AN INTRODUCTION TO THE NON-LINEAR FOURIER TRANSFORM

TERENCE TAO

Abstract. We review some standard material on the (Dirac) nonlinear Fourier transform, which is the scattering theory analogue of the more familiar linear Fourier transform. Particular emphasis is given on the close analogy between the two transforms.

1. The nonlinear Fourier transform - overview

Let F (x) be a complex Schwartz function on IR. We shall define the Fourier transform F̂(k) by

F̂(k) = − ∫_{−∞}^{∞} exp(2ikx)F (x) dx;

this normalization may appear strange, but will be convenient later on. Thus, for instance, if F is a finite combination of Dirac masses

F = ∑_{n=1}^{N} c_{n}δ_{x_{n}}

then F̂ is given (formally, at least) as an exponential sum:

F̂(k) = − ∑_{n=1}^{N} c_{n} exp(2ikx_{n}).  (1)

We observe the following well-known properties of this transform:

• (Linearity) For any F_{1} and F_{2}, we have

{ \atop F_{1} + F_{2}}(k) = F̂_{1}(k) + F̂_{2}(k).  (2)

• (Homogeneity) For any F and c ∈ IR, we have

ĉF(k) = cF̂(k).

• (Conjugation symmetry) If F̃(x) := \overline{F} (x), then

ˆF̃(k) = \overline{F̂(−k)}.

In particular, if F is real, then F̂(−k) = \overline{F̂(k)}.

• (Translation symmetry) If F̃(x) := F (x − x_{0}) for some x_{0} ∈ IR, then

ˆF̃(k) = F̂(k) exp(2ikx_{0}).

• (Modulation symmetry) If F̃(x) := F (x) exp(−2ik_{0}x) for some x_{0} ∈ IR, then

ˆF̃(k) = F̂(k − k_{0}).

• (Scale invariance) If F̃(x) := λ^{−1}F (x/λ) for some λ ∈ IR\{0}, then

ˆF̃(k) = F̂(λk).

• (Schwarz mapping) If F is Schwartz, then so is F̂, and conversely.

• (Plancherel) We have

∫_{−∞}^{∞} |F̂(k)|^{2} dk = π ∫_{−∞}^{∞} |F (x)|^{2} dx.  (3)

In particular the map F → F̂ is bounded on L^{2}, hence continuous on L^{2}. (In fact it is a homeomorphism).

• (Riemann-Lebesgue) We have

∥F̂∥_{∞} ≤ ∥F ∥_{1},

and as a consequence, the map F → F̂ is bounded from L^{1} to L^{∞}, and hence continuous. In fact it is continuous from L^{1} to C_{0}.

• (Hausdorff-Young) For any 1 ≤ p ≤ 2 we have

∥F̂∥_{p′} ≤ C_{p}∥F ∥_{p}

and thus the map F → F̂ is bounded and continuous from L^{p} to L^{p′}.

• (Paley-Wiener) If F is in L^{1} and supported on an interval [x_{1}, x_{2}], then F̂(k) is entire, and obeys the bounds

|F̂(k + it)| ≤ C_{F} exp(−2x_{1}t)

and

|F̂(k − it)| ≤ C_{F} exp(2x_{2}t)

for k ∈ R, t ≥ 0. In particular, if F is supported on [0, ∞) (resp. (−∞, 0]), then F̂(k) extends holomorphically to the upper (resp. lower) half-plane, with F̂(k + it) → 0 as t → +∞ (resp. t → −∞).

• (Menshov-Paley-Zygmund + Carleson) For any 1 ≤ p ≤ 2 we have

∥ sup_{N∈IR} { \atop F χ_{(−∞,N]}}(k)∥_{L_{k}^{p′}} ≤ C_{p}∥F ∥_{L_{x}^{p}}.

As a consequence, we have the pointwise convergence property

{ \atop F χ_{[−N,N]}}(k) → F̂(k)

for a.e. k ∈ IR, for any F ∈ L^{p}.

• (Explicit solution of Schrödinger) If F (t, x) solves the Schrödinger equation

iF_{t} + F_{xx} = 0; F (0, x) = F_{0}(x)

then we have

F̂ (t)(k) = exp(−4ik^{2}t)F̂_{0}(k).  (4)

• (Explicit solution of Airy) If F (t, x) solves the Airy equation

F_{t} = F_{xxx}; F (0, x) = F_{0}(x)

then we have

F̂ (t)(k) = exp(8ik^{3}t)F̂_{0}(k)  (5)

One can recover F from its Fourier transform by the inversion formula

πF (x) = ∫_{−∞}^{∞} F̂(k) exp(−2ikx) dk.  (6)

In particular, from (4) and (5) we see that we can use the Fourier transform and its inverse to explicitly solve the Schrödinger and Airy equations.

Now we define the Riesz projections H_{+}, H_{−} by the formula

H_{+}F̂(k) := { \atop F χ_{[0,∞)}}(k)

and

H_{−}F̂(k) := { \atop F χ_{(−∞,0]}}(k);

these are well-defined on, say, L^{2}, because of the invertibility of the Fourier transform on L^{2}. These operators are of course closely related to the Hilbert transform.

We remark the following properties of H_{+} and H_{−}, for Schwartz functions f(k):

• (Factorization) We have

f = H_{−}f + H_{+}f.

• (Analyticity) H_{+}f extends analytically to the upper half plane, and H_{−}f extends analytically to the lower-half plane, with both functions tending to 0 at infinity.

• (Asymptotics) As |k| → ∞ on the upper half-plane, we have

H_{+}(F̂)(k) = F (0)/2ik + O(1/|k|^{2})

and similarly as |k| → ∞ on the lower half-plane, we have

H_{−}(F̂)(k) = −F (0)/2ik + O(1/|k|^{2})

• (Boundedness) For any 1 < p < ∞, we have

∥H_{±}f∥_{p} ≤ C_{p}∥f∥_{p}.

Actually, the first two properties are enough to characterize the operators H_{+} and H_{−}. From the third property we see that we can use H_{+} or H_{−} to invert the Fourier transform at 0. By using translation invariance (which then corresponds to conjugating the H_{+} and H_{−} operators by a modulation) we thus see that we can use the Hilbert transform to invert the Fourier transform. (Of course, given (6), this may seem pointless, but when we move to the nonlinear Fourier transform this observation is key, as pointed out by [8].).

