INVERSE SCATTERING AND INTEGRABLE SYSTEMS

TERENCE TAO

Abstract. Some notes on the Dirac and Schrödinger scattering and inverse scattering equations, and their relations to the KdV, mKdV, and NLS equations.

1. Introduction

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

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

with this normalization, we have the inversion formula

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

and the Plancherel formula

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

Also observe that because F is real, we have the constraint F̂(−k) = \overline{F̂(k)}.

One can then solve the free Airy equation

V_{t} = V_{xxx}; V (0, x) = V_{0}(x)

by the formula

V̂ (t)(k) = exp(8ik^{3}t)V̂_{0}(k)

Thus one can explicitly solve the Airy equations by applying the Fourier transform, applying the appropriate phase modulation, and then inverting the Fourier transform.

The purpose of these notes is to introduce two non-linear variants of the Fourier transform, namely the Schrödinger scattering transform and the Dirac scattering transform, together with their inverse scattering transforms, in order to solve the KdV equation

V_{t} = V_{xxx} + 6V V_{x}; V (0, x) = V_{0}(x)

the (defocussing) mKdV equation

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

More precisely, the Schrödinger scattering transform and its inverse can be used to solve KdV (similar to how the Fourier transform solves Airy), while the Dirac scattering transform and its inverse can be used to solve mKdV^{1}.

1A variant of the Dirac scattering transform, in which a crucial sign is changed to flip the underlying group from SU(1, 1) ≡ SL_{2}(C) to SU(2)) can also be used to solve the cubic NLS equation. We won’t discuss it here.

The two scattering transforms are linked via the Miura transform, which on the level of scattering just reflects the simple fact that Dirac equations can be thought of as a special type of Schrödinger equation. However the Schrödinger equation contains further subtleties arising from bound states (which correspond to solitons and multisolitons in the KdV equation); the focussing mKdV equation does not have this problem. Since the Dirac transform is easier, we will discuss it first.

2. The Dirac scattering transform

Fix a real potential function F (x), which for simplicity we assume to be Schwartz, and consider the Dirac operator L = L(F) defined by

L := ( \begin{matrix}0 & −i(\frac{d}{dx} + F (x)) \\ i(−\frac{d}{dx} + F (x)) & 0 \end{matrix} ) ,

thus

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

Observe that L is self-adjoint. We consider the eigenfunction equation

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

for some complex number k (which will usually either lie on the real line or upper halfplane). By making the variation of constants ansatz

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

we can rewrite the eigenfunction equation as

\frac{d}{dx} ( \begin{matrix}a(k, x) \\ b(k, x) \end{matrix} ) = F (x) ( \begin{matrix}0 & exp(−2ikx) \\ exp(2ikx) & 0 \end{matrix} ) ( \begin{matrix}a(k, x) \\ b(k, x) \end{matrix} ) .  (2)

This ODE can be solved as

( \begin{matrix}a(k, x) \\ b(k, x) \end{matrix} ) = G_{±∞}(k, x) ( \begin{matrix}a(k, ±∞) \\ b(k, ±∞) \end{matrix} ) .

where ± is an arbitrary sign and the a fundamental solution G_{±∞}(k, x) from ±∞ is the 2 × 2 matrix solving the ODE

\frac{d}{dx}G_{±∞}(k, x) = F (x) ( \begin{matrix}0 & exp(−2ikx) \\ exp(2ikx) & 0 \end{matrix} ) G_{±∞}(k, x)

with initial data G_{±∞}(k, ±∞) = Id, where Id is the 2 × 2 identity matrix. The fundamental solutions G_{+∞} and G_{−∞} are related by the matrix identity

G_{−∞}(k, x) = G_{+∞}(k, x)G_{+∞}(k, −∞)^{−1}.  (3)

We will refer to G_{+∞}(k, −∞) as the scattering matrix of T (F), or as the scattering transform of F. Intuitively, this describes how a (C^{2}-valued) plane wave ϕ of frequency k gets distorted as it passes from x = −∞ to x = +∞ through the potential F while obeying the Dirac equation ϕ_{t} = iLϕ. (This can be made rigorous, maybe if I have time I’ll say how. And yes, I mean from −∞ to +∞ rather than the other way around).

One can think of G_{+∞}(k, −∞) as the continuous infinite product

G_{+∞}(k, −∞) = lim_{Δx→0} ∏_{x=−∞}^{∞}(1 + Δx ( \begin{matrix}0 & F (x) exp(−2ikx) \\ F (x) exp(2ikx) & 0 \end{matrix} ) ,

where the product is from x = −∞ to x = +∞ in steps of Δx, multiplied from left to right. This should be compared with the Riemann sum formulation of the Fourier transform

F̂(k) = lim_{Δx→0} ∑_{x=−∞}^{∞} ΔxF (x) exp(−2ikx)

or perhaps a bit more suggestively,

exp(F̂(k)) = lim_{Δx→0} ∏_{x=−∞}^{∞}(1 + ΔxF (x) exp(−2ikx)).

