Received 11 October 2008 Received in revised form 15 July 2010 Accepted 21 July 2010 Available online 25 July 2010 Communicated by J. Bronski
abstract We study the nonlinear Schrödinger equation with sequences of initial data that converge to a Dirac mass, and study the asymptotic behaviour of solutions. In doing so we find a connection to previously known long time asymptotics. We demonstrate a type of universality in the behaviour of solutions for real initial data, and we also show how this universality breaks down for examples of initial data that are not purely real. © 2010 Elsevier B.V. All rights reserved.
Keywords: Integrable systems Nonlinear partial differential equations Riemann–Hilbert analysis Scattering and inverse scattering theory
1. Introduction The goal of this paper is to analyze solutions of the defocussing nonlinear Schrödinger (NLS) equation with sequences of initial data that converge to a Dirac mass. One such sequence of initial data (among several that we study) leads us to the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0
(1)
1 x ϕ(x, t = 0) = f , ϵ ϵ
(2)
for a general class of functions f (y) which will be specified later. For convenience, we often assume that R f (y)dy = 1 so that 1ϵ f ϵx converges weakly to a unit Dirac mass as ϵ → 0. However, all the formulae which we derive are valid when f (y) has a finite non-zero mass. We use a combination of relatively classical results concerning the asymptotic behaviour of the integrable NLS equation together with some more involved Riemann–Hilbert analysis. To the best of
∗ Corresponding author at: Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-900, Brazil. E-mail addresses: [email protected] (J.P. Newport), [email protected] (K.D.T.-R. McLaughlin). 0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.07.006
our knowledge we are presenting the first analysis with sequences of initial data converging to Dirac masses, universality, and the breakdown thereof. Here and in what follows, by weak convergence we mean convergence in the sense of distributions with respect to a single variable, say, y. So a sequence fn (y) converges weakly to f (y) if φ(y)fn (y)dy → φ(y)f (y)dy for all smooth and compactly supported functions φ(y). It turns out that the asymptotic behaviour of solutions to this initial value problem is universal in the sense that the leading order asymptotic description, as ϵ → 0, depends only on the mass of f (y), so long as f (y) is real. If f (y) is a general complex function, universality fails, as the asymptotic description depends critically on the choice of f (y). The rest of this section is devoted to a brief look at the linear Schrödinger equation with Dirac mass initial data, and an overview of the results found. The rest of the paper is organized as follows. In Section 2 we will summarize the Scattering and Inverse Scattering Theory associated to the NLS equation. It is a nonlinear analogue of the Fourier theory used to solve the linear problem. In Section 3 we find a connection between long time asymptotics and sequences of initial data that converge to a Dirac mass, as the parameter ϵ → 0. We also study how the reflection coefficient scales with ϵ , and what effect a shift on the spectral variable will have. Theorems 1 and 2 are proved in this section. In Section 4 we prove universality of solutions with real initial data (Theorem 3), and characterize how universality fails when complex initial data is admitted. In
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
Section 5 we study the behaviour of the initial value problems (1)–(2) when t and ϵ tend to zero and prove Theorem 4. 1.1. Linear Schrödinger equation Intuition about the nonlinear problem can be obtained by considering the linear problem. We solve the initial value problem: iqt + qxx = 0
(3)
q(x, t = 0) = δ(x).
(4)
Using the Fourier transform
∫
1
F {f }(k) = fˆ (k) = √
∞
f (x)e−ikx dx, 2π −∞ we see that, formally, the solution to our equation is q(x, t ) =
1
∫
2π
∞
2
eikx−ik t dk.
(5)
However, the integrand in (5) is purely oscillatory, and we must decide upon a proper interpretation of the integral. To achieve this we introduce a regularization to our transformed data to ensure that the inverse transform of our initial data, and therefore the solution, converges. Let 1 2 qˆ ϵ0 = √ e−ϵ k . 2π Now, our solution (5) is ∫ 1 2 2 qϵ (x, t ) = eikx−ik t −ϵ k dk. 2π R Straightforward calculations yield: qϵ (x, t ) =
1
π
e− i 4
−x2
e 4(it +ϵ) . √ √ 2 π t − iϵ In the limit as ϵ → 0 we find the solution is
(6)
1 π (7) √ e− i 4 e . 2 πt Consider the asymptotics of our solution, as t → 0. Let h(x) be any smooth test function, and define ∫ ∫ 1 ix2 π I(x, t ) = q(x, t )h(x)dx = √ e−i 4 e 4t h(x)dx. 2 πt R R As t → 0, the dominant contribution from the integral is at x = 0. Stationary phase analysis yields: ix2 4t
q(x, t ) =
There is a transformation embodied by r (z ), called the reflection coefficient, which is the nonlinear analogue of the Fourier Transform. Under this transformation, the reflection coefficient linearizes the NLS equation. This function is uniquely associated to an initial condition, and is further discussed in Section 2. Theorem 1. Let r˜ (λ) be the reflection coefficient corresponding to the initial data f (y), where f (y) is a Schwartz class function with finite non-zero mass. Consider the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 1 x , ϕ(x, t = 0; ϵ) = f
ϵ
−∞
√
I(x; s) = h(0) + O( t ). Thus, our solution converges weakly to the Dirac delta function when t → 0. Remark 1. With ϵ = 0, the solution (7) may be thought of as a sequence of functions indexed by t. This sequence converges weakly to a Dirac mass as t → 0. A modification of this calculation shows that the order of limits does not matter. Indeed, if the vector (t , ϵ) → 0 in formula (6), then qϵ (x, t ) converges weakly to a Dirac mass. By contrast, in the nonlinear case, we find limitations on how (t , ϵ) → 0. 1.2. Results We are studying the behaviour of the NLS equation with sequences of initial data that converge to a Dirac mass for several reasons. Whereas the scattering and inverse scattering theory found herein are well established; existence, uniqueness and long time asymptotics are known for initial data in a weighted Sobolev space [1], the Dirac mass is a distribution that is outside this class of functions. In this paper we use the above machinery to investigate sequences of solutions corresponding to sequences of initial conditions (normally of the form 1ϵ f ϵx ) that converge to a Dirac mass.
2051
ϵ
x , and t > 0 bounded and let M be a positive constant, z0 = −ϵ 4t away from zero. Then for all x where |z0 | ≤ M, the solutions have the following asymptotic behaviour:
1
ix2
ϕ(x, t ; ϵ) = √ e 4t −iν ln(8t ) ϵ 2iν u(z0 ) + O (|ϵ log ϵ|) t
(8)
as ϵ → 0. The function ν is defined as:
ν(z0 ) =
2 −1 ln 1 − r˜ (z0 ) . 2π
(9)
The function u is defined in terms of its modulus and phase:
ν
−1 ln(1 − |˜r (z0 )|2 ) 4π ∫ z0 1 π arg u(z0 ) = ln(z0 − s)d ln(1 − |˜r (s)|2 ) + π −∞ 4 + arg Γ (iν) − arg r˜ (z0 ). |u(z0 )|2 =
2
=
(10)
(11)
Remark 2. A more refined version of this result appears in Section 5. With some additional constraints, we will be able to study the behaviour when t → 0, in a manner dependent on ϵ . Remark 3. It is a useful fact that Theorem 1 remains true with the same error term if the reflection coefficient is known to be entire, and exponentially bounded in the plane. This is the case, for example, when the function f (y) is compactly supported, even if the function has jump discontinuities. We will later construct exactly such an example for the purposes of breaking universality. A more general result is found in Theorem 2, which involves sequences of initial data with variable phase. We have listed these theorems separately to distinguish the effect of the scalings on the reflection coefficient. This can be seen more readily in the proofs of these theorems, contained in Section 3. Theorem 2. Let r˜ (λ) be the reflection coefficient corresponding to the initial data f (y), where f (y) is a Schwartz class function with finite non-zero mass. Consider initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 x 1 ϕ(x, t = 0; ϵ) = e−2iαx f .
ϵ
ϵ
−ϵ(x+4α t )
Let M be a positive real constant, λ0 = , t > 0 be bounded 4t away from zero, and α ∈ R be a constant. The for all values x where |λ0 | ≤ M the solutions have the following asymptotic behaviour:
ϕ(x, t ; ϵ) = e−2iαx−4iα
2t
[
1
ix2
√ e 4t −iν ln(8t ) ϵ 2iν u(λ0 ) + O (ϵ log ϵ)
]
t
with ν defined in (9), and u(λ0 ) defined in (10) and (11). Remark 4. As with Theorem 1 this result also holds true for any function f (y) whose reflection coefficient is entire and exponentially bounded in the plane.
