The effective equation method
In this chapter we present a general method of constructing the effective equation which describes the behaviour of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behaviour of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three– and four–wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. In the case of the NLS equation we use next some heuristic approximation from the arsenal of wave turbulence to show that under the iterated limit “the volume goes to infinity”, taken after the limit “the amplitude of oscillations goes to zero”, the energy spectrum of solutions for the effective equation is described by a Zakharov-type kinetic equation. Evoking the Zakharov ansatz we show that stationary in time and homogeneous in space solutions for the latter equation have a power law form. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanology.
It is well known that solutions of linear evolution PDEs in finite volume are superpositions of normal modes of oscillations (in most cases of interest these are the Fourier modes). When a nonlinearity is added as a perturbation, different modes start to interact and the solutions of the equation can be approximated by suitable power series expansions, provided that the nonlinearity is sufficiently small (or, in other words, the PDE is weakly nonlinear). In such cases, the equation can be written as
where is the linear operator, denotes the nonlinearity and is a small parameter, . The equation may contain a stochastic force, and in that case it reads
(the scaling of the random force by the factor is the most natural, see below). We will show that the limiting, as , exchange of energy between the modes may be described by replacing the original system with a suitable effective equation. This result may be regarded as a PDE-version of the Bogolyubov averaging principle (see BogMitr ()) which implies a similar property for distribution of energy between the oscillating modes for small-amplitude oscillations in finite-dimensional nonlinear systems.
The mentioned above convergence that holds as and various properties of the corresponding effective equations have been rigorously established (see K10 (); Kuk11 (); KM13 (); KM13KZ (); bplane (); wnpdes () and the discussion in wnpdes ()). The treatments of the deterministic and stochastic equations are similar, but the results, obtained in the stochastic case, are significantly stronger: while the deterministic effective equation controls the dynamics only on time intervals of order , in the presence of stochastic forcing the corresponding effective equation also approximates the stationary measure for the original equation, thus controlling the asymptotical in time behaviour of solutions when . Moreover, in the absence of forcing we only get information concerning the exchange of energy between the modes, whereas in the stochastically forced case, the stationary measure for the effective equation controls both the energies and the phases of the normal modes of solutions.
Below we explain how to construct the effective equations for eq. (1) and eq. (2) from the resonant terms of the nonlinearities. We will discuss two examples: the nonlinear Schrödinger equation and the Charney-Hasegawa-Mima equation on the plane. These two equations display completely different types of energy exchange between modes, and we will explain why this happens.
0.2 How to construct the effective equation
We consider hamiltonian PDEs, whose linear parts have imaginary pure point spectra and are diagonal in Fourier modes. Written in terms of the complex Fourier coefficients (also called waves), the equations which we study read
Here are real numbers and is a polynomial nonlinearity in of certain order , of the form
where are some complex coefficients, is the complex conjugate of and
We always assume that “the nonlinearity does not pump energy in the system”:
(in most case of interest the l.h.s. vanishes).
The quantities , and are called, respectively, the wave action, wave energy and wave phase. The relation between and , i.e. the function , is called the dispersion relation, or dispersion function.
The weakly nonlinear regime corresponds to solutions of small amplitude . We will study it in the presence of damping and, possibly, a random force, whose magnitude is controlled by another parameter, call it . So, instead of (3), we will consider
where controls the damping term, controls the forcing and the parameter is introduced to consider at the same time the forced and non-forced cases. The ’s are complex white noises, independent from each other.111 That is, , , where are standard independent real Wiener processes. The factors and in front of the damping and the dissipation are so chosen that, in the limit of , the solutions stay of order one, uniformly in .
Note that, while controls the size of the solutions, is the time-scale on which the forcing acts significantly, as it is the time needed for the standard deviations of the processes to become of order one. If and the system (6) is deterministic, still its time-scale is since, as we explain below, its solutions with initial data of order one stay of order one for , while for much bigger values of time they are very small since in view of (5) their -norms satisfy
We will consider the regime
(where is the degree of ), and study solutions of the equation with given initial conditions on the time-scale , examining them under the limit . Passing to the slow time (so that time corresponds to of order 1), eq. (6) becomes
where the upper dot stands for .