Thus the scattering transform is a non-commutative version of the (exponentiated) Fourier transform.

It is easy to verify that G_{±∞}(k, x) must have the structure

G_{±∞}(k, x) = ( \begin{matrix}a_{±∞}(k, x) & \overline{b_{±∞}(\overline{k}, x)} \\ b_{±∞}(k, x) & \overline{a_{±∞}(\overline{k}, x)} \end{matrix} )

for some complex-valued functions a_{±∞}(k, x), b_{±∞}(k, x) solving the Cauchy problem

\begin{aligned}∂_{x}a_{±∞}(k, x) & = F (x) exp(−2ikx)b_{±∞}(k, x) \\ ∂_{x}b_{±∞}(k, x) & = F (x) exp(2ikx)a_{±∞}(k, x) \\ a_{±∞}(k, ±∞) & = 1 \\ b_{±∞}(k, ±∞) & = 0. \end{aligned}  (4)

Observe that a_{+∞}(k, x), b_{+∞}(k, x) exp(−2ikx), b_{−∞}(k, x) and a_{−∞}(k, x) exp(2ikx) are analytic on the upper half-plane {Imk ≥ 0}. Formally, this follows from the asymptotic expansions

a_{+∞}(k, x) = 1 + ∑_{n≥2, even} ∫_{x<x_{1}<...<x_{n}} exp(2ik(x_{n} − x_{n−1} + … + x_{2} − x_{1})) ∏_{j=1}^{n} F (x_{j}) dx_{j}

b_{+∞}(k, x) exp(−2ikx) = − ∑_{n≥1, odd} ∫_{x<x_{1}<...<x_{n}} exp(2ik(x_{n} − x_{n−1} + … + x_{1} − x)) ∏_{j=1}^{n} F (x_{j}) dx_{j}.

a_{−∞}(k, x) exp(2ikx) = ∑_{n≥1, odd} ∫_{x>x_{1}>...>x_{n}} exp(2ik(x − x_{1} + … + x_{n−1} − x_{n})) ∏_{j=1}^{n} F (x_{j}) dx_{j}

b_{−∞}(k, x) = 1 + ∑_{n≥2, even} ∫_{x>x_{1}>...>x_{n}} exp(2ik(x_{1} − x_{2} + … + x_{n−1} − x_{n})) ∏_{j=1}^{n} F (x_{j}) dx_{j}

where we observe that the phase factor in parentheses is always positive (and so the exponential term decays as Imk → +∞.

Note in particular that to top order, a_{+∞}(k, x) ≈ 1 and b_{+∞}(k, x) ≈ − ∫_{x}^{∞} exp(2iky)F (y) dy, so b_{+∞} can be viewed as some sort of nonlinear partial Fourier transform of F (with b_{+∞}(k, −∞) being the complete nonlinear Fourier transform).

For arbitrary complex k we observe the identity

\frac{d}{dx}(|a_{+∞}(k, x)|^{2} − |b_{+∞}(k, x) exp(−2ikx)|^{2}) = −4Im(k)|b|^{2} exp(4Im(k)x).

In particular, for k real, we have the conservation law

|a_{+∞}(k, −∞)|^{2} − |b_{+∞}(k, −∞)|^{2} = |a_{+∞}(k, x)|^{2} − |b_{+∞}(k, x)|^{2} = |a_{+∞}(k, +∞)|^{2} − |b_{+∞}(k, +∞)|^{2} = 1. (5)

For k in the upper half-plane, the above identity implies the monotonicity formula (assuming F is compactly supported, or decaying extremely rapidly)

|a_{+∞}(k, x)|^{2} − |b_{+∞}(k, x) exp(−2ikx)|^{2} ≥ 1.

In particular, we see that a_{+∞}(k, x) is never zero, and in fact we have the inequality

|b_{+∞}(k, x) exp(−2ikx)| < |a_{+∞}(k, x)|  (6)

for all k in the upper half-plane. A similar argument gives

|a_{−∞}(k, x) exp(2ikx)| < |b_{−∞}(k, x)|

for all k in the upper half-plane.

From (5) and (2) we have the differential inequality

|∂_{x}b_{+∞}| ≲ |F (x)|(1 + |b_{+∞}|),

which by Gronwall’s inequality gives us the inequality

∥ log(1 + |b_{+∞}|)∥_{L_{k,x}^{∞}} ≲ ∥F ∥_{1}  (7)

(where k is restricted to be real). This is the nonlinear analogue^{2} of the p = 1 Hausdorff-Young inequality ∥F̂∥_{∞} ≲ ∥F ∥_{1} (at least when specialized to x = −∞; otherwise it is the p = 1 nonlinear version of Paley-Menshov-Zygmund). Of course we immediately obtain similar bounds for a and R thanks to (5).