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It is interesting to ask if the asymptotic behaviour of solutions in Theorem 1 depends on the choice of initial data, or is universal. We characterize universality in the following theorem. Theorem 3. For real initial data, f (y), the associated reflection coefficient satisfies r (0) = tanh
∫
f (y)dy . R
Therefore, the asymptotic description in (8) is unique in that it only depends on the mass of the function f (y). Remark 5. This universality is easily broken by leaving the class of real initial data. In Section 4 we provide a one parameter family of examples which break universality. The examples are piecewise constant, compactly supported functions, for which we may directly apply long time asymptotic results since the reflection coefficient is entire (see Remark 10). It is also interesting to see how the NLS equation regularizes singular data, such as of the form in (2). Of particular interest the asymptotic description of solutions when t → 0. Theorem 4. Let ϕ(x, t ) be the solution to the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0
(12)
1 x ϕ(x, t = 0; ϵ) = f , (13) ϵ ϵ and let t = T ϵ 2+β , with T ∈ R a constant. Then ϕ(x, t ) = ϕ(x, T ϵ 2+β ) converges weakly to a Dirac mass centered at x = 0 when ϵ (and therefore, t) tends to zero, for β > 0. Remark 6. When −2 < β < 0, oscillatory behaviour results in ϕ(x, t ; ϵ) not having a weak limit when ϵ → 0. When t = T ϵ 2 as above (i.e. β = 0), the solution
ϕ(x, T ϵ 2 ) =
1
ϵ
ψ
x ϵ
,T
converges weakly to a Dirac mass whose weight is R ψ(y, T )dy. Note that this is possibly different than the weight of R f (y)dy. A detailed analysis can be found in Section 5.
Thus, there is a critical value of β which separates the region where the solutions converge to the initial conditions from the region in which the solutions are rapidly oscillating, and therefore do not converge to the initial conditions. Both the non-uniqueness of the asymptotic behaviour of solutions, and the aspects of convergence to initial conditions are in stark contrast to the solutions for the linear equation. Remark 7. It is well known that the NLS equation can be used to model signal pulses in optical fibres. A somewhat surprising aspect of our results concerning universal and non-universal asymptotic behaviour of the solutions is the following. For essentially any initial pulse profile that is very narrow and very tall, the leading order asymptotic description of how it evolves down the fiber is universal, only depending on the total integral of the function, provided the phase is constant. However, this universal behaviour can be broken, in an analytically tractable way, by introducing a chirped phase. It would be very interesting to seek a chirped phase to optimize pulse characteristics down the fiber transmission line. Remark 8. The integrable defocussing nonlinear Schrödinger equation, within the class of Schrödinger equations considered in [2], appears to be a critical case for Dirac mass initial data. In this direction, the results in [2] pertain to power law nonlinearities below a critical value that depends on the dimension of the equation being considered. In dimension one that critical exponent yields the integrable NLS equation, and their results concerning Dirac mass initial data do not apply to the integrable case.
2. An introduction to scattering and inverse scattering theory In this section we will summarize the scattering and inverse scattering theory associated to the defocusing nonlinear Schrödinger equation: iϕt + ϕxx − 2|ϕ|2 ϕ = 0. We assume our initial data is Schwartz class. The scattering and inverse scattering theory we will require is a nonlinear version of the Fourier method for solving linear partial differential equations. One typically finds scattering data via the direct spectral transform. This scattering data has a very simple evolution in time. In order to reconstruct the solution at later times we must go from the evolved scattering data back to the potential, which is achieved using Riemann–Hilbert methods. This scattering and inverse scattering theory have been studied in great detail; in [3,4], for example. One can also find this theory in [5]. The Lax pair associated with the NLS equation is the pair of linear operators:
∂ 0 −ϕ +i ϕ 0 ∂x 2 ∂ −|ϕ| ϕx B = 2zI +i . −ϕx |ϕ|2 ∂x L = iσ3
(14)
(15)
If a 2 × 2 matrix function Ψ = Ψ (x, t , z ; ϕ) exists so that LΨ = z Ψ
(16)
∂ Ψ = BΨ , ∂t
(17)
the compatibility of partial derivatives implies that ϕ solves the NLS equation. Analysis of (16), called Scattering Theory, yields a map from initial data to a function, r (z ), called the reflection coefficient. For Schwartz class initial data, the map from the initial condition to the reflection coefficient is well defined, and the reflection coefficient is known to be Schwartz class. For details about the direct scattering theory and inverse scattering theory via Gelfand–Levitan–Marchenko equations, see [4], and for details about the inverse scattering theory via the Riemann–Hilbert approach, see [6] or [7]. It should be noted that this map is well defined under weaker conditions on the initial data (see [7]). The procedure typically used to reconstruct the potential from r (z ) as it evolves with the NLS equation is called the inverse scattering transform and can be solved with the Riemann–Hilbert approach. The Riemann–Hilbert Problem that needs to be solved is as follows. We wish to find a matrix M that has the following properties: M (z ) is analytic off the real axis M (z ) = I +
M1
M2
+ 2 + ··· z → ∞ z 2 1 − |r |2 −r (z )e−2izx−4iz t M+ (z ) = M− (z ) 2 r (z )e2izx+4iz t 1 z
(18) (19) z ∈ R. (20)
If we can find such an M, the solution to the NLS equation is embedded in M as shown in the following theorem. Theorem 5. Assume ϕ(x, t = 0) is Schwartz class. Then the solution ϕ(x, t ) to the NLS equation is:
ϕ = 2i(M1 (x, t ))12 , where (M1 )12 is the upper right entry in the matrix M1 found in the asymptotics of the solution of (18)–(20).
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
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This can be proved by first observing that Ψ = Me−izxσ3 is a solution to the first Lax equation, (16), and substituting the asymptotics for M into (16).
t = ϵ2τ
Remark 9. Theorem 5 is a well known fact and is true under weaker conditions on the potential. In [1] the authors require only that the initial data be in the weighted Sobolev space H 1,1 .
one can see that ψ solves the initial value problem
Retrieving the solution to the NLS equation from the solution of the RHP is called Inverse Scattering Theory. There are several ways to achieve this retrieval; the more classical approach involves the Gelfand–Levitan–Marchenko equations (see [4]), but one can also use more recently developed Riemann–Hilbert approach described here (and in more detail in [7]). We will use the Riemann–Hilbert characterization to to study asymptotics as t → 0 in Section 5. 3. Several approaches to finding evolution under the NLS equation for sequences approximating Dirac masses In this section we will study the evolution under the NLS equation of particular sequences of initial data that converge to a Dirac mass. This will be done in two separate ways. The first is to study the NLS equation directly, done in Section 3.1. We also study the NLS equation using the associated Lax pair and the reflection coefficient in Section 3.2. Each of these methods reveals a connection between long time asymptotics and sequences of initial conditions that converge to a Dirac mass. We also study a different sequence of initial data in Section 3.3 with a variable phase. In this section, we use previously known long time asymptotics (see [8], and the references therein) for the NLS equation to find our solution. For the initial value problem: iψτ + ψyy − 2|ψ|2 ψ = 0
ψ(y, τ = 0) = f (y), (21) with f (y) being a Schwartz class function, the behaviour of solutions is given by 1
ψ(y, τ ) = √ e τ
iy2 4τ
−iν(z0 ) ln(8τ )
u(z0 ) + O
log τ
τ
,
(22)
as τ → ∞. The function ν defined in (9), and z0 = The function u(z0 ) is defined in (10) and (11), and is a function of the reflection coefficient, r˜ (λ), associated with initial condition f (y). This asymptotic result is valid for more general classes of initial data. In fact, we will later use this result when the function f (y) is piece-wise constant and compactly supported (which results in the reflection coefficient being entire). For the purposes of this section Schwartz class functions are sufficient. The proper sense of the asymptotic result contained in (22) is as follows: Given M > 0, there is a T0 > 0 and a constant C > 0 so that forall y and τ satisfying τ > T0 and |z0 | ≤ M, the error term O logτ τ satisfies −y . 4τ
O log τ ≤ C log τ . τ τ
iϕt + ϕxx − 2|ϕ| ϕ = 0
(23)
1 x ϕ(x, t = 0; ϵ) = f ,
(24)
ϵ
iψτ + ψyy − 2|ψ|2 ψ = 0
ψ(y, τ = 0) = f (y). Note that we have scaled out the ϵ parameter. This allows us to use the long time asymptotics described at the beginning of this section. Recall, the long time asymptotics (22), are valid when τ → ∞ and when z0 is bounded. If we revert to the x and t variables, we see that z0 = − ϵ4tx and τ = ϵt2 . Thus, these asymptotics hold when ϵ → 0, for x in a compact set, and t bounded away from zero. We will see in Section 5 that we can in fact let t → 0. We use ψ(y, τ ) and it’s asymptotics to find the behaviour of ϕ(x, t )
ϕ(x, t ; ϵ) =
1
ϵ
ψ
1
= √ e
x
t
, ϵ ϵ2
ix2 4t
−iν ln(8t ) 2iν
ϵ
t
u(z0 ) + O
ϵ t
log
t
ϵ2
.