The purpose of this note is to discuss the non-linear Fourier transform F → \overbrace{F}, and the counterpart operators the non-linear Riesz projections f → H_{±}f, which obey an analogue of virtually every one of the properties mentioned above. The non-linear Fourier transform is also known as the (Dirac) scattering transform and has been intensely studied, both for its application to integrable systems, and also to the spectral theory of Dirac operators. See e.g. [1].

As a particular application of this transform, let us remark right now that just as one can use the linear Fourier transform and its inverse to solve the linear Schrödinger and Airy equations, one can use the nonlinear Fourier transform and its inverse to solve the defocusing mKdV equation

F_{t} + F_{xxx} = 6|F |^{2}F_{x}; F (0, x) = F_{0}(x)  (7)

and the defocusing NLS equation

iF_{t} + F_{xx} = 2|F |^{2}F; F (0, x) = F_{0}(x).  (8)

The Dirac nonlinear Fourier transform is a special case of a more general transform, the ANKS nonlinear Fourier transform, which includes as a special case the Schrödinger nonlinear Fourier transform^{1}. However the Dirac is by far the simplest, as the associated Dirac operator is antiselfadjoint and has no pure point spectrum. In terms of mKdV and NLS, this means that those equations do not have soliton or multisoliton solutions.

This note is mostly inspired by joint work with Christoph Thiele and Camil Muscalu, and also with Michael Christ and Jim Colliander. In addition the author thanks Jean-Pierre Kahane and Rowan Killip for many useful comments which have been incorporated, in some shape or form, into this note. We were also inspired by the work of Sylvester [8], which also strongly propounds the close analogy between the non-linear Fourier transform and the linear Fourier transform.

2. Definition of the non-linear Fourier transform

The nonlinear Fourier transform is defined using the matrix-valued Dirac operator L = L(F) defined^{2} by

L := ( \begin{matrix}D & −\overline{F} \\ F & −D \end{matrix} ) ,

where D := \frac{d}{dx} is the differentiation operator, thus

L ( \begin{matrix}ϕ \\ ψ \end{matrix} ) = ( \begin{matrix}ϕ_{x} − \overline{F} ψ \\ −ψ_{x} + F ϕ \end{matrix} ) .

Observe that L is anti-self-adjoint: L^{∗} = −L.

We consider the eigenfunction equation

L ( \begin{matrix}ϕ(k, x) \\ ψ(k, x) \end{matrix} ) = ik ( \begin{matrix}ϕ(k, x) \\ ψ(k, x) \end{matrix} ) ,

where ϕ, ψ are smooth functions on IR (with boundary conditions at x = ±∞ to be specified later), and k is an arbitrary complex number (although since L is anti-self-adjoint, it will be most natural to consider k real). Since this is a first-order system of two unknowns, we expect two linearly independent sets of solutions; we shall fix boundary conditions (at x = +∞) later. (Basically, we will set ϕ(x) ∼ e^{ikx} and ψ(x) ∼ 0 as x → +∞).

1A note on references: Many of the results below were first proved for the Schrödinger nonlinear transform, which is actually somewhat more complicated technically than the Dirac transform. Thus the works cited in this note may not treat the Dirac case directly, but in all cases their argument can be transferred to the Dirac case (and in fact becomes a little simpler upon doing so).

2Many other formulations of this operator are possible, but they are all conjugate to each other. This particular formulation has the advantage of being real when F is real.

Fix k. We can rewrite the eigenfunction equation as the system

ϕ_{x} − \overline{F} ψ = ikϕ
−ψ_{x} + F ϕ = −ikψ.

In the free case F ≡ 0, the general solution to this system is

ϕ(k, x) = a(k) exp(ikx)
ψ(k, x) = b(k) exp(−ikx)

where a(k) = a[F](k) and b(k) = b[F](k) are complex quantities. In the general case, it is thus natural to make the ansatz

ϕ(k, x) = a(k, x) exp(ikx)
ψ(k, x) = b(k, x) exp(−ikx),

at which point the eigenfunction equation becomes

\begin{aligned}a_{x}(k, x) & = \overline{F (x)} exp(−2ikx)b(k, x) \\ b_{x}(k, x) & = F (x) exp(2ikx)a(k, x). \end{aligned}  (9)

or in matrix form

∂_{x}G = W G

where G = G(k, x) is the matrix

G(k, x) := ( \begin{matrix}a(k, x) & \overline{b(k, x)} \\ b(k, x) & \overline{a(k, x)} \end{matrix} )

and W = W (k, x) is the matrix

W (k, x) := ( \begin{matrix}0 & \overline{F (x)} exp(−2ikx) \\ \overline{F (x)} exp(−2ikx) & 0 \end{matrix} ) .  (10)

At this point it is natural to introduce the three-dimensional non-compact, non-commutative Lie group

SU(1, 1) := \{( \begin{matrix}a & \overline{b} \\ b & \overline{a} \end{matrix} ) : |a|^{2} − |b|^{2} = 1\}

with associated Lie algebra

su(1, 1) := \{( \begin{matrix}ix & y − is \\ y + is & −ix \end{matrix} ) : x, y, s ∈ IR\}.

Observe that the matrix W lies in su(1, 1) if k is real. Thus if we set the initial condition^{3}

G(k, +∞) := Id := ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ∈ SU(1, 1)

then we have G(k, x) ∈ SU(1, 1) for all x and all real k, or in other words we have the energy conservation law

|a(k, x)|^{2} − |b(k, x)|^{2} = 1

3This is justified if F is compactly supported, or has sufficiently fast decay at +∞ (actually, integrability near +∞ will suffice, thanks to Gronwall’s inequality).

for all k, x ∈ IR. (This can also be deduced directly from (9), or from the original

eigenfunction equation by computing the Wronskian of ( \begin{matrix}ϕ \\ ψ \end{matrix} ) and the dual solution

( \overline{ψ} ; \overline{ϕ} )). In terms of a and b, the boundary conditions are that a(k, +∞) = 1 and

b(k, −∞) = 0.

The non-linear Fourier transform ^{4} \overbrace{F}(k) ∈ SU(1, 1) for any k ∈ IR is defined as

\overbrace{F}(k) := G(k, −∞).

Thus

\overbrace{F}(k) = ( \begin{matrix}a(k) & \overline{b(k)} \\ b(k) & \overline{a(k)} \end{matrix} )

where a(k) := a(k, −∞) and b(k) := b(k, −∞).