The nonlinear Fourier transform has some invariances similar to that of the linear Fourier transform. Because F is real, it is easy to check that a(−k, x) = \overline{a(k, x)}, b(−k, x) = \overline{b(k, x)}, and hence R(−k) = \overline{R(k)} (just as F̂(−k) = \overline{F̂(k)}). If one dilates F (x) to λF (λx), then a(k, x) becomes a(k/λ, λx) and similarly for b. If one translates F (x) to F (x − x_{0}), then a(k, x) becomes a(k, x − x_{0}), but b(k, x) is modulated to b(k, x − x_{0}) exp(2ikx_{0}), and R(k) becomes R(k) exp(2ikx_{0}). (Conversely, modulations transform to translations, but one has to generalize the above transform to complex F (which basically means replacing F with \overline{F} in the equation for b).

2Another application of Gronwall in fact gives that the scattering transform is continuous from L^{1} to L^{∞}.

3. Contour integration

Now we digress a little on contour integration. From Cauchy’s theorem we have

Lemma 3.1. Let f(k) be a function analytic on the upper half plane {k : Im(k) ≥ 0} which has the asymptotics f(k) = A/ik + O(1/|k|^{2}) as |k| → ∞ on this half-plane, for some constant A. Then we have

p.v. ∫_{−∞}^{∞} f(k) dk := lim_{R→+∞} ∫_{−R}^{R} f(k) dk = −πA.  (8)

Indeed, one just uses Cauchy’s theorem to shift the contour to the upper semicircle and then uses the asymptotics. One can also interpret this lemma using the Fourier transform. Formally, the Fourier transform f̂(ξ) of f(k) is supported on the half-line {ξ : ξ ≥ 0} and behaves like −2πAχ_{[0,∞)}(ξ) near the origin. Evaluating this at 0 in the principal value sense we obtain −πA.

There is one odd thing about this lemma. Suppose A is real; then the leading term A/ik of the asymptotics for f(k) is imaginary on the real axis. However the expression −πA in (8) is clearly real. The resolution to this paradox is that the leading term A/ik cancels itself out (asymptotically, at least) in the principal value integral. Perhaps a better way to view (8) is to take real parts to obtain

∫_{−∞}^{∞} \frac{f(k) + \overline{f(k)}}{2} dk = πA.  (9)

Observe that the integrand now decays like O(1/|k|^{2}) instead of O(1/|k|) and so there is no need for a principal value.

How does the above discussion relate to inverse scattering theory? Well, from Lemma 3.1 and the asymptotics (in k) of eigenfunctions, it will be very easy to express the potential function (V for Schrödinger, or F for Dirac) as a principal value integral (in k) of an expression involving eigenfunctions. However, such a formula is not directly useful, because it is difficult to bound that integral in any reasonable manner (given that the integrand only decays like 1/k and is thus not in L^{1}). But by taking real parts as in (9).

4. Asymptotics

Let F be a Schwartz function. For each fixed x that we claim the asymptotics in k

a_{+∞}(k, x) = 1 + O(1/|k|)
b_{+∞}(k, x) exp(−2ikx) = \frac{F (x)}{2ik} + O(1/|k|^{2}).

This can be proven by the above ODE (2) and integration by parts (first using (7) to get a and b are bounded). One can in fact get a full asymptotic development of a and b but this is all that we shall need.

From Lemma 3.1 we can obtain a principal value representation

− \frac{π}{2} F (x) = p.v. ∫ b_{+∞}(k, x) exp(−2ikx) dk.  (10)

While this does look somewhat like a nonlinear version of the Fourier inversion formula, it is not directly useful because of the principal value. As indicated in the previous section we shall need to use some conjugation to make the integrand integrable.

Similarly the functions a_{−∞}, b_{−∞} obey the asymptotics

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

on the upper half-plane. Furthermore, from (3) and some algebra we have the relationships

\begin{aligned}\frac{a_{−∞}(k, x)}{a_{+∞}(k, −∞)} & = R(k)a_{+∞}(k, x) + \overline{b_{+∞}(k, x)} \\ \frac{b_{−∞}(k, x)}{a_{+∞}(k, −∞)} & = R(k)b_{+∞}(k, x) + \overline{a_{+∞}(k, x)}. \end{aligned}  (11)

for k on the real line, where the reflection co-efficient R(k) is defined by

R(k) := −\overline{b_{+∞}(k, −∞)}/a_{+∞}(k, −∞).