We claim that, in the limit when (or, equivalently, ) goes to zero, the distribution of the energies on times of order one is described by an effective equation whose nonlinearity is constituted by resonant terms of the nonlinearity (see (13) below).
It is easier to understand the role of resonances and the form of the effective equation by passing to the interaction representation ( cf. BogMitr (); ZLF92 (); Naz ()), i.e., by performing the time-dependent change of variables from to
which transforms (8) to
where denotes the nonlinearity, written in the -variables. That is
Noting that the collection of the processes is another set of standard independent complex white noises, we re-write eq. (10) as
In the sum defining , the terms for which the resonance conditions
are satisfied (called the resonant terms) under the limit behave completely differently from others terms (called the nonresonant terms). Namely, the nonresonant terms oscillate faster and faster, whereas the resonant terms do not. We will say that a set of -vectors forms a resonance if relations (12) are satisfied, if , and the set does not equal the set . The collection of all resonances is called the resonant set.
In the spirit of the finite-dimensional averaging, following the Bogolyubov averaging principle (see BogMitr (), the behaviour of solutions of (11) under the limit is obtained by replacing the nonlinearity with its time average, i.e. with
Since for any real number we have
then only the resonant terms survive in the averaged nonlinearity. We write their sum as
This suggests to take for the effective equation the following system:
Indeed, it is proved in wnpdes () (also see K10 (); Kuk11 (); KM13 ()) that, if the original equation (6) is well posed on time intervals , then eq. (13) describes the limiting behaviour of the variables (and, as well, the distribution of energy since ) in the time-scale , for any sufficiently regular initial data. This holds both in the presence and in the absence of the random forcing (i.e., both for and ). Moreover, in the forced case we also control the limiting behaviour of the stationary solutions for eq. (6). So if the equation (6) and the effective equation both are mixing, then we control the behaviour of all solutions for (6) under the iterated limit Remarkably, in this case the effective equation describes not only the limiting behaviour of the actions, but also that of the angles. I.e., it completely controls the limiting distribution of solutions. So if is a functional on the space of sequences , satisfying some mild restriction on its growth as the norm of goes to infinity, and is any solution for (6), then
0.3 Structure of resonances
We intend to use the effective equations as a tool to investigate the energy transport in different physically relevant PDEs. We will show that the limiting, as , energy transport for any specific equation depends on the structure of the resonances (which, in turn, is determined by the form of the dispersion function ).
Three possibilities can occur:
1) The resonant set is empty. Then if the degree of the nonlinearity is even, the effective equation is linear. If is odd, the equation may contain nonlinear integrable terms of the form . But these terms do not contribute to the dynamics of the wave actions. So in any case different modes do not exchange energy, and no energy transport to high frequencies occurs.
Now assume that the resonant set is not empty. We say that integer vectors are equivalent if there exist vectors , such that the relations (12) hold. This equivalence divides to clusters, formed by elements which can be joined by chains of equivalences (see Kar (); Kar94 () for a discussion of the role of resonant clusters in weak turbulence).
The two remaining cases are:
2) All resonances are connected, so the whole is a single cluster. In this case, in the limit when the volume of the space-domain goes to infinity, under some additional assumptions a new type of kinetic equation can be derived, the energy transport takes place and power law stationary spectra, which depend only on the form of the dissipation, can be obtained.
3) All resonances are divided to non intersecting clusters. Now the energy transfer should be studied separately within each cluster. If sizes of the clusters are bounded, then no energy transport to high frequencies occurs.
See Kuk11 (); hgdcds () for the case 1), KM13 (); KM13KZ () for the stochastic case 2) and hgjdde () for the deterministic case, and see bplane () for the stochastic case 3). See wnpdes () for discussion and for theorems, applicable in all three cases, deterministic and stochastic.
Note that many examples of systems which fall to type 2) are given by equations (6) with completely resonant spectra , i.e. with spectra of the form , where are integers. Averaging theorems for completely resonant deterministic equations (6) with were discussed in FGH (); GG12 (); hgjdde (); also see wnpdes ().
Below we discuss examples for the case 2) when all resonances are connected (Section 0.4), and for the case 3) when the resonances make non intersecting finite clusters (Section 0.5). For more examples of systems of types 2) and 3) see Kar ().