(25)
Fixing t > 0 bounded away from zero completes the proof of the Theorem 1. In this subsection we have established asymptotic descriptions of solutions for sequences of initial data converging to a Dirac mass. The behaviour obtained above is not uniformly valid in t. In Section 5 we study the behaviour of these solutions when t → 0. Note that in the (x, t ) variables, the point z0 scales with ϵ . In the next section we will see why this is the case. 3.2. Sequences approximating Dirac mass initial data, studied through the reflection coefficient In this subsection we will consider how the reflection coefficient scales with ϵ , when the initial data scales as in the previous subsection. Proposition 1. Suppose r˜ (λ) is the reflection coefficient associated with the initial condition f (y). Then the reflection coefficient associated with the initial conditions
ϕ(x, t = 0) =
1 x f ,
ϵ
ϵ
is r (z ) = r˜ (ϵ z ). Proof. Let t = 0, and consider the first Lax equation LΨ = z Ψ , which we rewrite for the readers convenience:
∂ 0 Ψ +i ϕ ∂x
−ϕ
0
Ψ = zΨ .
We will look for solutions to the following initial value problem:
ϵ
ϵϕ = ψ
iσ3
3.1. Asymptotic behaviour via long time asymptotics for a particular sequence of initial data
2
x = ϵy
Assume that the potential is of the form ϕ(x) = 1ϵ ψ ϵx , ϵt2 as in the previous section. If we define y = x/ϵ and τ = t /ϵ 2 then the first Lax equation becomes iσ3
∂ 0 Ψ +i ψ(y, τ ) ∂y
−ψ(y, τ ) 0
Ψ = ϵzΨ .
Recall, the second Lax equation:
where f (y) is a Schwartz class function. Here we present the proof of Theorem 1. Proof. Denote ϕ as the solution to the initial value problems (23)–(24). Using the variable changes
Ψt = 2z Ψx + i
−|ϕ|2 −ϕx
ϕx Ψ + 2iz 2 Ψ σ3 . |ϕ|2
Note. This equation has an extra term which is often omitted, since it may be removed by an integrating factor.
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Using the same change of variables one sees that
ψy Ψ + 2i(ϵ z )2 Ψ σ3 . |ψ|2
−|ψ|2 Ψτ = 2(ϵ z )Ψy + i −ψy
Summarizing, we started with a solution, Ψ , to the Lax pair, in the (x, t ) variables, with ϕ as the potential, and spectral parameter z. Through a change of variables, we showed that Ψ was also a solution to the Lax pair in the (y, τ ) variables, with ψ as the potential, and spectral parameter ϵ z. Thus, when the initial condition scales like 1ϵ f ϵx this amounts to re-scaling of the spectral variable so that the reflection coefficient is r (z ) = r˜ (ϵ z ). 3.3. Sequences of initial data with variable phase In the previous section we considered scaled initial data that led to a scaled reflection coefficient. We will now consider what effect a shift will have on the spectral variable z. We shall consider a modified sequence of initial conditions, however we still use the variable ϕ to denote the solutions. Proposition 2. Suppose r˜ (λ) is the reflection coefficient associated with the initial condition f (y). Then the reflection coefficient associated with the initial conditions
ϕ(x, t = 0) =
1 −2iα x e f
x
, ϵ with α ∈ R a constant, is r (z ) = r˜ (ϵ(z − α)). Proof. Suppose ϕ solves the NLS equation with x 1 ϕ(x, t = 0) = e−2iαx f . ϵ ϵ ϵ
The solution will have the following form:
ϕ(x, t ) =
1 −2iα x−4iα 2 t e ψ
x + 4α t
ϵ
ϵ
,
t
ϵ2
.
Indeed, let y = x/ϵ and τ = t /ϵ 2 . Inserting this form of ϕ into the NLS equation yields an equation for ψ : iψτ − 4iαϵψy + ψyy − 2|ψ|2 ψ = 0. Let
we see that Ψ˜ solves the following Lax equations: iσ3
∂ 0 Ψ˜ + i ψ ∂ yˆ
−ψ
0
Ψ˜ = ϵ(z − α)Ψ˜ ,
and
−|ψ|2 Ψ˜ τˆ = 2ϵ(z − α)Ψ˜ yˆ + i −ψ yˆ
ψyˆ Ψ˜ + 2i(ϵ(z − α))2 Ψ˜ σ3 , |ψ|2
with spectral parameter ϵ(z − α). Summarizing, we started with the a solution to the Lax pair, Ψ , with ϕ as the potential in the (x, t ) variables, and spectral parameter z. We derived a solution, Ψ˜ , to the Lax pair with ψ as the potential in the (ˆy, τˆ ) variables with spectral parameter λ = ϵ(z − α). Now, in the variables yˆ , τˆ and λ, we will denote the reflection coefficient corresponding to ψ by r˜ (λ). The explicit transformations presented above now show that the reflection coefficient corresponding to ϕ is given by r (z ) = r˜ (ϵ(z − α)). Using the long time asymptotics (22) completes the proof of Theorem 2. 4. Characterization of non-uniqueness In this section we use the asymptotic description of solutions (8), to study the universality of solutions corresponding to initial conditions of the form 1ϵ f ϵx . Recall that ϕ has the asymptotic description: 1
ix2
ϕ(x, t ) = √ e 4t −iν0 ln(8t /ϵ ) u(z0 ) + O (|ϵ log ϵ|). 2
t
The function u(z0 ) is a function of the reflection coefficient evaluated at the stationary phase point, and can be written u(r (z0 )). It should be noted that because our initial data is C ∞ , this implies that the reflection coefficient r (z ) is smooth, which in turn implies that u(z ) is smooth. Moreover, if we consider t bounded away from 0 and x fixed, then z0 converges uniformly to 0, we can expand in powers of ϵ , and rewrite u(r (z0 )) = u(r (0)) + ϵ u1 (r (z0 ), ϵ), with u1 being the correction term. Thus,
yˆ = y + 4αϵτ
1
ix2
τˆ = τ .
ϕ(x, t ) = √ e 4t −iν0 ln(8t /ϵ ) u(r (0)) + O (|ϵ log ϵ|).
We find that ψ(ˆy, τˆ ) solves the NLS initial value problem:
The reflection coefficient is dependent on the choice of f (y). It is possible that the asymptotic descriptionofsolutions depends on this choice, even though the sequence 1ϵ f ϵx converges to a Dirac mass. It is a straightforward calculation to show that for real initial data, f (y), the reflection coefficient evaluated at z = 0, satisfies
iψτˆ + ψyˆ yˆ − 2|ψ|2 ψ = 0
ψ(ˆy, τˆ = 0) = f (ˆy). Now that we know the form of the potential, we study the Lax equations to see how the spectral variable scales. Since ϕ solves the NLS equation there exists Ψ solving the equations iσ3
∂ 0 Ψ +i ϕ ∂x
−ϕ
0
r (0) = tanh
Ψ = zΨ
ϕx Ψ + 2iz 2 Ψ σ3 . |ϕ|2
yˆ = x/ϵ + 4ϵα τˆ
τˆ = t /ϵ 2 ϵ
x + 4α t
ϵ
2 2 2 Ψ˜ = Ψˆ e−(4iϵ z α−2iϵ α )σ3 τˆ ,
,
f (y)dy .
Thus, the first order asymptotic description is universal in the sense that it only depends on the mass of f (y). We want to know if this universality holds when the function f (y) is complex. Consider the family of functions:
2 Define Ψˆ = eiα xσ3 +2iα t σ3 Ψ . After making the transformations
1 2 ϕ = e−2iαx−4iα t ψ
∫ R
and
−|ϕ|2 Ψt = 2z Ψx + i −ϕx
2
t
t
ϵ
2
f (y) =
0 1 − ia 1 + ia 0
y<−
1
[ 2 1 y ∈ − ,0 2 ] 1 y ∈ 0, 2 1 y> . 2
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
This function is complex valued (for a ̸= 0), and has unit mass. An algebraic calculation shows that the reflection coefficient satisfies
√
√
and
τ = T ϵβ .
(28)
a2
We now characterize the behaviour of solutions as ϵ → 0, for different values of β . Note that for β < −2, we have that t → ∞ as ϵ → 0. Thus, we do not consider these values of β . For the rest of this section, we assume that T > 0 is a constant.