This is also essentially a nonlinear Fourier transform of F (in that it is equal to the Fourier transform of F to top order), and also has the advantage of being bounded on the real line (indeed we have |R(k)| < 1 thanks to (6) or (5)).

From earlier sections we saw that a_{+∞}(k, −∞) had no zeroes on the upper half-plane. In particular we see that the function a_{−∞}(k, x) exp(2ikx)/a_{+∞}(k, −∞) is analytic on the upper half-plane, with asymptotics

a_{−∞}(k, x) exp(2ikx)/a_{+∞}(k, −∞) = \frac{F (x)}{−2ik} + O(1/|k|^{2}).

Thus from Lemma 3.1 again we have

\frac{π}{2} F (x) = p.v. ∫ a_{−∞}(k, x) exp(2ikx)/a_{+∞}(k, −∞) dk.

However, from conjugating (10) we have

−\frac{π}{2} F (x) = p.v. ∫ \overline{b_{+∞}(k, x)} exp(−2ikx) dk;

subtracting the two and using (11) we obtain the inverse scattering formula

πF (x) = ∫_{−∞}^{∞} R(k)a_{+∞}(k, x) exp(2ikx) dk.  (12)

Note that if a is bounded and R is integrable then the above integrand is indeed integrable.

As an exercise one can find the linear analogues of these formulae (i.e. assume F is very small and discard all terms quadratic or higher in F) and verify that one just gets the ordinary Fourier inversion formula from the introduction.

By combining (12) with (2) we can recover a_{+∞}, b_{+∞}, and F from R, by starting at x = +∞ and evolving backwards in x. If R is integrable then a simple Picard iteration technique (iterating a_{+∞}, b_{+∞}, and F in L^{∞} of x and k, restricting k to be real, of course) will work locally in x, however to extend the iteration globally in x we need some global L^{∞} bound on F (x) - basically, we need some non-linear analogue of the Hausdorff-Young inequality ∥F ∥_{∞} ≲ ∥F̂∥_{1}. This is the purpose of the next section.

5. Logarithmic inverse scattering formula

Consider the function

log(\frac{a_{+∞}(k, x) + b_{+∞}(k, x) exp(2ikx)}{a_{+∞}(k, x)}).

From (6) we see that the expression inside the logarithm stays strictly within unit distance of 1, and so we can take a logarithm without worrying about branch cuts or the singularity at 0. Thus this function is analytic on the upper half-plane, and has the asymptotics \frac{F (x)}{2ik} + O(1/|k|^{2}). Thus we have

p.v. ∫_{−∞}^{∞} log(\frac{a_{+∞}(k, x) + b_{+∞}(k, x) exp(2ikx)}{a_{+∞}(k, x)}) = −\frac{π}{2} F (x).

Similar reasoning gives

p.v. ∫_{−∞}^{∞} log(\frac{b_{−∞}(k, x) + a_{−∞}(k, x) exp(−2ikx)}{b_{−∞}(k, x)}) = +\frac{π}{2} F (x).

Meanwhile, if we conjugate the first identity we obtain

p.v. ∫_{−∞}^{∞} log(\frac{\overline{a_{+∞}(k, x)} + \overline{b_{+∞}(k, x)} exp(−2ikx)}{\overline{a_{+∞}(k, x)}}) = −\frac{π}{2} F (x).

Subtracting this from the second identity, we obtain

p.v. ∫_{−∞}^{∞} log(\frac{b_{−∞}(k, x) + a_{−∞}(k, x) exp(−2ikx)}{\overline{a_{+∞}(k, x)} + \overline{b_{+∞}(k, x)} exp(−2ikx)})/\frac{b_{−∞}(k, x)}{\overline{a_{+∞}(k, x)}}) dk = πF (x).

Inserting (11) and canceling the common factor of a_{+∞}(k, −∞), we obtain

p.v. ∫_{−∞}^{∞} log(1+R(k)\frac{b_{+∞}(k, x) + a_{+∞}(k, x) exp(−2ikx)}{\overline{a_{+∞}(k, x)} + \overline{b_{+∞}(k, x)} exp(−2ikx)})−log(1+R(k)\frac{b_{+∞}(k, x)}{\overline{a_{+∞}(k, x)}}) dk = πF (x).

Observe that the factor

\frac{b_{+∞}(k, x) + a_{+∞}(k, x) exp(−2ikx)}{\overline{a_{+∞}(k, x)} + \overline{b_{+∞}(k, x)} exp(−2ikx)}

is unimodular, while the factor \frac{b_{+∞}(k,x)}{\overline{a_{+∞}(k,x)}} has magnitude less than 1. Thus from this identity we immediately obtain the nonlinear Hausdorff-Young inequality

∥F ∥_{∞} ≲ ∥ log(1 − |R|)∥_{L^{1}}.