To understand this operator, first observe that if F ≡ 0 then G ≡ Id, so that \overbrace{F} ≡ Id. Thus, if F is small, then we expect G to be approximately equal to Id, so the ODE G_{x} = W G should be approximated by G_{x} = W. Of course, the ODE G_{x} = W, G(k, +∞) = Id can be explicitly solved by the fundamental theorem of calculus, to obtain

\overbrace{F}(k) = G(k, −∞) ≈ ( \begin{matrix}1 & \overline{F̂(k)} \\ F̂(k) & 1 \end{matrix} )

or, in what amounts to the same thing^{5} (but with both sides of the approximation manifestly in SU(1, 1))

\overbrace{F}(k) ≈ exp ( \begin{matrix}0 & \overline{F̂(k)} \\ F̂(k) & 0 \end{matrix} ))

Thus, for small F, we have the heuristic Born approximation

a(k) ≈ 1 + |F̂(k)|^{2}/2; b(k) ≈ F̂(k),  (11)

and one can think of \overbrace{F} as sort of a nonlinear exponentiated version of F̂. In particular, the linear Fourier transform is the linearization of the nonlinear Fourier transform at the origin.

exp ( \begin{matrix}0 & re^{−iθ} \\ re^{iθ} & 0 \end{matrix} ) = ( \begin{matrix}cosh r & (sinh r)e^{−iθ} \\ (sinh r)e^{iθ} & cosh r \end{matrix} ) .

4This is not the only choice of nonlinear Fourier transform. For instance, there is a similar transform corresponding to the group SU(2) which can be used to analyze the focussing NLS and mKdV equations.

More generally, there are a wide class of linear operators whose scattering transform acts like a nonlinear Fourier transform and which are connected to integrable systems; see e.g. [1].

5For future reference it may help to observe the explicit exponentiation rule

One can make this more rigorous, by going back to (9) and writing it in integral form (taking advantage of the boundary condition at +∞):

\begin{aligned}a(k, x) & = 1 − ∫_{y>x} \overline{F (y)} exp(−2iky)b(k, y) dy \\ b(k, x) & = − ∫_{y>x} F (y) exp(2iky)a(k, y) dy. \end{aligned}  (12)

Expanding this out as a Neumann series, we obtain

a(k, x) =1 + ∑_{n=1}^{∞} ∫_{x_{1}>...>x_{2n}>x} F (x_{1})\overline{F (x_{2})} … F (x_{2n−1})\overline{F (x_{2n})}
exp(2ik(x_{1} − x_{2} + … + x_{2n−1} − x_{2n})) dx_{1} … dx_{2n}
b(k, x) exp(−2ikx) = − ∑_{n=1}^{∞} ∫_{x_{1}>...>x_{2n−1}>x} F (x_{1})\overline{F (x_{2})} … F (x_{2n−1})
exp(2ik(x_{1} − x_{2} + … + x_{2n−1} − x)) dx_{1} … dx_{2n−1}.

Note that if F is Schwartz (or even just in L^{1}, that the summands here are bounded by ∥F ∥_{1}^{2n}/(2n)! and ∥F ∥_{1}^{2n−1}/(2n − 1)! respectively, so the series is indeed uniformly convergent and one can justify the fact that these multilinear expansions do solve the integral equation (and hence the differential equation (9)). Also we now see that if F is bounded in L^{1}, then the Born approximation (11) can be made a little more quantitative as

a(k) = 1 + |F̂(k)|^{2}/2 + O(∥F ∥_{1}^{4}); b(k) = F̂(k) + O(∥F ∥_{1}^{3}).

From the above multilinear expansions, if F is compactly supported (or exponentially rapidly decreasing), we see that for fixed x, the functions a(k, x), b(k, x) exp(−2ikx) are analytic for k in the upper half-plane {k : Imk ≥ 0} (because the phases (x_{1} − x_{2} + … + x_{2n−1} − x_{2n}), (x_{1} − x_{2} + … + x_{2n−1} − x) are non-negative). We also see the following analogue of the Paley-Wiener property (which seems to have first been observed in [5]):

• (Non-linear Paley-Wiener) If F is in L^{1} and is supported on an interval [x_{1}, x_{2}], then b(k) is entire, and obeys the bounds

|b(k + it)| ≤ sinh(∥F ∥_{1}) exp(−2x_{1}t)

and

|b(k − it)| ≤ sinh(∥F ∥_{1}) exp(2x_{2}t)

for k ∈ R, t ≥ 0, while a(k) extends to the upper half-plane and obeys the bounds

|a(k + it)| ≤ cosh(∥F ∥_{1}).

In particular, if F is supported on [0, ∞) (resp. (−∞, 0]), then b(k) extends holomorphically to the upper (resp. lower) half-plane, with b(k + it) → 0 as t → +∞ (resp. t → −∞).

Unfortunately the map F → \overbrace{F} is not linear, in the sense that the additivity property (2) fails. However it is multiplicative in a restricted sense:

• (Ordered multiplicativity) If F_{1} and F_{2} are compactly supported, and the support of F_{1} lies to the left of that of F_{2} (e.g. F_{1} is supported on (−∞, x_{0}] and F_{2} is supported on [x_{0}, ∞) then

\overbrace{F_{1} + F_{2}}(k) = \overbrace{F_{1}}(k)\overbrace{F_{2}}(k).

Proof Let W_{1}, W_{2} be the matrices associated to F_{1}, F_{2} by (10), and let G_{1}, G_{2} solve the associated ODE ∂_{x}G_{1} = W_{1}G_{1}, ∂_{x}G_{2} = W_{2}G_{2} with G_{1}(k, +∞) = G_{2}(k, ∞) = Id. Then if we define G(k, x) := G_{2}(k, x) for x ≥ x_{0} and G(k, x) = G_{1}(k, x)G_{2}(k, x_{0}) =

G_{1}(k, x)\overbrace{F_{2}}(k) for x ≤ x_{0} we see that G solves G_{x} = (W_{1} + W_{2})G with G(k, +∞) = Id, and the claim follows.

One may amuse oneself by verifying that this ordered multiplicativity property is consistent with the nonlinear Paley-Wiener property. It is also possible to verify this multiplicativity property from the multilinear expansions although it is not entirely trivial (it is comparable to verifying the multiplicative property e^{x}e^{y} = e^{x+y} using power series). This should also reinforce the intuition that the nonlinear Fourier transform is a non-linear exponentiated version of the linear Fourier transform (using products instead of sums).