Remark 10. Since this initial data is compactly supported, the reflection coefficient is entire. In fact, it may be computed explicitly, for all z as follows:
Proposition 3. Let t = T ϵ 2+β with −2 < β ≤ −1. The solutions ϕ do not possess a weak limit when ϵ → 0.
r (0) =
r (z ) =
1+
a2
2055
sinh
1+
. √ (1 + ia) cosh 1 + a2 − ia
eiz (ew − 1) (−az (ew − 1) + w(ew + 1)) 1 + 2a2 ew + ia(ew − 1)2 − z 2 + iz w + e2w (1 − z 2 − iz w)
,
√
with w = 1 + a2 − z 2 . Moreover, for any compactly supported data the reflection coefficient (which is bounded by one on the real axis) may grow no worse than exponentially off the axis (this is observed trivially in the example given above). One consequence of these basic properties of the reflection coefficient (for compactly supported initial data) is that the long time asymptotic results used in this paper remain valid. In particular, for the present example, even though the data is discontinuous, the long time asymptotics hold true. We have shown that the reflection coefficient depends critically on the choice of a. This in turn means that the first order asymptotic description of solutions depends on a, even though for any choice of a, the sequence 1ϵ f ϵx converges to a Dirac mass (whose weight is independent of a). Thus, we have shown how to break universality.
In this section we study the behaviour of the asymptotic description of solutions when t → 0. Recall, in the linear problem, the solution converged to a Dirac mass when (t , ϵ) → 0. However, for the same to hold in the nonlinear problem, we require additional constraints. In Section 3.1 we studied the initial value problems (23)–(24). We rewrite it here for convenience: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 1 x . ϕ(x, t = 0; ϵ) = f
ϵ
We found (recall (25)) that: 1
ϕ(x, t ; ϵ) = √ e
ix2 4t
−iν ln(8t ) 2iν
ϵ
t
u(z0 ) + O
ϵ t
log
t
ϵ2
.
This really means that there exists constants T0 , M and C so that t ≥ T0 and |z0 | < M implies ϵ2
2 ϕ(x, t ; ϵ) − √1 e ix4t −iν ln(8t ) ϵ 2iν u(z0 ) ≤ C ϵ log t . 2 t ϵ t
ϕ(x, t ; ϵ) = √
T ϵ 2+β
e
ix2 4T ϵ 2+β
+iν ln(8T ϵ β )
+ O ϵ −1−β log ϵ β −xϵ −1−β 4T
Remark 11. Note that in the case t = T ϵ 2+β with −1 < β < 0, we cannot draw a conclusion concerning weak convergence using the asymptotic description (26). Although we have τ = T ϵ β going to infinity as desired, z0 = − 4x ϵ −1−β is only bounded when ϵ → 0 for |x| ≤ M ϵ 1+β , for some constant M. It should be is possible to study the long time asymptotics for x large, and therefore z0 going to infinity. However, this would require in depth Riemann–Hilbert analysis that is beyond the scope of this paper. Proposition 4. Let t = T ϵ 2 . The solutions ϕ(x, t ; ϵ) converges to a constant multiple of a Dirac mass when ϵ → 0.
ϕ(x, t ) =
1
ψ(y, τ ) 1 x ,T . = ψ ϵ ϵ ϵ
Since T is constant, this a scaled function of x only, which clearly converges (in the weak sense) to C (T )δ(x). Note that C (T ) = ψ(y, T )dy is a constant depending on T . In general, we do not R know the value of this constant. However, if T → 0, we know C (T ) → R f (y)dy. Recall that Theorem 4 states that for t = T ϵ 2+β with β > 0, the solutions ϕ converge to a Dirac mass as ϵ → 0. Since τ = T ϵ β is going to zero as ϵ → 0, one requires a uniform estimate on the rate at which the solution ψ(y, τ ) converges to its initial data. There are certainly several ways to obtain this estimate. We do so via the Riemann–Hilbert formulation of the inverse scattering theory associated to the NLS equation. Proof. Let M represent the solution to the RHP (18)–(20). M satisfies the following jump relation:
with V (t ) defined in (20). Now, let N denote the solution of the same RHP, evaluated at t = 0. By a standard procedure, define E = MN −1 .
u(z0 ) (26)
with z0 =
β
eiν0 log 8T ϵ oscillates rapidly as ϵ approaches 0, this sequence does not have a weak limit.
M+ = M− V (t ),
This is stronger than the result in Theorem 1 since we can study this behaviour as t → 0, as we will see below. Under the rescaling t = T ϵ 2+β with β < 0, we have that 1
β
converging to a Dirac mass) multiplied by eiν ln(8T ϵ ) . Note that ν(z0 ) → ν0 (independent of x) when ϵ → 0. Since the factor
Proof. Since τ = T is not going to infinity, we cannot use long time asymptotics. However,
5. Small time asymptotics
ϵ
Proof. Clearly, for −2 < β ≤ −1, z0 is bounded. Also, τ = T ϵ β goes to infinity as desired. Thus, the above asymptotics are valid. Inspection of (26) shows that as ϵ tends to 0, we clearly have a product of a standard Dirac sequence (i.e. a sequence of functions
(27)
(29)
There exists an expansion found below for E, obtained by studying small norm RHP’s (for a discussion of small norm RHP’s, see [7]), as outlined below. E solves a RHP with identity asymptotics, and has jumps on the real axis, of the form: E+ = E− J
2056
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
as λ → ∞, with
with J =
−1 E− E+
= N− V (t )V (0)−1 N−−1 . For convenience, let J˜ = V (t )V (0) We use the change of variables
−1
y=
(30)
ˆ E1 (y, T ϵ 2+β ) ≤ C ϵ β .
(31)
From (29), we have
.
ˆ = Eˆ Nˆ M =
x
Eˆ 1
I+
λ
ϵ
t = T ϵ 2+β
+O
1
I+
λ2
Nˆ 1
+O
λ
1
λ2
.
And so
λ = ϵz.
M =
I+
Denote
ˆ (y, T ϵ 2+β , λ) = M (x, t , z ) M
ϵz
+O
1
I+
z2
N1
ϵz
+O
1 z2
.
ψ = f (y) + ϵ β g (y).
Nˆ (y, λ) = N (x, z ). In the λ-plane we know N is bounded, and we show J˜ = V (T ϵ 2+β ) V (0)−1 is small in norm. It is straightforward to verify the following ˜ formula for J: −2iλy
|r (λ)| 2 β −r (λ)e2iλy+4iλ T ϵ 2 β × e−4iλ T ϵ − 1 .
−r (λ)e 1 − |r (λ)| 2 β −|r (λ)|2 e4iλ T ϵ
2
E1
ˆ 12 = ψ . By the estimates on E, and what we know about Then, 2iM N, we have
Eˆ (y, T ϵ 2+β , λ) = E (x, t , z )
J˜ = I +
2
And, so
ϕ=
x 1 x f + ϵβ g , ϵ
ϵ
x
ϵ
with g ϵ bounded on compact sets.
Thus we have characterized the behaviour of solutions for t = T ϵ 2+β as ϵ → 0. In the case where β > 0, we found the solutions converge to the initial conditions 1ϵ f ϵx .
Define 2 β s = e2iλy r (λ) e4iλ T ϵ − 1
Acknowledgements
found in the (2, 1) entry of the matrix above. We will find estimates for different norms of this term. An analogous proof shows the other terms in the matrix will have similar estimates. Notice that 2 β e4iλ T ϵ − 1 =
4iλ2 T ϵ β
∫
ex dx.
0
We would like to acknowledge several individuals who aided us in our research. Discussions with Marco Bertola at the Centre de Recherches Mathematiques, as well as Peter Miller and Gregory Forest, were very useful. The research of K.D.T.-R.M. was supported in part by the National Science Foundation under grants DMS0415496, DMS-0200749, and DMS-0800979. The research of J. Newport was also supported by the National Science Foundation under grant DMS-0415496.
Thus, for all λ ∈ R, References
2 β 4iλ T ϵ − 1 ≤ 4λ2 T ϵ β . e And so,
|s| ≤ 4T ϵ β λ2 r (λ) .
(32)
Straightforward calculations show that
||s||L1 (λ) = O (ϵ β ) ||s||L2 (λ) = O (ϵ β ) ||s||L∞ (λ) = O (ϵ β ). Note that all of these estimates were possible because we know r (λ) is Schwartz class. Using basic properties of small norm RHP’s, we find that a unique solution Eˆ exists and satisfies: Eˆ = I +
Eˆ 1 (y, T ϵ 2+β )
λ
+O
1
λ2
,
[1] Percy Deift, Xin Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weight Sobolev space, Comm. Pure Appl. Math. 56 (8) (2003) 1029–1077. [2] V. Banica, L. Vega, On the Dirac delta as initial condition for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré (C) Non Linéaire Anal. 25 (4) (2008) 697–711. [3] R. Beals, R.R. Coifmann, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984) 39–90. [4] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulating of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69. [5] Jason Newport, Solutions to the nonlinear Schrödinger equation with Dirac mass initial data, Ph.D. Thesis, University of North Carolina at Chapel Hill, 2008. [6] P.A. Deift, A.R. Its, X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, in: Important Developments in Soliton Theory, in: Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, pp. 181–204. [7] Percy Deift, Xin Zhou, Long-time behaviour of the non-focusing nonlinear Schrödinger equation—a case study, in: New Lectures in Mathematical Sciences, vol. 5, 1994. [8] P.A. Deift, X. Zhou, Long-time asymptotics for integrable systems. Higher order theory, Comm. Math. Phys. 165 (1994) 175–191.