Once again, one can amuse oneself by verifying that the above “logarithmic inverse scattering formula” collapses to the Fourier inversion formula for small F and R (the second logarithmic term is second-order in F, while the complicated factor in front of R(k) in the first logarithmic term collapses to exp(−2ikx) to zeroth order).

6. Nonlinear Plancherel

One can refine the asymptotics for a_{+∞}(k, x) to

a_{+∞}(k, x) = 1 + \frac{1}{−2ik} ∫_{x}^{∞} F^{2}(y) dy + O(1/|k|^{2}).

To see this, start with the integral identity

a_{+∞}(k, x) = 1 − ∫_{x}^{∞} F (y)b_{+∞}(k, y) exp(−2iky) dy

coming from (2) and insert the asymptotics for b_{+∞}(k, y) exp(−2iky) (assuming of course that F is Schwartz). Actually one can keep iterating this process to get arbitrarily high order asymptotics for a_{+∞} and b_{+∞} but we shall not do so here.

In particular we have

log a_{+∞}(k, −∞) = \frac{1}{−2ik} ∫_{∞}^{∞} F^{2}(y) dy.

Since the left-hand side is analytic on the upper half-plane (recall (6)) we obtain from Lemma 3.1 that

p.v. ∫_{−∞}^{∞} log a_{+∞}(k, −∞) dk = \frac{π}{2} ∫_{∞}^{∞} F^{2}(y) dy.

Taking real parts and using (5) we obtain

∫_{−∞}^{∞} log(1 + |b_{+∞}(k, −∞)|^{2}) dk = π ∫_{∞}^{∞} F^{2}(y) dy.

This is the nonlinear analogue of the Plancherel theorem (1). One can also get higher order versions of this which correspond to the linear formulae

∫_{−∞}^{∞} |(2k)^{s}F̂(k)|^{2} dk = π ∫_{−∞}^{∞} |∂_{x}^{s}F (x)|^{2} dx,

at least when s is a non-negative integer, by multiplying log a_{+∞}(k, −∞) by k^{2s} and using the appropriate residue theorem. The expressions on the right become the conserved quantities of mKdV. We will not pursue this matter further here.

7. Lax pairs - general theory

We now digress on the general theory of Lax pairs. Suppose that for each time t, L(t) and P (t) are self-adjoint and skew-adjoint operators respectively which obey the Lax pair equation

\frac{d}{dt}L(t) = [P (t), L(t)].

Then we observe that if ϕ(0) is an eigenfunction of L(0) with eigenvalue λ

L(0)ϕ(0) = λϕ(0)

and we evolve ϕ(t) by the equation

\frac{d}{dt}ϕ(t) = P (t)ϕ(t)

then (formally, at least) ϕ(t) is an eigenfunction of L(t) with eigenvalue λ:

L(t)ϕ(t) = λϕ(t).

Indeed, we can formally compute

\frac{d}{dt}(L(t)ϕ(t) − λϕ(t)) = [P (t), L(t)]ϕ(t) + L(t)P (t)ϕ(t) − λP (t)ϕ(t)
= P (t)(L(t)ϕ(t) − λϕ(t))

and so the claim follows (morally at least) from Gronwall’s inequality.

Thus the Lax pair equation preserves the spectral properties of L(t) in time, and the eigenfunctions evolve according to the companion operator P (t). Actually, it is easy to see that we may replace P (t) by P (t) − iτ for any real constant τ and still retain all of the above analysis (basically because iτ is skew-adjoint and commutes with L(t). Indeed we could even let τ vary in time if desired). On the level of ϕ, this just amounts to introducing an arbitrary phase shift to ϕ.

8. Lax pair formulation of mKdV

We now apply this formalism to the mKdV equation

F_{t} = F_{xxx} + 6F^{2}F_{x}.

Suppose F is a real-valued solution to this equation. For any time t, the function F (t) is a real-valued function of x, and so we can form the self-adjoint operator

L(t) := ( \begin{matrix}0 & −i(\frac{d}{dx} + F (t, x)) \\ i(−\frac{d}{dx} + F (t, x)) & 0 \end{matrix} )

as before. Observe that L squares as

L^{2}(t) = ( \begin{matrix}\frac{d^{2}}{dx^{2}} + V (t, x) & 0 \\ 0 & \frac{d^{2}}{dx^{2}} + W (t, x) \end{matrix} )

where V := F_{x} + F^{2} and W := −F_{x} + F^{2} are the Miura transform and adjoint Miura transform of F respectively.

