Weakly nonlinear Schrödinger equation with
random initial data
It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution of the nonlinear Schrödinger equation yields then a stochastic process stationary in and . If denotes the strength of the nonlinearity, we prove that the space-time covariance of has a limit as for , with fixed and sufficiently small. The limit agrees with the prediction from kinetic theory.
- 1 Introduction
- 2 Kinetic limit and main results
- 3 Duhamel expansion
- 4 Diagrammatic representation
- 5 Resolution of the momentum constraints
- 6 Expansion parameters and classification of graphs
- 7 Main lemmata
- 8 Partially paired and higher order graphs
- 9 Fully paired graphs
- 10 Completion of the proof of the main theorem
- A Nearest neighbor interactions
The nonlinear Schrödinger equation (NLS) governs the evolution of a complex valued wave field and reads
Here are the “hopping amplitudes” and we assume that they satisfy
has an exponentially decreasing upper bound.
We consider only the dispersive case . Usually the NLS is studied in the continuum setting, where is replaced by and the linear term is . It will become evident later on why for our purposes the spatial discretization is a necessity.
The NLS is a Hamiltonian system. To see this, we define the canonical degrees of freedom , , via . Their Hamiltonian function is obtained by substitution in
It is easy to check that the corresponding equations of motion,
are identical to the NLS. In particular, we conclude that the energy is conserved, for all . Also the -norm is conserved, in this context also referred to as particle number ,
In this work we are interested in random initial data. From a statistical physics point of view a very natural choice is to take the initial -field to be distributed according to a Gibbs measure for and , which physically means that the wave field is in thermal equilibrium. Somewhat formally the Gibbs measure is defined through
Here is the inverse temperature and the chemical potential. The partition function is a constant chosen so that (1.5) is a probability measure. To properly define the Gibbs measure one has to restrict (1.5) to some finite box , which yields a well-defined probability measure on . The Gibbs probability measure on is then obtained in the limit . The existence of this limit is a well-studied problem . If is sufficiently small and sufficiently negative, then the Gibbs measure exists. The random field , , distributed according to , is stationary with a rapid decay of correlations. It is also gauge invariant in the sense that in distribution for any .
Of course, -almost surely it holds and . Thus one has to define solutions for the NLS with initial data of infinite energy. This has been accomplished for standard anharmonic Hamiltonian systems by Lanford, Lebowitz, and Lieb , who prove existence and uniqueness under a suitable growth condition at infinity for the initial data. These arguments extend to the Hamiltonian system (1.3). It remains to prove that the so-defined infinite volume dynamics is well approximated by the finite volume dynamics with periodic boundary conditions. Very likely such a result can be achieved using the methods developed in . For our purposes it is more convenient to circumvent the issue by proving estimates which are uniform in the volume.
Let us briefly comment why the more conventional continuum NLS,
poses additional difficulties. The Gibbs measure at finite volume is a perturbed Gaussian measure which is singular at short distances. Thus the construction of the dynamics requires an effort. Furthermore, the limit is a fundamental problem of constructive quantum field theory and is known to be difficult . To establish the existence of the dynamics for such singular initial data has not even been attempted.
In the present context the most basic quantity is the stationary covariance
where denotes expectation with respect to . The existence of such a function follows from the translation invariance of the measure, and one would like to know its qualitative dependence on . For deterministic infinitely extended Hamiltonian systems, such as the NLS, establishing the qualitative behavior of equilibrium time correlations is known to be an extremely difficult problem with very few results available, despite intense efforts. For linear systems one has an explicit solution in Fourier space, see below. But already for completely integrable systems, like the Toda chain, not much is known about time correlations in thermal equilibrium.
It is instructive first to discuss the linear case, , for which purpose we introduce Fourier transforms. For let us denote its Fourier transform by
, and the inverse Fourier transform by
with , a parametrization of the -dimensional torus. (We will use arithmetic relations on . These are defined using the arithmetic induced on the torus via its definition as equivalence classes , i.e., by using “periodic boundary conditions”.) In particular, we set
The function is the dispersion relation of our discretized linear Schrödinger equation. It follows from the assumptions on that
and its periodic extension is a real analytic function.
In Fourier space the energy is given by
where is a formal Dirac -function, used here to simplify the notation for the convolution integral. Clearly, . The NLS after Fourier transform reads
For , is a Gaussian measure with mean zero and covariance
provided . Under our assumptions on the Gaussian field has exponential mixing. For the time-dependent equilibrium covariance one obtains
Clearly, is a solution of the linear wave equation for exponentially localized initial data and thus spreads dispersively.