Contents lists available at ScienceDirect
Physica D journal homepage: www.elsevier.com/locate/physd
Solutions to the nonlinear Schrödinger equation with sequences of initial data converging to a Dirac mass J.P. Newport a,c,∗ , K.D.T.-R. McLaughlin b,c a
Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, NC 27599, United States
b
Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721-0089, United States
c
Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-900, Brazil
article
info
Article history: Received 11 October 2008 Received in revised form 15 July 2010 Accepted 21 July 2010 Available online 25 July 2010 Communicated by J. Bronski
abstract We study the nonlinear Schrödinger equation with sequences of initial data that converge to a Dirac mass, and study the asymptotic behaviour of solutions. In doing so we find a connection to previously known long time asymptotics. We demonstrate a type of universality in the behaviour of solutions for real initial data, and we also show how this universality breaks down for examples of initial data that are not purely real. © 2010 Elsevier B.V. All rights reserved.
Keywords: Integrable systems Nonlinear partial differential equations Riemann–Hilbert analysis Scattering and inverse scattering theory
1. Introduction The goal of this paper is to analyze solutions of the defocussing nonlinear Schrödinger (NLS) equation with sequences of initial data that converge to a Dirac mass. One such sequence of initial data (among several that we study) leads us to the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0
(1)
1 x ϕ(x, t = 0) = f , ϵ ϵ
(2)
for a general class of functions f (y) which will be specified later. For convenience, we often assume that R f (y)dy = 1 so that 1ϵ f ϵx converges weakly to a unit Dirac mass as ϵ → 0. However, all the formulae which we derive are valid when f (y) has a finite non-zero mass. We use a combination of relatively classical results concerning the asymptotic behaviour of the integrable NLS equation together with some more involved Riemann–Hilbert analysis. To the best of
∗ Corresponding author at: Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-900, Brazil. E-mail addresses: [email protected] (J.P. Newport), [email protected] (K.D.T.-R. McLaughlin). 0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.07.006
our knowledge we are presenting the first analysis with sequences of initial data converging to Dirac masses, universality, and the breakdown thereof. Here and in what follows, by weak convergence we mean convergence in the sense of distributions with respect to a single variable, say, y. So a sequence fn (y) converges weakly to f (y) if φ(y)fn (y)dy → φ(y)f (y)dy for all smooth and compactly supported functions φ(y). It turns out that the asymptotic behaviour of solutions to this initial value problem is universal in the sense that the leading order asymptotic description, as ϵ → 0, depends only on the mass of f (y), so long as f (y) is real. If f (y) is a general complex function, universality fails, as the asymptotic description depends critically on the choice of f (y). The rest of this section is devoted to a brief look at the linear Schrödinger equation with Dirac mass initial data, and an overview of the results found. The rest of the paper is organized as follows. In Section 2 we will summarize the Scattering and Inverse Scattering Theory associated to the NLS equation. It is a nonlinear analogue of the Fourier theory used to solve the linear problem. In Section 3 we find a connection between long time asymptotics and sequences of initial data that converge to a Dirac mass, as the parameter ϵ → 0. We also study how the reflection coefficient scales with ϵ , and what effect a shift on the spectral variable will have. Theorems 1 and 2 are proved in this section. In Section 4 we prove universality of solutions with real initial data (Theorem 3), and characterize how universality fails when complex initial data is admitted. In
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
Section 5 we study the behaviour of the initial value problems (1)–(2) when t and ϵ tend to zero and prove Theorem 4. 1.1. Linear Schrödinger equation Intuition about the nonlinear problem can be obtained by considering the linear problem. We solve the initial value problem: iqt + qxx = 0
(3)
q(x, t = 0) = δ(x).
(4)
Using the Fourier transform
∫
1
F {f }(k) = fˆ (k) = √
∞
f (x)e−ikx dx, 2π −∞ we see that, formally, the solution to our equation is q(x, t ) =
1
∫
2π
∞
2
eikx−ik t dk.
(5)
However, the integrand in (5) is purely oscillatory, and we must decide upon a proper interpretation of the integral. To achieve this we introduce a regularization to our transformed data to ensure that the inverse transform of our initial data, and therefore the solution, converges. Let 1 2 qˆ ϵ0 = √ e−ϵ k . 2π Now, our solution (5) is ∫ 1 2 2 qϵ (x, t ) = eikx−ik t −ϵ k dk. 2π R Straightforward calculations yield: qϵ (x, t ) =
1
π
e− i 4
−x2
e 4(it +ϵ) . √ √ 2 π t − iϵ In the limit as ϵ → 0 we find the solution is
(6)
1 π (7) √ e− i 4 e . 2 πt Consider the asymptotics of our solution, as t → 0. Let h(x) be any smooth test function, and define ∫ ∫ 1 ix2 π I(x, t ) = q(x, t )h(x)dx = √ e−i 4 e 4t h(x)dx. 2 πt R R As t → 0, the dominant contribution from the integral is at x = 0. Stationary phase analysis yields: ix2 4t
q(x, t ) =
There is a transformation embodied by r (z ), called the reflection coefficient, which is the nonlinear analogue of the Fourier Transform. Under this transformation, the reflection coefficient linearizes the NLS equation. This function is uniquely associated to an initial condition, and is further discussed in Section 2. Theorem 1. Let r˜ (λ) be the reflection coefficient corresponding to the initial data f (y), where f (y) is a Schwartz class function with finite non-zero mass. Consider the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 1 x , ϕ(x, t = 0; ϵ) = f
ϵ
−∞
√
I(x; s) = h(0) + O( t ). Thus, our solution converges weakly to the Dirac delta function when t → 0. Remark 1. With ϵ = 0, the solution (7) may be thought of as a sequence of functions indexed by t. This sequence converges weakly to a Dirac mass as t → 0. A modification of this calculation shows that the order of limits does not matter. Indeed, if the vector (t , ϵ) → 0 in formula (6), then qϵ (x, t ) converges weakly to a Dirac mass. By contrast, in the nonlinear case, we find limitations on how (t , ϵ) → 0. 1.2. Results We are studying the behaviour of the NLS equation with sequences of initial data that converge to a Dirac mass for several reasons. Whereas the scattering and inverse scattering theory found herein are well established; existence, uniqueness and long time asymptotics are known for initial data in a weighted Sobolev space [1], the Dirac mass is a distribution that is outside this class of functions. In this paper we use the above machinery to investigate sequences of solutions corresponding to sequences of initial conditions (normally of the form 1ϵ f ϵx ) that converge to a Dirac mass.
2051
ϵ
x , and t > 0 bounded and let M be a positive constant, z0 = −ϵ 4t away from zero. Then for all x where |z0 | ≤ M, the solutions have the following asymptotic behaviour:
1
ix2
ϕ(x, t ; ϵ) = √ e 4t −iν ln(8t ) ϵ 2iν u(z0 ) + O (|ϵ log ϵ|) t
(8)
as ϵ → 0. The function ν is defined as:
ν(z0 ) =
2 −1 ln 1 − r˜ (z0 ) . 2π
(9)
The function u is defined in terms of its modulus and phase:
ν
−1 ln(1 − |˜r (z0 )|2 ) 4π ∫ z0 1 π arg u(z0 ) = ln(z0 − s)d ln(1 − |˜r (s)|2 ) + π −∞ 4 + arg Γ (iν) − arg r˜ (z0 ). |u(z0 )|2 =
2
=
(10)
(11)
Remark 2. A more refined version of this result appears in Section 5. With some additional constraints, we will be able to study the behaviour when t → 0, in a manner dependent on ϵ . Remark 3. It is a useful fact that Theorem 1 remains true with the same error term if the reflection coefficient is known to be entire, and exponentially bounded in the plane. This is the case, for example, when the function f (y) is compactly supported, even if the function has jump discontinuities. We will later construct exactly such an example for the purposes of breaking universality. A more general result is found in Theorem 2, which involves sequences of initial data with variable phase. We have listed these theorems separately to distinguish the effect of the scalings on the reflection coefficient. This can be seen more readily in the proofs of these theorems, contained in Section 3. Theorem 2. Let r˜ (λ) be the reflection coefficient corresponding to the initial data f (y), where f (y) is a Schwartz class function with finite non-zero mass. Consider initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 x 1 ϕ(x, t = 0; ϵ) = e−2iαx f .
ϵ
ϵ
−ϵ(x+4α t )
Let M be a positive real constant, λ0 = , t > 0 be bounded 4t away from zero, and α ∈ R be a constant. The for all values x where |λ0 | ≤ M the solutions have the following asymptotic behaviour:
ϕ(x, t ; ϵ) = e−2iαx−4iα
2t
[
1
ix2
√ e 4t −iν ln(8t ) ϵ 2iν u(λ0 ) + O (ϵ log ϵ)
]
t
with ν defined in (9), and u(λ0 ) defined in (10) and (11). Remark 4. As with Theorem 1 this result also holds true for any function f (y) whose reflection coefficient is entire and exponentially bounded in the plane.