We also (apparently rather arbitrarily) introduce the anti-selfadjoint operator

P (t) := ( \begin{matrix}4\frac{d^{3}}{dx^{3}} + 3(\frac{d}{dx}V (t, x) + V (t, x)\frac{d}{dx}) & 0 \\ 0 & 4\frac{d^{3}}{dx^{3}} + 3(\frac{d}{dx}W (t, x) + W (t, x)\frac{d}{dx}) \end{matrix} );

this operator may seem arbitrary but it can be interpreted as an approximation to the pseudo-differential operator 4i|L|^{3} = 4i(L^{2})^{3/2}, discarding all terms lower than zeroth order. A tedious computation then shows that if F obeys mKdV, then the Lax pair equation (??) is indeed satisfied.

Now let k be an arbitrary real, and let

Φ(t, k, x) := ( \begin{matrix}a(t, k, x) exp(ikx) + b(t, k, x) exp(−ikx) \\ a(t, k, x) exp(ikx) − b(t, k, x) exp(−ikx) \end{matrix} )

be defined at time t = 0 by a(0, k, x) = a_{+∞}(0, k, x), b(0, k, x) = b_{+∞}(k, x), where a_{+∞}(t, …) and b_{+∞}(t, …) are the scattering transform of F (t). Let Φ evolve according to the ODE

Φ′(t) = (P (t) − 4(ik)^{3})Φ(t).

Then since L(0)Φ(0) = kΦ(0), we see from the previous section that L(t)Φ(t) = kΦ(t) for all time t. Also, since P (t) is just 4\frac{d^{3}}{dx^{3}} outside of the support of F, we see that (if F is sufficiently rapidly decreasing)

a_{t}(t, k, ±∞) = 0; b_{t}(t, k, ±∞) = 8ik^{3}b(t, k, ±∞).

In particular we have

a(t, k, +∞) = 1
a(t, k, −∞) = a_{+∞}(0, k, −∞)
b(t, k, +∞) = 0
b(t, k, −∞) = b_{+∞}(0, k, −∞) exp(8ik^{3}t).

By definition of a_{+∞} and b_{+∞}, we thus see that

a_{+∞}(t, k, −∞) = a_{+∞}(0, k, −∞)

and

b_{+∞}(t, k, −∞) = b_{+∞}(0, k, −∞) exp(8ik^{3}t).

In other words, the mKdV flow does not affect the a_{+∞} component of the scattering transform, but multiplies the b_{+∞} component by exp(8ik^{3}t). This should be compared with how the Airy flow affects the Fourier transform. In principle, this allows one to solve the mKdV equation via the scattering and inverse scattering transforms.

One can also construct the higher order members of the mKdV hierarchy in a similar manner, using |L|^{5/2} instead of |L|^{3/2}, etc. This will then correspond to something like exp(32ik^{5}t) on the scattering side, etc.

9. The Schrödinger scattering transform

The above scattering theory for the first-order Dirac operator can be largely extended to the second-order Schrödinger operator

L := −\frac{d^{2}}{dx^{2}} + V (x)

where V is a real Schwartz potential. For any complex k, we can solve the eigenfunction equation

Lϕ(k, x) = k^{2}ϕ(k, x)

with solutions which are asymptotically equal to exp(ikx) as x → +∞ and a(k) exp(ikx)+ b(k) exp(ikx) as x → −∞. For real k, one can refer to a(k) and b(k) as the Schrödinger scattering transform of V (x).

To compute a(k) and b(k) one can solve the system of ODE

a_{x}(k, x) = 1/(2ik)V (x)[a(k, x) + b(k, x) exp(−2ikx)]
b_{x}(k, x) = 1/(−2ik)V (x)[b(k, x) + a(k, x) exp(2ikx)].

with initial data a(k, +∞) = 1, b(k, +∞) = 0 and set a(k) := a(k, −∞), b(k) := b(k, −∞); compare with (2). For small V we have the WKB approximation

a(k) ≈ exp(−\frac{1}{2ik} ∫ V (y)dy)

and the Born approximation

b(k) ≈ \frac{1}{2ik} ∫ V (y) exp(2iky)dy = −\frac{1}{2ik} V̂ (k)

which can be seen from the above ODE by inserting the crude approximation a ≈ 1, b ≈ 0 in the right-hand side. Thus the Schrödinger scattering transform is also a sort of nonlinear Fourier transform, but with more complicated features.

In the special case that V is the Miura transform V = F_{x} +F^{2} of some real potential F, we see that the Schrödinger scattering transform for V is identical to the Dirac scattering transform of F, mainly because the Schrödinger operator L here is a component of the square of the Dirac operator in the previous section. Thus one can view the Schrödinger scattering transform as a generalization of the Dirac one. Note also that in the small V and small F limit, this connection is consistent with the linear fact that derivatives correspond to multiplication by −2ik on the Fourier side.

