MODULATION STABILITY - A VERY SIMPLE EXAMPLE

TERENCE TAO

Abstract. We display the techniques used to establish modulational stability for an extremely simple ODE.

1. Introduction

Consider the ODE

∂_{t}ϕ(t) = i|ϕ(t)|^{2}ϕ(t)  (1)

for a complex function ϕ : R → C, with initial data ϕ(0) = 1 + ε for some small parameter 0 ≤ ε < 1. This ODE can of course be solved explicitly as

ϕ(t) = (1 + ε)e^{i(1+ε)^{2}t} = (1 + ε)e^{i(2ε+ε^{2})t}e^{it};  (2)

in the limiting case ε = 0 we have the “ground state” solution R_{0}(t) := e^{it}. Thus to first order in ε we have the expansion

ϕ(t) = R_{0}(t)(1 + ε + 2iεt + O_{t}(ε^{2}))  (3)

where the error O_{t}(ε^{2}) is bounded by a constant multiple of ε^{2} for each fixed t. Thus we see that the deviation between u and the ground state R_{0} grows linearly in t, to first order in ε. The error term begins to dominate the main term once t ≳ 1/ε. However, we can do better by exploiting the phase invariance

ϕ(t) ↦ e^{iθ}ϕ(t)  (4)

of where θ is an arbitrary real parameter. Applying this to the ground state R_{0}(t), we obtain the more general ground states

R_{θ}(t) = e^{iθ}R_{0}(t) = e^{i(t+θ)}

for any real θ. We then replace the ansatz

ϕ(t) = R_{0}(t) + small error

that underlies (3), with an ansatz

ϕ(t) = R_{θ}(t) + small error

where we allow θ = θ(εt) to vary slowly in t (and more precisely to depend on the slow time variable T = εt). For instance, from (2) we have

ϕ(t) = R_{2εt}(t)(1 + ε + O_{t}(ε^{2}));  (5)

we have thus absorbed the first-order-in-ε linear growth in t into the phase parameter θ, which in this case is 2εt. This ansatz is significantly more stable; the error term begins to become significant only when t ∼ ε^{−2}, not when t ∼ ε^{−1}.

1991 Mathematics Subject Classification. 42B15.

Now let us perform the exercise of rederiving all the above observations without using the explicit solution (2). The reason for doing this is that the techniques we develop in doing so can be applied to significantly more complicated equations than (1), such as the nonlinear Schrödinger equation, for which explicit solutions are usually unavailable.

We first work without introducing the slowly varying parameter θ. Since ϕ(0) = 1 + ε is close to R_{0}(0) = 1, and ϕ and R_{0} both solve (1), we expect ϕ(t) to stay close to R(t) for later times t. To make this intuition rigorous, the most obvious thing to do is to use the simple ansatz

ϕ(t) = R_{0}(t) + εw(t).

Thus we have w(0) = 1, and a brief calculation shows that w evolves by the equation

∂_{t}w = i(2w + \overline{w}e^{2it} + ε2|w|^{2}e^{it} + εw^{2}e^{−it} + ε^{2}|w|^{2}w).

We write this as

∂_{t}w = i(2w + \overline{w}e^{2it}) + O_{w}(ε).

Apart from the O_{w}(ε) error, this is a linear equation; we refer to this as the linearised equation of (1) around the ground state R_{0}. It is however a somewhat unpleasant equation to solve due to the time-dependent factor e^{2it} in the linear term; note that naive applications of Gronwall’s inequality seem to only be able to show that w grows at most exponentially in time, which is a lousy upper bound since (3) indicates that w should in fact exhibit linear growth.

One reason for the time-dependent factor in the equation for w is that the phase symmetry (4) affects w in a time-dependent way (due to the time-dependent nature of R_{0}). One can eliminate this by choosing a slightly better ansatz, namely

ϕ(t) = R_{0}(t)(1 + εw(t)).

Then we again have w(0) = 1, and another brief calculation shows that w evolves by the equation

∂_{t}w = i(w + \overline{w} + ε2|w|^{2} + εw^{2} + ε^{2}|w|^{2}w).

The key feature here is the disappearance of any explicit dependence of the equation on t (i.e. w evolves by an autonomous equation). This is closely related to the fact that the phase invariance (4) now affects w in a time-independent manner (the R_{0} factors cancel each other out). We write this as

∂_{t}w = Lw + O(ε|w|^{2} + ε^{2}|w|^{3})

where L : C → C is the real-linear operator Lw = i(w + \overline{w}). This operator has a double eigenvalue at 0, with an eigenfunction i and a generalised eigenfunction 1:

Li = 0; L1 = 2i.