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J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
It is interesting to ask if the asymptotic behaviour of solutions in Theorem 1 depends on the choice of initial data, or is universal. We characterize universality in the following theorem. Theorem 3. For real initial data, f (y), the associated reflection coefficient satisfies r (0) = tanh
∫
f (y)dy . R
Therefore, the asymptotic description in (8) is unique in that it only depends on the mass of the function f (y). Remark 5. This universality is easily broken by leaving the class of real initial data. In Section 4 we provide a one parameter family of examples which break universality. The examples are piecewise constant, compactly supported functions, for which we may directly apply long time asymptotic results since the reflection coefficient is entire (see Remark 10). It is also interesting to see how the NLS equation regularizes singular data, such as of the form in (2). Of particular interest the asymptotic description of solutions when t → 0. Theorem 4. Let ϕ(x, t ) be the solution to the initial value problem: iϕt + ϕxx − 2|ϕ|2 ϕ = 0
(12)
1 x ϕ(x, t = 0; ϵ) = f , (13) ϵ ϵ and let t = T ϵ 2+β , with T ∈ R a constant. Then ϕ(x, t ) = ϕ(x, T ϵ 2+β ) converges weakly to a Dirac mass centered at x = 0 when ϵ (and therefore, t) tends to zero, for β > 0. Remark 6. When −2 < β < 0, oscillatory behaviour results in ϕ(x, t ; ϵ) not having a weak limit when ϵ → 0. When t = T ϵ 2 as above (i.e. β = 0), the solution
ϕ(x, T ϵ 2 ) =
1
ϵ
ψ
x ϵ
,T
converges weakly to a Dirac mass whose weight is R ψ(y, T )dy. Note that this is possibly different than the weight of R f (y)dy. A detailed analysis can be found in Section 5.
Thus, there is a critical value of β which separates the region where the solutions converge to the initial conditions from the region in which the solutions are rapidly oscillating, and therefore do not converge to the initial conditions. Both the non-uniqueness of the asymptotic behaviour of solutions, and the aspects of convergence to initial conditions are in stark contrast to the solutions for the linear equation. Remark 7. It is well known that the NLS equation can be used to model signal pulses in optical fibres. A somewhat surprising aspect of our results concerning universal and non-universal asymptotic behaviour of the solutions is the following. For essentially any initial pulse profile that is very narrow and very tall, the leading order asymptotic description of how it evolves down the fiber is universal, only depending on the total integral of the function, provided the phase is constant. However, this universal behaviour can be broken, in an analytically tractable way, by introducing a chirped phase. It would be very interesting to seek a chirped phase to optimize pulse characteristics down the fiber transmission line. Remark 8. The integrable defocussing nonlinear Schrödinger equation, within the class of Schrödinger equations considered in [2], appears to be a critical case for Dirac mass initial data. In this direction, the results in [2] pertain to power law nonlinearities below a critical value that depends on the dimension of the equation being considered. In dimension one that critical exponent yields the integrable NLS equation, and their results concerning Dirac mass initial data do not apply to the integrable case.
2. An introduction to scattering and inverse scattering theory In this section we will summarize the scattering and inverse scattering theory associated to the defocusing nonlinear Schrödinger equation: iϕt + ϕxx − 2|ϕ|2 ϕ = 0. We assume our initial data is Schwartz class. The scattering and inverse scattering theory we will require is a nonlinear version of the Fourier method for solving linear partial differential equations. One typically finds scattering data via the direct spectral transform. This scattering data has a very simple evolution in time. In order to reconstruct the solution at later times we must go from the evolved scattering data back to the potential, which is achieved using Riemann–Hilbert methods. This scattering and inverse scattering theory have been studied in great detail; in [3,4], for example. One can also find this theory in [5]. The Lax pair associated with the NLS equation is the pair of linear operators:
∂ 0 −ϕ +i ϕ 0 ∂x 2 ∂ −|ϕ| ϕx B = 2zI +i . −ϕx |ϕ|2 ∂x L = iσ3
(14)
(15)
If a 2 × 2 matrix function Ψ = Ψ (x, t , z ; ϕ) exists so that LΨ = z Ψ
(16)
∂ Ψ = BΨ , ∂t
(17)
the compatibility of partial derivatives implies that ϕ solves the NLS equation. Analysis of (16), called Scattering Theory, yields a map from initial data to a function, r (z ), called the reflection coefficient. For Schwartz class initial data, the map from the initial condition to the reflection coefficient is well defined, and the reflection coefficient is known to be Schwartz class. For details about the direct scattering theory and inverse scattering theory via Gelfand–Levitan–Marchenko equations, see [4], and for details about the inverse scattering theory via the Riemann–Hilbert approach, see [6] or [7]. It should be noted that this map is well defined under weaker conditions on the initial data (see [7]). The procedure typically used to reconstruct the potential from r (z ) as it evolves with the NLS equation is called the inverse scattering transform and can be solved with the Riemann–Hilbert approach. The Riemann–Hilbert Problem that needs to be solved is as follows. We wish to find a matrix M that has the following properties: M (z ) is analytic off the real axis M (z ) = I +
M1
M2
+ 2 + ··· z → ∞ z 2 1 − |r |2 −r (z )e−2izx−4iz t M+ (z ) = M− (z ) 2 r (z )e2izx+4iz t 1 z
(18) (19) z ∈ R. (20)
If we can find such an M, the solution to the NLS equation is embedded in M as shown in the following theorem. Theorem 5. Assume ϕ(x, t = 0) is Schwartz class. Then the solution ϕ(x, t ) to the NLS equation is:
ϕ = 2i(M1 (x, t ))12 , where (M1 )12 is the upper right entry in the matrix M1 found in the asymptotics of the solution of (18)–(20).
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
2053
This can be proved by first observing that Ψ = Me−izxσ3 is a solution to the first Lax equation, (16), and substituting the asymptotics for M into (16).
t = ϵ2τ
Remark 9. Theorem 5 is a well known fact and is true under weaker conditions on the potential. In [1] the authors require only that the initial data be in the weighted Sobolev space H 1,1 .
one can see that ψ solves the initial value problem
Retrieving the solution to the NLS equation from the solution of the RHP is called Inverse Scattering Theory. There are several ways to achieve this retrieval; the more classical approach involves the Gelfand–Levitan–Marchenko equations (see [4]), but one can also use more recently developed Riemann–Hilbert approach described here (and in more detail in [7]). We will use the Riemann–Hilbert characterization to to study asymptotics as t → 0 in Section 5. 3. Several approaches to finding evolution under the NLS equation for sequences approximating Dirac masses In this section we will study the evolution under the NLS equation of particular sequences of initial data that converge to a Dirac mass. This will be done in two separate ways. The first is to study the NLS equation directly, done in Section 3.1. We also study the NLS equation using the associated Lax pair and the reflection coefficient in Section 3.2. Each of these methods reveals a connection between long time asymptotics and sequences of initial conditions that converge to a Dirac mass. We also study a different sequence of initial data in Section 3.3 with a variable phase. In this section, we use previously known long time asymptotics (see [8], and the references therein) for the NLS equation to find our solution. For the initial value problem: iψτ + ψyy − 2|ψ|2 ψ = 0
ψ(y, τ = 0) = f (y), (21) with f (y) being a Schwartz class function, the behaviour of solutions is given by 1
ψ(y, τ ) = √ e τ
iy2 4τ
−iν(z0 ) ln(8τ )
u(z0 ) + O
log τ
τ
,
(22)
as τ → ∞. The function ν defined in (9), and z0 = The function u(z0 ) is defined in (10) and (11), and is a function of the reflection coefficient, r˜ (λ), associated with initial condition f (y). This asymptotic result is valid for more general classes of initial data. In fact, we will later use this result when the function f (y) is piece-wise constant and compactly supported (which results in the reflection coefficient being entire). For the purposes of this section Schwartz class functions are sufficient. The proper sense of the asymptotic result contained in (22) is as follows: Given M > 0, there is a T0 > 0 and a constant C > 0 so that forall y and τ satisfying τ > T0 and |z0 | ≤ M, the error term O logτ τ satisfies −y . 4τ
O log τ ≤ C log τ . τ τ
iϕt + ϕxx − 2|ϕ| ϕ = 0
(23)
1 x ϕ(x, t = 0; ϵ) = f ,
(24)
ϵ
iψτ + ψyy − 2|ψ|2 ψ = 0
ψ(y, τ = 0) = f (y). Note that we have scaled out the ϵ parameter. This allows us to use the long time asymptotics described at the beginning of this section. Recall, the long time asymptotics (22), are valid when τ → ∞ and when z0 is bounded. If we revert to the x and t variables, we see that z0 = − ϵ4tx and τ = ϵt2 . Thus, these asymptotics hold when ϵ → 0, for x in a compact set, and t bounded away from zero. We will see in Section 5 that we can in fact let t → 0. We use ψ(y, τ ) and it’s asymptotics to find the behaviour of ϕ(x, t )
ϕ(x, t ; ϵ) =
1
ϵ
ψ
1
= √ e
x
t
, ϵ ϵ2
ix2 4t
−iν ln(8t ) 2iν
ϵ
t
u(z0 ) + O
ϵ t
log
t
ϵ2
.