Much of the theory goes through for the Schrödinger case (for a detailed analysis, see the work of Deift and Trubowitz). For instance, one still has the conservation law |a(k)|^{2} − |b(k)|^{2} = 1 for real k, and a(k) is still analytic on the upper half-plane. The main additional difficulty is that a(k) can have zeroes on the upper imaginary axis k = iw; this corresponds to bound states of L, i.e. eigenfunctions Lϕ = −w^{2}ϕ which decay like exp(−wx) as x → +∞ and b(iw) exp(+wx) as x → −∞. Fortunately, for Schwartz V there can only be a finite number of these bound states^{3}. For each bound state k = iw, an important number is the residue c(iw) of the meromorphic function 1/a(k) at iw; many of the formulae derived above for the Dirac scattering transform using contour integration have counterparts in the Schrödinger theory, but with additional residue terms coming from each bound state k = iw. In particular one can invert the scattering transform and reconstruct V from the absolutely continuous portion a(k), b(k) of the scattering data, together with the pure point portion iw, c(iw) of the scattering data.

It turns out that V = F_{x} + F^{2} for some decaying function F if and only if L has no bound states^{4}. Informally, one can think of the Schrödinger theory as being the direct sum of the (Miura transform of the) Dirac theory, plus an additional discrete component coming from the theory of bound states. This corresponds to the spectral decomposition of L into the absolutely continuous component (on [0, ∞)) and the pure point spectrum (a finite number of points in (−∞, 0)).

One can make this partition into pure point and absolutely continuous more precise. If λ_{1} < 0 is the lowest bound state of L, so that one has an L^{2} eigenfunction ϕ with Lϕ = λ_{1}ϕ, then by energy minimizing arguments one can show that ϕ cannot change sign (so ϕ is always positive, say), and by setting F := ϕ_{x}/ϕ one obtains that

L = (\frac{d}{dx} + F)(−\frac{d}{dx} + F) + λ_{1}.

3Intuitively, this says that there are at most a finite number of obstructions to the quadratic form ∫ u_{x}^{2} + V u^{2} being positive definite if V is Schwartz - indeed u must be concentrated near the support of V and have low frequency in order to have a chance of this form being negative.

4For a proof of this, see Deift and Trubowitz. In the case of the circle the argument is simpler; see the next section

If one then considers the variant operator

L̃ = (−\frac{d}{dx} + F)(\frac{d}{dx} + F) + λ_{1}

(which is also a Schrödinger operator but with slightly different potential, basically replacing the Miura transform of F by the adjoint Miura transform of F), this operator has exactly the same spectral properties as L except that the lowest energy λ_{1} has been removed. (The remaining scattering data remains unchanged). By repeating this process one can remove the bound states one at a time and end up with a potential with no bound states (which is thus the Miura transform of something decaying). It is also possible to invert this procedure, and reinstate bound states provided that one knows what the residues c(iw) are. For details on all this, see Deift and Trubowitz.

One can interpret the scattering data (both pure point and absolutely continuous) formally in terms of the determinant^{5} det(k^{2} − L) - which is reasonable in light of the fact that the zeroes of det(k^{2} − L) should correspond to bound states. I won’t detail this connection here.

If V = V (t, x) evolves by the KdV equation

V_{t} = V_{xxx} + 6V V_{x}

then we have a Lax pair with L as above, and P equal to

P = 4\frac{d^{3}}{dx^{3}} + 3(V \frac{d}{dx} + \frac{d}{dx}V).

One can interpret P as an approximation to 4iL^{3/2}. By arguing as before we see that the scattering transform again evolves like

a(t, k) = a(0, k); b(t, k) = b(0, k) exp(8ik^{3}t),

while the residues c(iw) grow exponentially:

c(t, iw) = c(0, iw) exp(8w^{3}t).

In principle this allows one to solve KdV using the Schrödinger scattering and inverse scattering transforms (which I haven’t detailed here).

Each bound state corresponds to a soliton solution of KdV, and multiple bound states correspond to multisoliton solutions. (Indeed, the scattering transform of a multisolution is trivial on the a.c. portion of the transform (i.e. a(k) = 1 and b(k) = 0 for all k), with the only interesting activity occuring in the bound states. The residues c(iw) encodes the position of the soliton corresponding to energy −w^{2}. Thus every potential is in some sense a nonlinear combination of a multisoliton and a “dispersive component”; the dispersive component evolves under KdV similar to an Airy evolution (or to a mKdV evolution, which it is indeed linked to via Miura), while the solitons just move at constant speed (+4w^{2}, to be precise).

5Of course, as L is infinite dimensional this determinant does not make sense directly, however by using gadgets such as zeta functions, heat kernels, or log-determinants one can make this formula rigorous.

There is a nice article by Segal on this topic.

10. Miura on the circle

This section is not directly related to the previous discussion, as it is on the circle, but shares much of the same spirit.