As an illustration of the nonlinear Fourier transform^{6}, if F is a finite combination of Dirac masses

F = ∑_{n=1}^{N} c_{n}δ_{x_{n}}

with x_{1} < x_{2} < … < x_{n} then F̂ is given (formally, at least) as an exponential product:

\overbrace{F}(k) = ∏_{n=1}^{N} exp ( \begin{matrix}0 & −\overline{c_{n} exp(2ikx_{n})} \\ −c_{n} exp(2ikx_{n}) & 0 \end{matrix} ) ,  (13)

where the product is taken from left to right, i.e. ∏_{n=1}^{N} a_{n} := a_{1}a_{2} … a_{N}. This can be demonstrated by first using the ordered multiplicative property to reduce to a single Dirac mass, which then one approximates in the standard manner by step function potentials; we omit the details. The above formula (13) should be compared with (1).

We have the following analogues of the linear Fourier transform symmetry rules:

• (Phase Homogeneity) For any F and e^{iθ} ∈ IR, we have

\overbrace{e^{iθ}F}(k) = ( \begin{matrix}exp(−iθ) & 0 \\ 0 & 1 \end{matrix} ) \overbrace{F}(k) ( \begin{matrix}exp(iθ) & 0 \\ 0 & 1 \end{matrix} )

or in other words

a[e^{iθ}F](k) = a[F](k); b[e^{iθ}F](k) = b[F](k).

• (Conjugation symmetry) If F̃(x) := \overline{F} (x), then

\overbrace{F̃}(k) = \overbrace{F}(−k)^{∗},

or in other words

a[\overline{F}](k) = \overline{a[F](−k)}; b[\overline{F}](k) = \overline{b[F](−k)}.

6We are indebted to Jean-Pierre Kahane for this illuminating suggestion

In particular, if F is real, then \overbrace{F}(−k) = \overbrace{F}(k)^{∗}.

• (Translation symmetry) If F̃(x) := F (x − x_{0}) for some x_{0} ∈ IR, then

\overbrace{e^{iθ}F}(k) = ( \begin{matrix}exp(−ikx_{0}) & 0 \\ 0 & exp(ikx_{0}) \end{matrix} ) \overbrace{F}(k) ( \begin{matrix}exp(ikx_{0}) & 0 \\ 0 & exp(−ikx_{0}) \end{matrix} )

or equivalently

a[F̃](k) = a[F](k); b[F̃](k) = exp(2ikx_{0})b[F](k).

• (Modulation symmetry) If F̃(x) := F (x) exp(−2ik_{0}x) for some x_{0} ∈ IR, then

\overbrace{F̃}(k) = \overbrace{F}(k − k_{0})

or equivalently

a[F̃](k) = a[F](k − k_{0}); b[F̃](k) = b[F](k − k_{0}).

• (Scale invariance) If F̃(x) := λ^{−1}F (x/λ) for some λ ∈ IR\{0}, then

\overbrace{F̃}(k) = \overbrace{F}(λk)

or equivalently

a[F̃](k) = a[F](λk); b[F̃](k) = b[F](λk).

We leave these properties as easy exercises for the reader. Observe that they are consistent with the Born approximation (11) and with the Dirac delta examples (13) (recalling that Dirac deltas have the scaling λ^{−1}δ(x/λ) = δ(x)).

If F is Schwartz, then the a and b are uniformly bounded in k and x. By using (12) and an integration by parts we can obtain the preliminary bounds

b(k) = O(1/|k|); a(k) = 1 + O(1/|k|);

we shall obtain much better asymptotics later on.

From the matrix form of Gronwall’s inequality (or by using the multilinear expansions) we have the following result (basically due to Wiener):

• (Nonlinear Riemann-Lebesgue) If F is Schwartz, then

sup_{k} ∥\overbrace{F}∥_{op} ≤ exp ∥F ∥_{1},

and in particular we have the bounds

|a(k)| ≤ cosh(∥F ∥_{1}); |b(k)| ≤ sinh(∥F ∥_{1}).

Furthermore, we have Lipschitz continuity bounds of the form

sup_{k} ∥\overbrace{F_{1}}(\overbrace{F_{2}})^{−1}∥_{op} ≤ C(∥F_{1}∥_{1}, ∥F_{2}∥_{1})∥F_{1} − F_{2}∥_{1},

thus the operation of nonlinear Fourier transform can be extended continuously to L^{1}. From this and the above asymptotics, we in particular see that for any F in L^{1},

the function \overbrace{F}(k) is continuous and tends to Id as k → ±∞.

Actually, one can improve this result, and show that the non-linear Fourier transform not only maps L^{1} into L^{∞}, but it in fact (modulo the identity) maps L^{1} into the Wiener space A = {F̂ : F ∈ L^{1}} - i.e. the range of the non-linear Fourier transform on L^{1} is essentially the same as that of the linear Fourier transform. To see this, observe from the multilinear expansion formula that (if F is Schwartz, at least) that b = β̂, where β(x) = β[F](x) is the function

β(x) := ∑_{n=1}^{∞} ∫_{x_{1}>...>x_{2n−1}} F (x_{1})\overline{F (x_{2})} … F (x_{2n−1})δ(x_{1}−x_{2}+…+x_{2n−1}−x)) dx_{1} … dx_{2n−1}.

From this it is an easy matter to show that the map F → β[F] is bounded on L^{1}; in fact it is locally Lipschitz (a fact first observed by Hitrik [7]; in fact, the map is locally bilipschitz from L^{1} to L^{1}). A similar phenomenon occurs for the inverse Fourier transform of a (or more precisely of a − 1).

The non-linear Riemann-Lebesgue lemma has application to spectral theory. Because of Stone’s formula linking the Weyl m-function (which is closely related to the non-linear Fourier transform) and the spectral measure of L, it is (approximately) true that the a.c. spectrum of the Dirac operator L is essentially equal to the set of Lebesgue points of the

non-linear Fourier transform \overbrace{F}(k). (Making this statement rigorous and precise is a little technical, and will not be done here). In particular, we thus expect L^{1} potentials to have a.c. spectrum on all of the real line (which is indeed the case).

3. Asymptotics of a, and the non-linear Plancherel theorem

Let F be a compactly supported function, and fix a frequency k. Then, by Gronwall’s

inequality (i.e. the nonlinear Riemann-Lebesgue lemma), \overbrace{F} is uniformly bounded. Now we compute the asymptotics of a using (12), (9) and integration by parts:

a(k) = 1 − ∫ b(k, x)\overline{F (x)} exp(−2ikx) dx

= 1 + \frac{1}{−2ik} ∫ b_{x}(k, x)\overline{F (x)} exp(−2ikx) dx + \frac{1}{−2ik} ∫ b(k, x)\overline{F′(x)} exp(−2ikx) dx

= 1 + \frac{1}{−2ik} ∫ b_{x}(k, x)\overline{F (x)} exp(−2ikx) dx + O(1/|k|^{2})

= 1 + \frac{1}{−2ik} ∫ a(k, x)|F (x)|^{2} dx + O(1/|k|^{2})

= 1 + \frac{1}{−2ik} ∫ |F (x)|^{2} dx + O(1/|k|^{2})

where we have used the preliminary asymptotics a(k) = 1 + O(1/|k|), b(k) = O(1/|k|) mentioned earlier.