0.3.1 The equations
for , where the parameter is introduced in order to control different scaling for solution as the size of the torus varies.222More exactly, (14) is the damped/driven cubic NLS equation. See ZLF92 (); Naz () for the non-perturbed NLS equations. The dissipation is a linear operator, diagonal in the exponential base of functions on
As before, by we denote the the Fourier coefficients of :
If , the resonance condition (12) takes the form
All solutions for this system are such that , or . So in this case the resonant set is empty, and no energy cascade to high frequencies happens when . This is well known.
Now consider a higher-dimensional NLS equation, write it in the Fourier variables , , and pass to the slow time . Then, if the forcing and the dissipation are chosen in accordance with the prescription of the previous section (cf. (6)), the equation reads
Here are the eigenvalues of the dissipation operator. The eigenvalues of the operator , call them , follow the dispersion relation
Below we will see that if , then the whole forms a single cluster, so the equation fits the case 2).
An interesting example of the case 3) of isolated clusters is provided by the Charney–Hasegawa–Mima (CHM) equation on the plane (see bplane () and see ZLF92 (); Naz () for this equation with ), which we write as
Here the constant is called the Froude number and denotes the Jacobian determinant of the vector . The space-domain is a strip of horizontal size and vertical size one, under double periodic boundary conditions, i.e.,
Again we pass to the Fourier modes333Note that, due to the fact that the function is real, . and to the slow time to re-write the equation as
where , and the dispersion function has the form
while for CHM the effective equation is the system
It is clear that the behaviour of solutions for eqs. (20)-(21) is dictated by the structure of resonances since they determine the surviving terms of the nonlinearity. The geometric properties of the resonant set for the higher dimensional NLS equations are described in the following section, whereas the resonances for CHM are discussed in Section 0.3.3.
0.3.2 Structure of resonances for the NLS equation
In the case of 2 NLS equation each resonance is formed by four points of which have a simple geometrical characterization: they form the vertices of a rectangle. Indeed, if a quadruple satisfies (12) with , then on account of the second relation we have . So the polygonal is a parallelogram. Substituting in the first relation and using (17) we get
I.e., is orthogonal to . So is a rectangle in .
It is easy to see that for any vectors there is an integer rectangle of the form . So the equivalence, defined by the clusters of the 2d NLS equation makes a single cluster, and the equation falls in the case 2). A graphical illustration of some resonant quadruples and their connections in is displayed in fig. 1.
Similar all higher-dimensional NLS equations fall in case 2).
0.3.3 Structure of resonances for CHM
The structure of resonances for the CHM equation depends on the shape-factor . Below we discuss it, supposing for simplicity that the Froude number is kept fixed (see bplane () for the general case). We start with explicitly rewriting for the present case the resonance condition (12) (recall that , and ):
Solutions to these equations can be divided to different classes, according to how many numbers among and are non-zero:
If all three are zero, then any triad , , constitutes a solution. As vanish in this case (see (21)), such triads do not form a resonance.
If only one number is different from zero, then (22) admits no solution.
If only one among and vanishes, two subcases arise (as and play an exchangeable role):
if (which implies ), then and there are two solutions, one for , , and another for ;
if (which implies ), then and again there are two solutions, one for , , and another for .
All three are different from zero. This is the only case when the solutions depend on . Indeed, let us fix a triad and look for which values of it constitutes a resonance. The second line of (22) may be re-written as
where and are polynomials. In particular, . In view of the inequality , valid for nonvanishing and ,
where the use is made of the first line of (22). This shows that the second order polynomial in at the l.h.s. of (23) is nontrivial. So for any fixed triad , where , and are nonzero, relation (23) holds for at most two nonnegative values of .
The different types of resonances are represented in Fig. 2, where only the points above the horizontal axis are displayed (cf. footnote 3). There the resonances of type (iii) (which we will call standard resonances) are connected by solid lines: they form triangles symmetric with respect to the vertical axis , in which each point is connected with and , for any . The resonances of type (iv) (which we will call non-standard) are displayed as dashed lines: they constitute triangles in which none of the vertices lies on the vertical axis.