(25)
Fixing t > 0 bounded away from zero completes the proof of the Theorem 1. In this subsection we have established asymptotic descriptions of solutions for sequences of initial data converging to a Dirac mass. The behaviour obtained above is not uniformly valid in t. In Section 5 we study the behaviour of these solutions when t → 0. Note that in the (x, t ) variables, the point z0 scales with ϵ . In the next section we will see why this is the case. 3.2. Sequences approximating Dirac mass initial data, studied through the reflection coefficient In this subsection we will consider how the reflection coefficient scales with ϵ , when the initial data scales as in the previous subsection. Proposition 1. Suppose r˜ (λ) is the reflection coefficient associated with the initial condition f (y). Then the reflection coefficient associated with the initial conditions
ϕ(x, t = 0) =
1 x f ,
ϵ
ϵ
is r (z ) = r˜ (ϵ z ). Proof. Let t = 0, and consider the first Lax equation LΨ = z Ψ , which we rewrite for the readers convenience:
∂ 0 Ψ +i ϕ ∂x
−ϕ
0
Ψ = zΨ .
We will look for solutions to the following initial value problem:
ϵ
ϵϕ = ψ
iσ3
3.1. Asymptotic behaviour via long time asymptotics for a particular sequence of initial data
2
x = ϵy
Assume that the potential is of the form ϕ(x) = 1ϵ ψ ϵx , ϵt2 as in the previous section. If we define y = x/ϵ and τ = t /ϵ 2 then the first Lax equation becomes iσ3
∂ 0 Ψ +i ψ(y, τ ) ∂y
−ψ(y, τ ) 0
Ψ = ϵzΨ .
Recall, the second Lax equation:
where f (y) is a Schwartz class function. Here we present the proof of Theorem 1. Proof. Denote ϕ as the solution to the initial value problems (23)–(24). Using the variable changes
Ψt = 2z Ψx + i
−|ϕ|2 −ϕx
ϕx Ψ + 2iz 2 Ψ σ3 . |ϕ|2
Note. This equation has an extra term which is often omitted, since it may be removed by an integrating factor.
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Using the same change of variables one sees that
ψy Ψ + 2i(ϵ z )2 Ψ σ3 . |ψ|2
−|ψ|2 Ψτ = 2(ϵ z )Ψy + i −ψy
Summarizing, we started with a solution, Ψ , to the Lax pair, in the (x, t ) variables, with ϕ as the potential, and spectral parameter z. Through a change of variables, we showed that Ψ was also a solution to the Lax pair in the (y, τ ) variables, with ψ as the potential, and spectral parameter ϵ z. Thus, when the initial condition scales like 1ϵ f ϵx this amounts to re-scaling of the spectral variable so that the reflection coefficient is r (z ) = r˜ (ϵ z ). 3.3. Sequences of initial data with variable phase In the previous section we considered scaled initial data that led to a scaled reflection coefficient. We will now consider what effect a shift will have on the spectral variable z. We shall consider a modified sequence of initial conditions, however we still use the variable ϕ to denote the solutions. Proposition 2. Suppose r˜ (λ) is the reflection coefficient associated with the initial condition f (y). Then the reflection coefficient associated with the initial conditions
ϕ(x, t = 0) =
1 −2iα x e f
x
, ϵ with α ∈ R a constant, is r (z ) = r˜ (ϵ(z − α)). Proof. Suppose ϕ solves the NLS equation with x 1 ϕ(x, t = 0) = e−2iαx f . ϵ ϵ ϵ
The solution will have the following form:
ϕ(x, t ) =
1 −2iα x−4iα 2 t e ψ
x + 4α t
ϵ
ϵ
,
t
ϵ2
.
Indeed, let y = x/ϵ and τ = t /ϵ 2 . Inserting this form of ϕ into the NLS equation yields an equation for ψ : iψτ − 4iαϵψy + ψyy − 2|ψ|2 ψ = 0. Let
we see that Ψ˜ solves the following Lax equations: iσ3
∂ 0 Ψ˜ + i ψ ∂ yˆ
−ψ
0
Ψ˜ = ϵ(z − α)Ψ˜ ,
and
−|ψ|2 Ψ˜ τˆ = 2ϵ(z − α)Ψ˜ yˆ + i −ψ yˆ
ψyˆ Ψ˜ + 2i(ϵ(z − α))2 Ψ˜ σ3 , |ψ|2
with spectral parameter ϵ(z − α). Summarizing, we started with the a solution to the Lax pair, Ψ , with ϕ as the potential in the (x, t ) variables, and spectral parameter z. We derived a solution, Ψ˜ , to the Lax pair with ψ as the potential in the (ˆy, τˆ ) variables with spectral parameter λ = ϵ(z − α). Now, in the variables yˆ , τˆ and λ, we will denote the reflection coefficient corresponding to ψ by r˜ (λ). The explicit transformations presented above now show that the reflection coefficient corresponding to ϕ is given by r (z ) = r˜ (ϵ(z − α)). Using the long time asymptotics (22) completes the proof of Theorem 2. 4. Characterization of non-uniqueness In this section we use the asymptotic description of solutions (8), to study the universality of solutions corresponding to initial conditions of the form 1ϵ f ϵx . Recall that ϕ has the asymptotic description: 1
ix2
ϕ(x, t ) = √ e 4t −iν0 ln(8t /ϵ ) u(z0 ) + O (|ϵ log ϵ|). 2
t
The function u(z0 ) is a function of the reflection coefficient evaluated at the stationary phase point, and can be written u(r (z0 )). It should be noted that because our initial data is C ∞ , this implies that the reflection coefficient r (z ) is smooth, which in turn implies that u(z ) is smooth. Moreover, if we consider t bounded away from 0 and x fixed, then z0 converges uniformly to 0, we can expand in powers of ϵ , and rewrite u(r (z0 )) = u(r (0)) + ϵ u1 (r (z0 ), ϵ), with u1 being the correction term. Thus,
yˆ = y + 4αϵτ
1
ix2
τˆ = τ .
ϕ(x, t ) = √ e 4t −iν0 ln(8t /ϵ ) u(r (0)) + O (|ϵ log ϵ|).
We find that ψ(ˆy, τˆ ) solves the NLS initial value problem:
The reflection coefficient is dependent on the choice of f (y). It is possible that the asymptotic descriptionofsolutions depends on this choice, even though the sequence 1ϵ f ϵx converges to a Dirac mass. It is a straightforward calculation to show that for real initial data, f (y), the reflection coefficient evaluated at z = 0, satisfies
iψτˆ + ψyˆ yˆ − 2|ψ|2 ψ = 0
ψ(ˆy, τˆ = 0) = f (ˆy). Now that we know the form of the potential, we study the Lax equations to see how the spectral variable scales. Since ϕ solves the NLS equation there exists Ψ solving the equations iσ3
∂ 0 Ψ +i ϕ ∂x
−ϕ
0
r (0) = tanh
Ψ = zΨ
ϕx Ψ + 2iz 2 Ψ σ3 . |ϕ|2
yˆ = x/ϵ + 4ϵα τˆ
τˆ = t /ϵ 2 ϵ
x + 4α t
ϵ
2 2 2 Ψ˜ = Ψˆ e−(4iϵ z α−2iϵ α )σ3 τˆ ,
,
f (y)dy .
Thus, the first order asymptotic description is universal in the sense that it only depends on the mass of f (y). We want to know if this universality holds when the function f (y) is complex. Consider the family of functions:
2 Define Ψˆ = eiα xσ3 +2iα t σ3 Ψ . After making the transformations
1 2 ϕ = e−2iαx−4iα t ψ
∫ R
and
−|ϕ|2 Ψt = 2z Ψx + i −ϕx
2
t
t
ϵ
2
f (y) =
0 1 − ia 1 + ia 0
y<−
1
[ 2 1 y ∈ − ,0 2 ] 1 y ∈ 0, 2 1 y> . 2
J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
This function is complex valued (for a ̸= 0), and has unit mass. An algebraic calculation shows that the reflection coefficient satisfies
√
√
and
τ = T ϵβ .