Now we investigate the extension of a to the upper half-plane. Observe that if F is compactly supported, then the equations (9), (12) remain well-posed for complex k. We write k = s + it for s ∈ IR, t ≥ 0, and observe from (9) that

∂_{x}|a(s + it, x)|^{2} = 2Re\overline{F (x)} exp(−2isx) exp(2tx)b(s + it, x)\overline{a(s + it, x)}
∂_{x}|b(s + it, x)|^{2} = 2ReF (x) exp(2isx) exp(−2tx)a(s + it, x)\overline{b(s + it, x)}.

Thus

∂_{x}(|a(s + it, x)|^{2} − exp(4tx)|b(s + it, x)|^{2}) = −4t exp(4tx)|b(s + it, x)|^{2}.

In particular, |a(s + it, x)|^{2} − exp(4tx)|b(s + it, x)|^{2} is monotone decreasing. Since this quantity equals 1 at x = +∞, we thus see that

|a(s + it, x)| ≥ 1

and in particular that |a(k)| ≥ 1 for all k on the upper half-plane. In particular we have the fundamental fact

Lemma 3.1. 1/a(k) is holomorphic on the upper half-plane, and is bounded in magnitude by 1.

This fact is what makes the Dirac non-linear Fourier transform easier to analyze than other non-linear Fourier transforms such as the Schrödinger transform. (In the Schrödinger case, 1/a(k) is meromorphic rather than holomorphic, and the poles and residues of 1/a(k) must be included as part of the non-linear Fourier transform in order to invert the transform correctly).

Another consequence of the above analysis is that log a(k) is well-defined, is holomorphic on the upper half-plane, and has non-negative real part. Also, we have the asymptotics

log a(k) = \frac{1}{−2ik} ∫ |F (x)|^{2} dx + O(1/|k|^{2})

for all real k (and hence for all k on the upper half-plane, by applying appropriate forms of the maximum principle). From the Cauchy integral formula we thus have

p.v. ∫_{−∞}^{∞} log a(k) dk = \frac{π}{2} ∫ |F (x)|^{2} dx;

taking real parts we obtain

∫_{−∞}^{∞} log |a(k)| dk = \frac{π}{2} ∫ |F (x)|^{2} dx.

Since |a(k)|^{2} − |b(k)|^{2} = 1, we thus have • (Non-linear Plancherel) For any compactly supported F, we have

∫_{−∞}^{∞} log(1 + |b(k)|^{2}) dk = π ∫ |F (x)|^{2} dx.

This should be compared with the ordinary Plancherel formula via the Born approximation (11). It should also be compared with the formula

sup_{k} log(1 + |b(k)|) dk ≤ C ∫ |F (x)| dx

arising from the non-linear Riemann-Lebesgue formula.

In principle, the above formula should allow us to show that the non-linear Fourier transform can be defined on all of L^{2}, and should in fact be a homeomorphism from L^{2} to the range of the Fourier transform. This has in fact been achieved by Sylvester [8], in the special case that F is restricted to a half-line [0, ∞).

The non-linear Plancherel theorem is also a key ingredient in showing that Dirac operators with L^{2} potentials have a.e. absolutely continuous spectrum [4]. To obtain the stronger result that the eigenfunctions of such an operator are a.e. bounded, one would need a non-linear analogue of Carleson’s theorem, which is still open at this time of writing.

Actually, the non-linear Plancherel theorem can be generalized. Recall that the linear Plancherel theorem can be generalized to

∫_{−∞}^{∞} |F̂(k)|^{2}k^{n} dk = π ∫_{−∞}^{∞} \overline{F} (x)(∂_{x}/(−2i))^{n}F (x) dx

when n is a non-negative integer s = 0, 1, 2, … (and even to arbitrary n ∈ IR, if one defines ∂_{x}^{n} appropriately); of course one can use integration by parts to balance the derivatives if desired. Similarly, for any n = 0, 1, 2 …, there is some polynomial expression P_{n}(F, F′, …) of F and its derivatives (and conjugates thereof) such that

∫_{−∞}^{∞} log(1 + |b(k)|^{2})k^{n} dk = π ∫ ReP_{n}(F, F′, …)(x) dx;

this was first observed by Faddeev and Zakharov [6].

We sketch a proof as follows. It suffices to show an asymptotic series of the form

log a(k) ∼ \frac{1}{−2ik} ∑_{n=0}^{∞} k^{−n} ∫ P_{n}(F, F′, …) dx,

as k → +∞. This can be achieved by noting that (9) implies the system

∂_{x} log a(k, x) = \frac{b(k, x)}{a(k, x)}\overline{F (x)} exp(2ikx)
∂_{x} \frac{b(k, x)}{a(k, x)} = F (x) exp(−2ikx) − (\frac{b(k, x)}{a(k, x)})^{2}\overline{F (x)} exp(2ikx).

From the latter equation (and the fact that \frac{b(k,x)}{a(k,x)} both vanish at infinity) we can obtain an asymptotic expansion of the form

\frac{b(k, x)}{a(k, x)} ∼ exp(−2ikx) ∑_{n=1}^{∞} k^{−n}Q_{n}(F, F′, …)

for some polynomial functions Q_{n} which can be determined explicitly by the recursion

−2iQ_{n+1} := −∂_{x}Q_{n} − ∑_{m=1}^{n} \overline{F} Q_{m}Q_{n+1−m}; −2iQ_{1} := F

as is easily checked. Multiplying by \overline{F (x)} exp(2ikx) and integrating, one then obtains the desired expansion of log a, with P_{n} := −2i\overline{F} Q_{n+1}.

One consequence of these formulae is that b is rapidly decreasing if F is Schwartz. In fact it is not hard to conclude from this that the map F → b maps compactly supported functions to Schwartz functions (especially since b can be extended above and below the real axis in this case).