If , as general heuristics the nonlinearity should induce an exponential damping of . The physical picture is based on excitations of wave modes which interact weakly and are damped through collisions. Approximate theories have been developed in the context of phonon physics and wave turbulence, see e.g. [10, 23]. To mathematically establish such a time-decay is completely out of reach, at present, whatever the choice of the nonlinear wave equation.
To make some progress we will investigate here the regime of small nonlinearity, . The idea is not to aim for results which are valid globally in time, but rather to consider the first time scale on which the effect of the nonlinearity becomes visible. For small the rate of collision for two resonant waves is of order . Therefore, the nonlinearity is expected to show up on a time scale . This suggests to study the limit
Note that the location is not scaled. For this limit to exist, one has to remove the oscillating phase resulting from (1.14), which on the speeded-up time scale is rapidly oscillating, of order . In fact, a second rapidly oscillating phase of order will show up, which also has to be removed. Under suitable conditions on , we will prove that , with the removals just mentioned, has a limit for , at least for with some suitable . The limit function indeed exhibits exponential damping.
A similar result has been obtained a long time ago for a system of hard spheres in equilibrium and at low, but fixed, density . There the small parameter is the density rather than the strength of the nonlinearity. But the over-all philosophy is the same. To establish the decay of time-correlations in equilibrium at a fixed low density is an apparently very hard problem. Therefore, one looks for the first time scale on which the collisions between hard spheres have a visible effect. By fiat, hard spheres remain well localized in space, and on the time scale of interest only a finite number of collisions per particle are taken into account. In contrast, waves tend to delocalize through collisions. This is the reason why the problem under study has remained open. Our resolution uses techniques totally different from .
The limit , with fixed, together with a possible rescaling of space by a factor , is called kinetic limit, because the limit object is governed by a kinetic type transport equation. Formal derivations are discussed extensively in the literature, e.g., see [12, 18]. On the mathematical side, Erdős and Yau  study in great detail the linear Schrödinger equation with a random potential, extended to even longer time scales in [6, 7]. The discretized wave equation with a random index of refraction is covered in . For nonlinear wave equations the only related study is by Benedetto et al.  on the dynamics of weakly interacting quantum particles. They transform to multipoint Wigner functions, which leads to an expansion somewhat different from the one used here. We refer to  for a comparison. As in our contribution, Benedetto et al. have to analyze the asymptotics of high-dimensional oscillatory integrals. But in contrast, they have no control on the error term in the expansion.
Before closing the introduction, we owe the reader some explanations why a seemingly perturbative result requires so many pages for its proof. From the solution to (1.1) one can regard as some functional of the initial field ,
For given it depends only very little on those ’s for which . To make progress it seems necessary to first average the initial conditions over so that subsequently one can control the limit with , . Such an average can be accomplished by writing as a power series in , which is done through the Duhamel formula. For any we write
Here and , , . Using the product rule and the equations of motion (1) yields a formula relating the :th moment at time to the time-integral of a sum over :th moments at time . Iterating this equation leads to a (formal) series representation
where is a sum/integral over monomials of order in and . Since each time-derivative increases the degree of the monomial by two, we have
The first difficulty arises from the fact that the sum in (1.19) does not converge absolutely for any . Very roughly, is a sum of terms of equal size. The iterated time-integration yields a factor . However, for the approximately Gaussian average the :th moment grows also as . To be able to proceed one has to stop the series expansion at some large which depends on . A similar situation was encountered by Erdős and Yau  in their study of the Schrödinger equation with a weak random potential. We will use the powerful Erdős-Yau techniques as a guideline for handling the series in (1.19).
The stopping of the series expansion will leave a remainder term containing the full original time-evolution. Erdős and Yau control the error term in essence by unitarity of the time-evolution. For the NLS mere conservation of will not suffice. Instead, we use stationarity of . In wave turbulence  one is also interested in non-stationary initial measures, e.g., in Gaussian measures with a covariance different from . For such initial data we have no idea how to control the error term, while other parts of our proof apply unaltered.