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a2
We now characterize the behaviour of solutions as ϵ → 0, for different values of β . Note that for β < −2, we have that t → ∞ as ϵ → 0. Thus, we do not consider these values of β . For the rest of this section, we assume that T > 0 is a constant.
Remark 10. Since this initial data is compactly supported, the reflection coefficient is entire. In fact, it may be computed explicitly, for all z as follows:
Proposition 3. Let t = T ϵ 2+β with −2 < β ≤ −1. The solutions ϕ do not possess a weak limit when ϵ → 0.
r (0) =
r (z ) =
1+
a2
2055
sinh
1+
. √ (1 + ia) cosh 1 + a2 − ia
eiz (ew − 1) (−az (ew − 1) + w(ew + 1)) 1 + 2a2 ew + ia(ew − 1)2 − z 2 + iz w + e2w (1 − z 2 − iz w)
,
√
with w = 1 + a2 − z 2 . Moreover, for any compactly supported data the reflection coefficient (which is bounded by one on the real axis) may grow no worse than exponentially off the axis (this is observed trivially in the example given above). One consequence of these basic properties of the reflection coefficient (for compactly supported initial data) is that the long time asymptotic results used in this paper remain valid. In particular, for the present example, even though the data is discontinuous, the long time asymptotics hold true. We have shown that the reflection coefficient depends critically on the choice of a. This in turn means that the first order asymptotic description of solutions depends on a, even though for any choice of a, the sequence 1ϵ f ϵx converges to a Dirac mass (whose weight is independent of a). Thus, we have shown how to break universality.
In this section we study the behaviour of the asymptotic description of solutions when t → 0. Recall, in the linear problem, the solution converged to a Dirac mass when (t , ϵ) → 0. However, for the same to hold in the nonlinear problem, we require additional constraints. In Section 3.1 we studied the initial value problems (23)–(24). We rewrite it here for convenience: iϕt + ϕxx − 2|ϕ|2 ϕ = 0 1 x . ϕ(x, t = 0; ϵ) = f
ϵ
We found (recall (25)) that: 1
ϕ(x, t ; ϵ) = √ e
ix2 4t
−iν ln(8t ) 2iν
ϵ
t
u(z0 ) + O
ϵ t
log
t
ϵ2
.
This really means that there exists constants T0 , M and C so that t ≥ T0 and |z0 | < M implies ϵ2
2 ϕ(x, t ; ϵ) − √1 e ix4t −iν ln(8t ) ϵ 2iν u(z0 ) ≤ C ϵ log t . 2 t ϵ t
ϕ(x, t ; ϵ) = √
T ϵ 2+β
e
ix2 4T ϵ 2+β
+iν ln(8T ϵ β )
+ O ϵ −1−β log ϵ β −xϵ −1−β 4T
Remark 11. Note that in the case t = T ϵ 2+β with −1 < β < 0, we cannot draw a conclusion concerning weak convergence using the asymptotic description (26). Although we have τ = T ϵ β going to infinity as desired, z0 = − 4x ϵ −1−β is only bounded when ϵ → 0 for |x| ≤ M ϵ 1+β , for some constant M. It should be is possible to study the long time asymptotics for x large, and therefore z0 going to infinity. However, this would require in depth Riemann–Hilbert analysis that is beyond the scope of this paper. Proposition 4. Let t = T ϵ 2 . The solutions ϕ(x, t ; ϵ) converges to a constant multiple of a Dirac mass when ϵ → 0.
ϕ(x, t ) =
1
ψ(y, τ ) 1 x ,T . = ψ ϵ ϵ ϵ
Since T is constant, this a scaled function of x only, which clearly converges (in the weak sense) to C (T )δ(x). Note that C (T ) = ψ(y, T )dy is a constant depending on T . In general, we do not R know the value of this constant. However, if T → 0, we know C (T ) → R f (y)dy. Recall that Theorem 4 states that for t = T ϵ 2+β with β > 0, the solutions ϕ converge to a Dirac mass as ϵ → 0. Since τ = T ϵ β is going to zero as ϵ → 0, one requires a uniform estimate on the rate at which the solution ψ(y, τ ) converges to its initial data. There are certainly several ways to obtain this estimate. We do so via the Riemann–Hilbert formulation of the inverse scattering theory associated to the NLS equation. Proof. Let M represent the solution to the RHP (18)–(20). M satisfies the following jump relation:
with V (t ) defined in (20). Now, let N denote the solution of the same RHP, evaluated at t = 0. By a standard procedure, define E = MN −1 .
u(z0 ) (26)
with z0 =
β
eiν0 log 8T ϵ oscillates rapidly as ϵ approaches 0, this sequence does not have a weak limit.
M+ = M− V (t ),
This is stronger than the result in Theorem 1 since we can study this behaviour as t → 0, as we will see below. Under the rescaling t = T ϵ 2+β with β < 0, we have that 1
β
converging to a Dirac mass) multiplied by eiν ln(8T ϵ ) . Note that ν(z0 ) → ν0 (independent of x) when ϵ → 0. Since the factor
Proof. Since τ = T is not going to infinity, we cannot use long time asymptotics. However,
5. Small time asymptotics
ϵ
Proof. Clearly, for −2 < β ≤ −1, z0 is bounded. Also, τ = T ϵ β goes to infinity as desired. Thus, the above asymptotics are valid. Inspection of (26) shows that as ϵ tends to 0, we clearly have a product of a standard Dirac sequence (i.e. a sequence of functions
(27)
(29)
There exists an expansion found below for E, obtained by studying small norm RHP’s (for a discussion of small norm RHP’s, see [7]), as outlined below. E solves a RHP with identity asymptotics, and has jumps on the real axis, of the form: E+ = E− J
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J.P. Newport, K.D.T.-R. McLaughlin / Physica D 239 (2010) 2050–2056
as λ → ∞, with
with J =
−1 E− E+
= N− V (t )V (0)−1 N−−1 . For convenience, let J˜ = V (t )V (0) We use the change of variables
−1
y=
(30)
ˆ E1 (y, T ϵ 2+β ) ≤ C ϵ β .
(31)
From (29), we have
.
ˆ = Eˆ Nˆ M =
x
Eˆ 1
I+
λ
ϵ
t = T ϵ 2+β
+O
1
I+
λ2
Nˆ 1
+O
λ
1
λ2
.
And so
λ = ϵz.
M =
I+
Denote
ˆ (y, T ϵ 2+β , λ) = M (x, t , z ) M
ϵz
+O
1
I+
z2
N1
ϵz
+O
1 z2
.
ψ = f (y) + ϵ β g (y).
Nˆ (y, λ) = N (x, z ). In the λ-plane we know N is bounded, and we show J˜ = V (T ϵ 2+β ) V (0)−1 is small in norm. It is straightforward to verify the following ˜ formula for J: −2iλy
|r (λ)| 2 β −r (λ)e2iλy+4iλ T ϵ 2 β × e−4iλ T ϵ − 1 .
−r (λ)e 1 − |r (λ)| 2 β −|r (λ)|2 e4iλ T ϵ
2
E1
ˆ 12 = ψ . By the estimates on E, and what we know about Then, 2iM N, we have
Eˆ (y, T ϵ 2+β , λ) = E (x, t , z )
J˜ = I +
2
And, so
ϕ=
x 1 x f + ϵβ g , ϵ
ϵ
x
ϵ
with g ϵ bounded on compact sets.
Thus we have characterized the behaviour of solutions for t = T ϵ 2+β as ϵ → 0. In the case where β > 0, we found the solutions converge to the initial conditions 1ϵ f ϵx .
Define 2 β s = e2iλy r (λ) e4iλ T ϵ − 1
Acknowledgements
found in the (2, 1) entry of the matrix above. We will find estimates for different norms of this term. An analogous proof shows the other terms in the matrix will have similar estimates. Notice that 2 β e4iλ T ϵ − 1 =
4iλ2 T ϵ β
∫
ex dx.
0
We would like to acknowledge several individuals who aided us in our research. Discussions with Marco Bertola at the Centre de Recherches Mathematiques, as well as Peter Miller and Gregory Forest, were very useful. The research of K.D.T.-R.M. was supported in part by the National Science Foundation under grants DMS0415496, DMS-0200749, and DMS-0800979. The research of J. Newport was also supported by the National Science Foundation under grant DMS-0415496.
Thus, for all λ ∈ R, References
2 β 4iλ T ϵ − 1 ≤ 4λ2 T ϵ β . e And so,
|s| ≤ 4T ϵ β λ2 r (λ) .
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Straightforward calculations show that
||s||L1 (λ) = O (ϵ β ) ||s||L2 (λ) = O (ϵ β ) ||s||L∞ (λ) = O (ϵ β ). Note that all of these estimates were possible because we know r (λ) is Schwartz class. Using basic properties of small norm RHP’s, we find that a unique solution Eˆ exists and satisfies: Eˆ = I +
Eˆ 1 (y, T ϵ 2+β )
λ
+O
1
λ2
,
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