AN INTRODUCTION TO THE NON-LINEAR FOURIER TRANSFORM 13

4. The nonlinear Fourier transform and NLS

We can use the nonlinear Fourier transform to analyze the defocusing nonlinear Schrödinger equation (8), which we rewrite as

F_{t} = iF_{xx} − 2i|F |^{2}F.

Let us only consider Schwartz solutions to this equation. Since

F_{x} = [D, F]

and

F_{xx} = [D, F_{x}] = [D, [D, F]] = D^{2}F − 2DF D + F D^{2}

we can write NLS in operator form as

F_{t} = i(D^{2}F − 2DF D + F D^{2} − 2F \overline{F} F).

Introduce the self-adjoint operator

J := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

and observe that J^{2} = 1. A computation shows that

L^{3} = ( \begin{matrix}… & … \\ F D^{2} − DF D + D^{2}F − F \overline{F} F & … \end{matrix} ) ,

where the … entries are not of interest to us. Similarly we have

LJLJL = ( \begin{matrix}… & … \\ F D^{2} + DF D + D^{2}F + F \overline{F} F & … \end{matrix} ) .

Thus

3L^{3} − LJLJL = ( \begin{matrix}… & … \\ 2F D^{2} − 4DF D + 2D^{2}F − 4F \overline{F} F & … \end{matrix} ) = ( \begin{matrix}… & … \\ −2iF_{t} & … \end{matrix} ) .

Since 3L^{3} − LJLJL is manifestly anti-self-adjoint, we thus have

3L^{3} − LJLJL = ( \begin{matrix}… & −2i\overline{F_{t}} \\ −2iF_{t} & … \end{matrix} ) .

In particular, we have

[3L^{3} − LJLJL, J] = ( \begin{matrix}0 & 4i\overline{F_{t}} \\ −4iF_{t} & 0 \end{matrix} ) = −4iL_{t}.

Since

[L^{3}, J] = L^{3}J − JL^{3} = [L, L^{2}J + LJL + JL^{2}]

and

[LJLJL, J] = [L, JLJLJ]

we thus have

[L, 3L^{2}J + 3LJL + 3JL^{2} − JLJLJ] = −4iL_{t}.

Thus if we define P = P (L) to be the anti-self-adjoint second-order operator

P := \frac{−i}{4}(3L^{2}J + 3LJL + 3JL^{2} − JLJLJ)

then we can rewrite NLS in Lax pair formulation

L_{t} = [P, L].

In particular, if ϕ(0) is an eigenfunction of L(0) with (necessarily imaginary) eigenvalue ik:

L(0)ϕ(0) = ikϕ(0)  (14)

and one evolves ϕ in time by the equation

ϕ_{t} = P ϕ + cϕ  (15)

for some arbitrary constant c = c(k), then

∂_{t}(Lϕ − ikϕ) = L_{t}ϕ + Lϕ_{t} − ikϕ_{t}
= (P L − LP)ϕ + L(P + c)ϕ − ik(P + c)ϕ
= (P + c)(Lϕ − ikϕ)

so by Gronwall’s inequality we see (formally, at least) that

L(t)ϕ(t) = ikϕ(t)

for all time t.

We apply this with

ϕ(0, x) := ( \begin{matrix}a[F_{0}](k, x) exp(ikx) \\ b[F_{0}](k, x) exp(−ikx) \end{matrix} )

and c := −2ik^{2}; by definition of a and b we thus see that (14) is obeyed. We let ϕ(t) evolve by (15). Observe that for x close to ±∞, the operator P + c approaches −2iD^{2} − 2ik^{2}. Since

ϕ(0) ≈ ( \begin{matrix}exp(ikx) \\ 0 \end{matrix} ) as x → +∞

and

ϕ(0) ≈ ( \begin{matrix}a[F_{0}](k) exp(ikx) \\ a[F_{0}](k) exp(−ikx) \end{matrix} ) as x → −∞

we thus see that

ϕ(t) ≈ ( \begin{matrix}exp(ikx) \\ 0 \end{matrix} ) as x → +∞

and

ϕ(t) ≈ ( \begin{matrix}a[F_{0}](k) exp(ikx) \\ a[F_{0}](k) exp(−4ik^{2}t) exp(−ikx) \end{matrix} ) as x → −∞.

From the definition of a and b we thus see that

a[F (t)](k) = a[F_{0}](k); b[F (t)](k) = exp(−4ik^{2}t)b[F_{0}](k).

This is the nonlinear analogue of (4) (as can be seen using the Born approximation (11)). Thus if one can invert the nonlinear Fourier transform, one can solve NLS fairly explicitly.

It turns out that there are higher order analogues of this computation, in which the NLS equation (which is second order in space) is replaced by a certain fourth order, or sixth order, etc. equation; this gives the famous NLS hierachy of integrable systems, which on the nonlinear Fourier side are just given by multiplication by the appropriate phase. Assuming the injectivity of the nonlinear Fourier transform (which will be proven

AN INTRODUCTION TO THE NON-LINEAR FOURIER TRANSFORM 15

presently) we see from the above computation that all the flows in this hierarchy commute. See e.g. [3].

5. The nonlinear Fourier transform and mKdV

Just as the nonlinear Fourier transform can (in principle) solve NLS, it can also be used to solve the mKdV equation

F_{t} = −F_{xxx} + 6|F |^{2}F_{x}.

This appears to have first been observed in [9]. We can rewrite mKdV as

F_{t} = −D^{3}F + 3D^{2}F D − 3DF D^{2} + F D^{3} + 6F \overline{F} F.

Let J be as before. A computation shows that

L^{4} = ( \begin{matrix}… & … \\ D^{3}F − D^{2}F D + DF D^{2} − F D^{3} + F \overline{F} F D − F \overline{F} DF + F D\overline{F} F − DF \overline{F} F & … \end{matrix} )

= ( \begin{matrix}… & … \\ D^{3}F − D^{2}F D + DF D^{2} − F D^{3} − 2F \overline{F} F_{x} & … \end{matrix} ) .

Similarly we have L^{2}JLJL + LJL^{2}JL + LJLJL^{2}

= ( \begin{matrix}… & … \\ 3D^{3}F + D^{2}F D − DF D^{2} + 3F D^{3} − F \overline{F} F D + F \overline{F} DF − F D\overline{F} F + DF \overline{F} F & … \end{matrix} )

= ( \begin{matrix}… & … \\ 3D^{3}F + D^{2}F D − DF D^{2} + F D^{3} + 2F \overline{F} F_{x} & … \end{matrix} ) .