The central difficulty resides in which is a sum of rather explicit, but high-dimensional, dimension , oscillatory integrals. On top, because of the -function in (1), the integrand is restricted to a non-trivial linear subspace. In the limit , , , only a few oscillatory integrals have a non-zero limit. Summing up these leading oscillatory integrals results in the anticipated exponential damping. The major task of our paper is to discover an iterative structure in all remaining oscillatory integrals, in a way which allows for an estimate in terms of a few basic “motives”. Each of these subleading integrals is shown to contain at least one motive whose appearance leads to an extra fractional power of , thereby ensuring a zero limit.
In Section 2.1 we first give the mathematical definition of the above system in finite volume, and state in Section 2.2 the assumptions and main results. Their connection to kinetic theory is discussed in Section 2.3. The proof of the main result is contained in the remaining sections: we derive a suitable time-dependent perturbation expansion in Section 3, and develop a graphical language to describe the large, but finite, number of terms in the expansion in Section 4. The analysis of the oscillatory integrals in the expansion is contained in Sections 5–9. More detailed outline of the technical structure of the proof can be found in Section 3.1. The estimates are collected together and the limit of the non-zero terms is computed in Section 10 where we complete the proof of the main theorem. In an Appendix, we show that the standard nearest neighbor couplings in dimensions lead to dispersion relations satisfying all assumptions of the main theorem.
Acknowledgments. We would like to thank László Erdős and Horng-Tzer Yau for many illuminating discussions on the subject. The research of J. Lukkarinen was supported by the Academy of Finland.
2 Kinetic limit and main results
2.1 Finite volume dynamics
To properly define expectations such as (1.7), one has to go through a finite volume construction, which will be specified in this subsection.
the dimension an arbitrary positive integer. We apply periodic boundary conditions on , and let for all . Fourier transform of is denoted by , with the dual lattice and with
for all (or for all , which yields the periodic extension of ). The inverse transform is given by
where . For all , it holds . The arithmetic operations on are done periodically, identifying it as a parametrization of , the cyclic group of elements (for instance, for , we have then and .) Similarly, is identified as a subset of the -torus .
We will use the short-hand notations
as well as the similar but unrelated notation for “regularized” absolute values
Let us also denote the limit by . Let be defined as in (1.10). For the finite volume, we introduce the periodized through
Clearly, and for all .
After these preparations, we define the finite volume Hamiltonian for by
where and is the following discrete -function
Here denotes a generic characteristic function: , if the condition is true, and otherwise. for all , with and denoting the -norm.
Introducing, as before, the canonical conjugate pair through , and then applying the evolution equations associated to , we find that satisfies the finite volume discrete NLS
The Fourier-transform satisfies the evolution equation
The evolution equations have a continuously differentiable solution for all and for any given initial conditions , which follows by a standard fixed point argument and the conservation laws stated below. The energy is naturally conserved by the time-evolution. In addition, for all ,
The right hand side sums to zero if we sum over all . Therefore, for ,
and thus also is a constant of motion.
The initial field is taken to be distributed according to the finite volume Gibbs measure as explained in the introduction. We assume that its parameters are fixed to some values satisfying and , and we drop the dependence on these parameters from the notation. Then the Gibbs measure is
Expectation values with respect to the finite volume, perturbed measure are denoted by . Taking the limits and leads to a Gaussian measure. It is defined via its covariance function which has a Fourier transform
We denote expectations over this Gaussian measure by . Note that by the translation invariance of the finite volume Gibbs measure, there always exists a function such that for all ,
Since the energy and norm are conserved, the Gibbs measure is time stationary. In other words, for all and any integrable
In addition, since the dynamics and the Gibbs measure are invariant under periodic translations of , under the stochastic process is stationary jointly in space and time.
2.2 Main results
We have to impose two types of assumptions. Those in Assumption 2.2 are conditions on the dispersion relation . Assumption 2.1 is concerned with a specific form of the clustering of the Gibbs measure. In each case we comment on their current status.