Let T = IR/Z be the unit circle, and let C_{0}^{∞}(T) be the smooth functions of mean zero on the circle. Consider the linearized Miura transform f → f′ on the circle. This maps C^{∞}(T) to C^{∞}(T), but the map is neither surjective nor injective because derivatives always have mean zero, and constants have vanishing derivative. However, by adding a constant and restricting the domain to C_{0}^{∞} we can recover a bijection:

Proposition 10.1. For any real V ∈ C^{∞}(T), there exists a unique potential F ∈ C_{0}^{∞}(T) and a unique real number c such that V = F′ + c.

Indeed, c is just the mean of V, and F is the primitive of V − c, normalized to have mean zero.

We now give the nonlinear analogue of this for the Miura transform V = F′ + F^{2}.

Proposition 10.2. For any real V ∈ C^{∞}(T), there exists a unique potential F ∈ C_{0}^{∞}(T) and a unique real number c such that V = F′ + F^{2} + c.

For F and V small one can verify this by solving iteratively the integral equation F = D^{−1}(V − F^{2}), where D^{−1} is the inverse of the operator f ↦ f′, thought of as a map from C_{0}^{∞}(T) to C_{0}^{∞}(T). (In other words, the Fourier multiplier with symbol 1/(2ik) for k ≠ 0 and 0 for k = 0). However, to obtain existence and uniqueness for large V and F one needs a more “elliptic” approach. Proof Fix V. Consider the self-adjoint elliptic operator L := −\frac{d^{2}}{dx^{2}} + V on the circle. From standard theory this operator has discrete spectrum, with eigenvalues λ_{1} ≤ λ_{2} ≤ … tending to infinity. (To see this, observe that for large enough C, L + C is positive definite and invertible, and the inverse (L + C)^{−1} is compact self-adjoint, hence has discrete spectrum tending to 0). Let λ_{1} be the smallest eigenvalue of L; it can be defined variationally as

λ_{1} := inf{E(u) : u ∈ C^{∞}(T); ∫ u^{2} = 1}

where the energy E(u) is defined by

E(u) := ⟨Lu, u⟩ = ∫ u_{x}^{2} + V u dx.

We first observe that if V = F′ + F^{2} + c for some F ∈ C_{0}^{∞}(T), then we can factorize

L − c = (\frac{d}{dx} + F)(−\frac{d}{dx} + F)

and hence

E(u) = c ∫ u^{2} dx + ∫ (−u_{x} + F u)^{2} dx.

In particular we have λ_{1} ≥ c. Indeed we have λ_{1} = c, as can be seen by setting u := e^{G}, where G is the primitive of F chosen so that u is L^{2}-normalized.

This shows that c is uniquely determined by V (being equal to λ_{1} of L). It also shows that F is uniquely determined by V, since if we had two choices F, F̃ for F then by the above argument we see that a primitive of F must equal a primitive of F̃, a contradiction.

Now to prove existence. Of course we set c = λ_{1}; in fact by subtracting λ_{1} from V if necessary we may assume that c = λ_{1} = 0. Let u be a L^{2}-normalized zero eigenfunction for L; thus E(u) = 0, and E(v) ≥ 0 for all other potentials v (not necessarily L^{2}-normalized). Also we have u_{xx} = V u. Without loss of generality we may assume that u is positive somewhere. We in fact claim that u cannot change sign; for otherwise max(u, 0) would have equal or lower energy than u, and then by smoothing out max(u, 0) slightly at the zeroes we contradict the minimality of u. Also u cannot have a repeated zero anywhere since the local well-posedness of the second-order ODE u_{xx} = V u would then force u to be zero everywhere. Thus u is always positive. If we then set e^{G} := u and F := G′ (i.e. F is the logarithmic derivative F = u_{x}/u of u) then we see from the ODE u_{xx} = V u that F has the desired properties.

This proposition allows one to connect mKdV and KdV on the torus. Suppose that V is a real potential on the torus obeying the KdV equation

V_{t} = V_{xxx} + 6V V_{x}.

The quantity λ_{1} = λ_{1}(\frac{−d^{2}}{dx^{2}} + V (t)) is an invariant of the KdV flow, so we write c = λ_{1}. Setting V (t) = F_{x}(t) + F (t)^{2} + c as per the above proposition, we see that F evolves by the drifting mKdV equation

F_{t} = F_{xxx} + 6F^{2}F_{x} + 6cF_{x}.

If we make the Gallilean change of variables F (x, t) := F̃(x + 6ct, t) then we can erase the drift term:

F̃_{t} = F̃_{xxx} + 6F̃^{2}F̃_{x}.

note that all the F̃ functions have mean zero, which is consistent with the fact that the mean is preserved by the mKdV flow.

Department of Mathematics, UCLA, Los Angeles, CA 90095

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