Thus

5L^{4} − L^{2}JLJL − LJL^{2}JL − LJLJL^{2} = ( \begin{matrix}… & … \\ 2D^{3}F − 6D^{2}F D + 6DF D^{2} − 2F D^{3} − 12F \overline{F} F_{x} & … \end{matrix} )

= ( \begin{matrix}… & … \\ −2F_{t} & … \end{matrix} ) .

Since the left-hand side is manifestly self-adjoint, we thus have

5L^{4} − L^{2}JLJL − LJL^{2}JL − LJLJL^{2} = ( \begin{matrix}… & −2\overline{F_{t}} \\ −2F_{t} & … \end{matrix} ) .

In particular, we have

[5L^{4} − L^{2}JLJL − LJL^{2}JL − LJLJL^{2}, J] = ( \begin{matrix}0 & 4\overline{F_{t}}) \\ −4F_{t} & 0) \end{matrix} ) = −4L_{t}.

Since

[L^{4}, J] = L^{4}J − JL^{4} = [L, L^{3}J + L^{2}JL + LJL^{2} + JL^{3}]

and similarly

[L^{2}JLJL + LJL^{2}JL + LJL^{2}, J] = [L, JL^{2}JLJ + JLJL^{2}J + JLJLJL + LJLJLJ] we thus have [L, 5(L^{3}J + L^{2}JL + LJL^{2} + JL^{3}) − (JL^{2}JLJ + JLJL^{2}J + JLJLJL + LJLJLJ)] = −4L_{t}.

Thus if we define P = P (L) to be the anti-self-adjoint second-order operator

P := \frac{5}{4}(L^{3}J + L^{2}JL + LJL^{2} + JL^{3}) − \frac{1}{4}(JL^{2}JLJ + JLJL^{2}J + JLJLJL + LJLJLJ).

then we can rewrite NLS in Lax pair formulation

L_{t} = [P, L].

We now argue as in the previous section, with c := 4ik^{3}. For x → ±∞, the operator P +c approaches 4D^{3} + 4ik^{3}, and thus

a[F (t)](k) = a[F_{0}](k); b[F (t)](k) = exp(8ik^{3}t)b[F_{0}](k).

This is the nonlinear analogue of (5).

The mKdV equation is also part of a hierarchy, starting with the trivial transport equation u_{t} +u_{x} = 0, and in fact having one equation for every odd order of derivatives in space. This hierarchy commutes with itself and with the NLS hierarchy; see e.g. [3]. Also, one can see that the integral expressions ∫ P_{n} constructed earlier are conserved quantities of all of these flows; thus mKdV and NLS have an infinite number of conserved quantities (in fact, they have exactly the same set of conserved quantities!).

6. The nonlinear Riesz projections

If the nonlinear Fourier transform is injective, then we can define the nonlinear Riesz projections H_{±} by

H_{+}\overbrace{F}(k) := \overbrace{F χ_{[0,∞)}}(k)

and

H_{−}\overbrace{F}(k) := \overbrace{F χ_{(−∞,0]}}(k)

in complete analogy with the linear Riesz projections. At present it is not clear that these projections are well-defined, but this will be proven later in this section.

Let

f(k) = ( \begin{matrix}a(k) & \overline{b(k)} \\ b(k) & \overline{a(k)} \end{matrix} )

be a function of the form \overbrace{F}(k) for some Schwartz F; thus a(k) is analytic on the upper half plane and obeys |a(k)| ≥ 1 on this half-plane, while b(k) is Schwartz with |a(k)|^{2} − |b(k)|^{2} = 1. We write H_{±}f out in components as

H_{±}f(k) = ( \begin{matrix}a_{±}(k) & \overline{b_{±}(k)} \\ b_{±}(k) & \overline{a_{±}(k)} \end{matrix} )

Thus a_{±}(k) is also analytic on the upper half-plane with |a_{±}(k)| ≥ 1, and |a_{±}(k)|^{2} − |b_{±}(k)|^{2} = 1. (On the other hand, b_{±}(k) will not decay rapidly, because F χ_{±[0,∞)} is not smooth; more on this later).

From the ordered multiplicativity property and the non-linear Paley-Wiener property we have (assuming injectivity of the nonlinear Fourier transform)

• (Nonlinear factorization) We have

f = (H_{−}f)(H_{+}f).

In particular we have

a(k) = a_{−}(k)a_{+}(k) + \overline{b}_{−}(k)b_{+}(k)
b(k) = b_{−}(k)a_{+}(k) + \overline{a}_{−}(k)b_{+}(k)

• (Analyticity) a_{+}(k), a_{−}(k), b_{+}(k) extend analytically to the upper half plane, and b_{−}(k) extends analytically to the lower-half plane, with b_{±}(k) tending to 0 at infinity and a_{±}(k) tending to 1 at infinity.

Actually, one can take these two properties (and the properties of a_{±}, b_{±} mentioned earlier) to essentially be the definition of the non-linear Riesz projections; it turns out that for any f the factorization f = (H_{−}f)(H_{+}f) is unique. We will prove this presently, but for now we observe that this allows us to invert the nonlinear Fourier transform. For, by using (12) and an integration by parts we have

• (Asymptotics) As |k| → ∞ on the upper half-plane, we have

b_{+}(k) = F (0)/2ik + O(1/|k|^{2})

and similarly as |k| → ∞ on the lower half-plane, we have

b_{−}(k) = −F (0)/2ik + O(1/|k|^{2}).

Thus if we can solve this factorization problem uniquely (which is an instance of a Riemann-Hilbert^{7} problem), then the nonlinear Riesz projections are well-defined, and we

can recover F (0) from f = \overbrace{F}. By modulation invariance we can thus recover F (x_{0}) for

any x_{0} ∈ IR, thus recovering F from \overbrace{F}. (This procedure was essentially first performed by Gelfand, Levitan, and Marchenko). In particular we see that the non-linear Fourier transform is injective and hence invertible.

So we need to solve the Riemann-Hilbert problem. We rewrite the factorization as

H_{−}f = f(H_{+}f)^{−1}

which expands as

a_{−}(k) = \overline{a}_{+}(k)a(k) − \overline{b}(k)b_{+}(k)
b_{−}(k) = \overline{a}_{+}(k)b(k) − \overline{a}(k)b_{+}(k).

We use the fundamental fact that a(k) is invertible on the upper half-plane, to rewrite this as

\frac{a_{−}(k)}{a(k)} − 1 = \overline{a_{+}(k)} − 1 − R(k)b_{+}(k)
\frac{\overline{b}_{−}(k)}{a(k)} = R(k)a_{+}(k) − \overline{b}_{+}(k).