To exploit this we express w in terms of the generalised eigenfunctions as w = u+iv, with u, v real, and obtain the equations

∂_{t}u = 0 + O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3})
∂_{t}v = 2u + O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3})

with initial data u(0) = 1, v(0) = 0. Ignoring all the ε terms, we now see the linear growth of w emerging: the real part of u stays constant at 1, while the imaginary part grows at the constant speed 2. (Compare this with (3).) Indeed, writing the above system of ODE in integral form as

u(t) = 1 + O(∫_{0}^{t} ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3} dt′)
v(t) = ∫_{0}^{t} 2u(t′) dt′ + O(∫_{0}^{t} ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3} dt′)

and using a continuity method, one can easily establish (for ε small) that

u(t) = O(1); v(t) = O(1 + t) for all t ≪ 1/ε^{1/3}.

One can even push this to t ≪ 1/ε^{1/2} by being more careful with the error terms, noting that the ε|v|^{2} term can be replaced by ε|u||v|. But one cannot get up to the expected time of 1/ε by this method. For that we need a more sophisticated ansatz. The key point is to eliminate the “secular^{1} mode” v, which causes the linear growth. To eliminate a degree of freedom, we must first introduce another; and the most natural degree of freedom to introduce arises from the phase invariance (4), by using the ansatz

ϕ(t) = R_{θ(εt)}(t)(1 + εw(t))  (6)

for some real function θ. We are now using three degrees of freedom (the real function θ and the complex function w) to describe a system (ϕ) with only two degrees of freedom. This gives us a “budget” of one degree of freedom to spend at some later point in the argument to eliminate some troublesome problem (in this case, the secular mode v). If we insert this ansatz into the equation we obtain

iθ′(εt) + ∂_{t}w = i(w + \overline{w} + ε2|w|^{2} + εw^{2} + ε^{2}|w|^{2}w)

or using the coordinates w = u + iv

∂_{t}u = O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3})
θ′(εt) + ∂_{t}v = 2u + O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3}).

Note that while the derivative of θ appears in this system, θ itself does not. This is ultimately due to the phase invariance (4), which can be viewed as a translation of θ by a constant while leaving w unchanged. This allows us to partially decouple θ from the system, as we shall shortly see.

The above system consists of only two real constraints on three unknowns θ, w, making the system under-determined. To make the system determined again one needs to add a constraint. One such constraint is to force v = 0, i.e. to force w to be real. (This is essentially viewing (6) in polar coordinates.) From the inverse function theorem we see that such a constraint is attainable, at least if ϕ(t) is close to the unit circle {z : |z| = 1} (there is of course an issue with the singularity at ϕ = 0). The system then simplifies to

∂_{t}u = O(ε|u|^{2} + ε^{2}|u|^{3})
θ′(εt) = 2u + O(ε|u|^{2} + ε^{2}|u|^{3}).

We also have initial data u(0) = 1, θ(0) = 0. Note the partial decoupling of θ; we can solve for u by itself first, and then once u is constructed one can then solve for θ′. The secular mode v has been eliminated, and now standard continuity methods allow us to obtain the preliminary bounds u(t) = O(1), θ(t) = O(t) for times 0 ≤ t ≪ ε^{−1}. In fact we have the more precise estimates u(t) = 1 + O(εt) and θ(εt) = 2εt + O(ε^{2}t^{2}) in this time range, as can be seen by bootstrapping the preliminary bounds (inserting them into the right-hand side of the system and seeing what that implies for the left-hand side), which already gives (5).

1These are modes which grow at slow time scales such as t ∼ 1/ε rather than fast time scales such as t ∼ 1. They are typically occur when error terms grow polynomially rather than exponentially.

Another way to spend the degree of freedom is to drop v as before, and solve for θ by dropping all terms of order ε or higher. Note when ε = 0 we obtain the modulation equations

∂_{t}u = 0
θ′ = 2u

and thus θ(εt) = 2εt. We now return to the ε ≠ 0 system with this particular choice of θ and get the system

∂_{t}u = O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3})
∂_{t}v = 2(u − 1) + O(ε|u|^{2} + ε|v|^{2} + ε^{2}|u|^{3} + ε^{2}|v|^{3}).

With this ansatz, we still have the secular mode v, but it is not as dangerous as before. Indeed a continuity method shows that u = 1 + O(εt) and v = O(εt + εt^{2}) for all 0 < t ≪ 1/ε^{1/2}; if we observe as before that the ε|v|^{2} term in the u evolution can in fact be replaced by ε|u||v| then one can stretch the same estimates up to time 0 < t ≪ 1/ε^{2/3}. This is not as good as what one gets when one decides to eliminate v completely, but is better than when we did not have θ at all.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

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