Assumption 2.1 (Equilibrium correlations)
Let and be given. We take the initial conditions to be distributed according to the Gibbs measure which is assumed to be -clustering in the following sense: We assume that there exists and , independent of , such that for and all one has the following bound for the fully truncated correlation functions (i.e., cumulants)
where , . We also assume a comparable convergence of the two-point correlation functions for ,
In the present proof, valid for , we do not use the full strength of the bound in (2.18), namely, we could omit the prefactor . However, the prefactor could be needed in any proof which concerns . In contrast, we do make use of the prefactor in (2.19). The second condition can equivalently be recast in terms of as
Technically, Assumption 2.1 refers to the clustering of a weakly coupled massive two-component -theory. Such problems have a long tradition in equilibrium statistical mechanics and are handled through cluster expansions, e.g., see [19, 20]. The difficulty with Assumption 2.1 resides in the precise - and -dependence of the bounds. Motivated by our work, the issue was reinvestigated for the equilibrium measure (2.14) in the contribution of Abdesselam, Procacci, and Scoppola , in which they prove Assumption 2.1 for hopping amplitudes of finite range and with zero boundary conditions, i.e., setting for . The authors ensure us that their results remain valid also for periodic boundary conditions, thereby establishing Assumption 2.1 for a large class of hopping amplitudes.
For the main theorem we will need properties of the linear dynamics, , which can be thought of as implicit conditions on .
Assumption 2.2 (Dispersion relation)
Suppose , and satisfies all of the following:
The periodic extension of is real-analytic and .
(-dispersivity). Let us consider the free propagator
We assume that there are such that for all ,
(constructive interference). There exists a set consisting of a union of a finite number of closed, one-dimensional, smooth submanifolds, and a constant such that for all , , and ,
where is the distance (with respect to the standard metric on the -torus, ) of from .
(crossing bounds). Define for , , and ,
We assume that there is a measurable function so that constants , , for the following bounds can be found.
For any , , , and , the following bounds are satisfied:
For all we have
and if also , , , and , and we denote , then
where is defined by
We prove in Appendix A that the nearest neighbor interactions satisfy all of the above assumptions for , if we use , , and the function
with a certain constant depending only on and . Presumably a larger class of ’s could be covered, but this needs a separate investigation.
The estimates in Appendix A in fact imply that also for the dispersion relation of the nearest neighbor interactions satisfies assumptions DR1–DR3. However, even if also DR4 could be checked, this would not be sufficient to generalize the result to since is used to facilitate the analysis of constructive interference effects in Sec. 8.2. The present estimates require that the co-dimension of the bad set is at least three which for would allow only a finite collection of bad points. As we have no examples of such dispersion relations, we have assumed throughout the proof. Nevertheless, by more careful analysis of the constructive interference effects we expect the results to generalize to interactions in . Again, this remains a topic for further investigation. ∎
We wish to inspect the decay of the space-time covariance on the kinetic time scale . More precisely, given some test-functions , with a compact support, we study the expectation of a quadratic form,
where , , and is obtained from similarly. Since we assume the test-functions to have a compact support, and are, in fact, independent of for all large enough lattice sizes. In addition, , and for all . To get a finite limit, it will be necessary to cancel the rapidly oscillating factors. To this end, let us define
Differentiating the expectation value and applying Assumption 2.1 shows that
Then the task is to control the limit of the quadratic form
As discussed in the introduction, we expect that the infinite volume limit of exists, but since proving this would have been a diversion from our main results, we have stated the main theorem in a form which does not need this property. Clearly, if the limit does exist, then (2.36) implies the stronger result
Independently of the convergence issue, the theorem implies that, if is sufficiently large, is small enough, and is not too large, we can approximate
where the “renormalized dispersion relation” is given by
We point out that , as by explicit computation
(We prove in Section 7.4 that the integral in (2.4) and the positive measure in (2.2) are well-defined for any satisfying items DR1 and DR2 of Assumption 2.2.) If , then the term yields the exponential damping in , both forward and backwards in time, and if for all , then on the kinetic scale the covariance has an exponential bound .
The restriction to finite times with appears artificial since the limit equation is obviously well-defined for all . In fact, as can be inferred from the proof given in Sec. 10, if we collect only the terms having a non-zero contribution to the limit, the expansion is not restricted by such a finite radius of convergence. However, the bounds used to control the remaining terms are not summable beyond certain radius. As a comparison, let us observe that even the perturbation expansions of solutions to nonlinear kinetic equations, such as (2.45) below, have generically only a finite radius of convergence. ∎
2.3 Link to kinetic theory
To briefly explain the connection of our result to the kinetic theory for weakly nonlinear wave equations, we assume that the initial data , , are distributed according to a Gaussian measure, , with mean zero and covariance
is stationary under the dynamics, but nonstationary for . Since translation and gauge invariance are preserved in time, necessarily
The central claim of kinetic theory is the existence of the limit
where is the solution of the spatially homogeneous kinetic equation
with initial conditions