7The fact that Hilbert lent his name to both the Riemann-Hilbert problem and the Hilbert transform is not a coincidence!

where we define the reflection coefficient^{8}, R(k) as

R(k) := \frac{\overline{b}(k)}{a(k)}.

The functions \frac{a_{−}(k)}{a(k)} − 1 and \frac{\overline{b}_{−}(k)}{a(k)} are analytic on the upper half-plane and decay to 0 at infinity, by assumption. So we can apply H_{−} to cancel them, to obtain

\begin{aligned}\overline{a}_{+} & = 1 + H_{−}(R(k)b_{+}(k)) \\ \overline{b}_{+} & = H_{−}(R(k)a_{+}(k)) \end{aligned}  (16)

or in other words

( \begin{matrix}a_{+}(k) & b_{+}(k) \\ \overline{b_{+}(k)} & \overline{a_{+}(k)} \end{matrix} ) = Id + T ( \begin{matrix}a_{+}(k) & b_{+}(k) \\ \overline{b_{+}(k)} & \overline{a_{+}(k)} \end{matrix} )  (17)

where T is the matrix-valued operator

T := ( \begin{matrix}0 & H_{+}\overline{R} \\ H_{−}R & 0 \end{matrix} )

and we think of R as the operator of multiplication by R(k).

Observe that since b is Schwartz and |a|^{2} − |b|^{2} = 1, we see that ∥R∥_{L_{k}^{∞}} is strictly less than 1, and thus H_{−}R and H_{+}\overline{R} are contractions on L^{2}. This allows us to invert the operator (1 − T) by a Neumann series. (The first term in this Neumann series, after the identity, consists of H_{−}R and its conjugate^{9}, which is already in L^{2}). Thus we can uniquely solve for a_{+} and b_{+}, and one can obtain a_{−} and b_{−} by a similar procedure. This allows us to define the nonlinear Riesz projections and hence (by the asymptotics as k → +∞, and by exploiting modulation invariance) allow us to invert the nonlinear Fourier transform.

With a lot more work one can show that this inverse nonlinear Fourier transform does indeed map Schwartz functions to Schwartz functions. It would be of substantial interest if one could use this transform to show that the nonlinear Fourier transform was a homeomorphism on L^{2}, since this, combined with the Lax pair theory, would show that the mKdV and NLS hierarchies were globally well-posed in L^{2}. (Currently, this is only known for the NLS equation itself, see Tsutsumi [10]).

A little more effort with the above formula reveals that F (0) can essentially be written as the trace of (1 − T)^{−1}. This trace in turn can be written as a log derivative of a certain determinant, see e.g. [3]. This leads to rather compact formulae for the inverse nonlinear Fourier transform, involving the log determinant of some Fredholm operator, but it is not clear whether these formulae are so tractable to work with analytically, especially when the potential F and its transform are quite rough (e.g. only in L^{2}).

8This co-efficient can be interpreted as the coset \overbrace{F}(k)U(1), where U(1) is the maximal torus of SU(1, 1) thought of as acting on the right. If one wanted to recover a_{−} and b_{−} by this type of procedure one would use the other reflection co-efficient U(1)\overbrace{F}(k), which is essentially b(k)/a(k).

9This shows that when R is small, the (components of the) non-linear Riesz projections approach the linear Riesz projections of R. This is consistent with the Born approximation, suggests that R ≈ F̂.

The system (16) can be recast by taking the Fourier transform of all sides. Since a_{+}, b_{+} can be extended to the upper half-plane, they must have the representation

a_{+}(k) = 1 + ∫_{0}^{∞} A(x)e^{2ikx} dx

b_{+}(k) = ∫_{0}^{∞} B(x)e^{2ikx} dx;

Since the right-hand side of the b_{+} equation is B(0)/(−2ik) + O(1/|k|^{2}) (assuming that B is smooth and decaying) we thus see that

F (0) = B(0).

We also write the reflection co-efficient in Fourier as

R(k) = ∫IR \overline{r(x)}e^{−2ikx} dx.

Then the system (16) becomes the system

A(x) = ∫_{0}^{∞} \overline{B(y)}r(x + y) dy

B(x) = r(x) + ∫_{0}^{∞} \overline{A(y)}r(x + y) dy

or in matrix form

( \begin{matrix}A(x) & B(x) \\ \overline{B(x)} & \overline{A(x)} \end{matrix} ) = ( \begin{matrix}0 & r(x) \\ \overline{r(x)} & 0 \end{matrix} ) + ∫_{0}^{∞} ( \begin{matrix}0 & r(x + y) \\ \overline{r(x + y)} & 0 \end{matrix} ) ( \begin{matrix}A(y) & B(y) \\ \overline{B(y)} & \overline{A(y)} \end{matrix} ) dy.

This equation is known as the Gelfand-Levitan-Marchenko equation; since F (0) = B(0), it allows one (in principle) to recover F (0) from R. A similar procedure lets one recover F (x_{0}) for all x_{0} (basically one needs to translate r by x_{0}).

References

[1] M. Ablowitz, D. Kaup, A. Newell, H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems. Studies in Appl. Math. 53 (1974), no. 4, 249–315.

[2] R. Beals, R. Coifman, Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37 (1984), no. 1, 39–90.

[3] H. Blohm, Solution of nonlinear equations by trace methods, Nonlinearity 13 (2000), no. 6, 1925– 1964.

[4] P. Deift, R. Killip, On the absolutely continuous spectrum of one-dimensional Schrodinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341–347.

[5] P.Deift; E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251.

[6] V.E. Zakharov, L.D. Faddeev, Korteweg-de Vries equation: a completely integrable Hamiltonian system, Funktsional’nyi Analiz i Ego Prilozheniya 7 (1971), 18-27.

[7] M. Hitrik, Properties of the scattering transform on the real line, J. Math. Anal. Appl. 258 (2001), 223–243.

[8] J. Sylvester, D. Winebrenner, Linear and nonlinear inverse scattering, SIAM J. Appl. Math. 59 (1999), 669–699.

[9] S. Tanaka, Some remarks on the modified Korteweg-de Vries equation, Publ. Res. Inst. Math. Sci. 8 (1972/73), 429–437.

[10] Y. Tsutsumi, L^{2} solutions for nonlinear Schrodinger equations and nonlinear groups, Funk. Ekva. 30 (1987), 115–125.

20 TERENCE TAO

Department of Mathematics, UCLA, Los Angeles, CA 90095

E-mail address: tao@math.ucla.edu