Since each non-standard resonance appears only for at most two values of , then by removing (at most) a countable set of ’s we kill all of them. Let us denote this removed set . The set of remaining values of , for which no non-standard resonance appear, can be regarded as “typical”. Accordingly we will refer to as typical values of (or as to the typical case). Below in Section 0.5 we show that in the typical case all resonances are divided to non-intersecting clusters of size at most 3, thus fitting the third option, considered in Section 0.3.
0.4 NLS: the power-law energy spectrum
Effective equation (20) for NLS (which, as we have seen, determines the energy spectrum) is not easy to handle since its completely connected resonance structure (see Section 0.3.2) makes impossible to split it to simpler subsystems (on contrary to the CHM equation, see Section 0.5). We present here a way to investigate the behaviour of solutions of (20) when the size of the box goes to infinity, based on certain traditional for the wave turbulence heuristic approximation (see Naz (); ZLF92 (); New ()), following our work KM13KZ (). This will lead us to a wave kinetic (WK) equation of the form, usually encountered in the wave turbulence. The treatment follows closely the paper KM13KZ (), to which the reader can refer for further details.
0.4.1 The limit
From the point of view of mathematics, the limit when the size of the torus tends to infinity in equation (20) presents a serious problem. In particular, for what concerns the definition of a possible limiting stochastic equation. Instead of trying to resolve this difficulty, for any finite we will study the expectations of the actions of solutions for the corresponding equation (the function is called called the wave-action spectrum), and then pass to the limit as only for these quantities. 444In the case of the non-forced equation the expectations should be taken with respect to the distribution of the initial data, while for the forced equation – with respect to the distribution of the forcing (and, possibly, of the initial data).
We fix and, by making use of Ito’s formula for , get from (20) that
Now we pass to the expected values, and define the moment of of order as
where stands for the expected value at time , i.e., for any measurable function . Then from the system (24) we get
This system is not closed since it involves the moments of order 4. By applying again Ito’s formula, we can express the time derivative of moments of any order as a function of the moments of order and those of order . The coupled system, containing the equations for all moments, is called the chain of moments equation (see MY ()).555Notice that, since the equation which we consider has a cubic nonlinearity, equations for moments of even order are decoupled from those for moments of odd order. Systems of this kind are usually treated by approximating moments of high order by suitable functions of lower order moments in order to get a closed system of equations. We will show that if the quasi-stationary and quasi-Gaussian approximations (see below) are chosen to close the system of moment equations, then under the limit we recover a modified version of the WK equation.
To study the sum in the r.h.s. of (26), we notice that if the Krönecker deltas are different from zero because equals to one vector among and is equal to another, then the moment is real and does not contribute to the sum. So we may assume that , . In this case we calculate the fourth order moments in the r.h.s. of (26) through Ito’s formula (see KM13KZ ()) and get
We make now the first approximation by neglecting the term containing the time derivative at the l.h.s. of (27). This can be justified, if is large enough, by the quasi-stationary approximation (cf. Section 2.1.3 in ZLF92 ()). Namely, let us write equation (27) as
Notice that since all ’s are positive, then the linear differential equation in the l.h.s. is exponentially stable. Assume that as a function of is almost constant during time-intervals, sufficient for relaxation of the differential equation. Then
We insert this in (26) and get
We then apply the second approximation, generally accepted in the WT (see ZLF92 (); Fal (); Naz (); New ()) which enables us to transform the previous relation to a closed equation for the second order moments. This consists in the quasi–Gaussian approximation, i.e., in the assumption that the higher-order moments (25) can be approximated by polynomials of the second-order moments, as if the random variables were independent complex Gaussian variables. In particular,
At this point we pass in equation (28), closed using the relation (29), to the limit . This can be done by approximating sums with integrals if, instead of parametrising the modes by integer vectors , we parametrise them by vectors from the shrunk lattice . Accordingly we define
and note that since the restriction, imposed by the Krönecker deltas, is homogeneous, then it does not change under this re-parametrisation. Abusing notation, we will drop the tildes in the rest of the Section, but will use the parametrisation by points of .
We denote by the sum, given by the second and third lines of (28), written in the new parametrisation, and note that it splits into a finite number of sums like
Here is a manifold in , defined as
where stands for one among the vectors , and – for a permutation of the set .666Note that the relations, defining are not independent. It is easy to see that since every is a regular function, then when passing from the sums to integrals in the limit , each term as a function of becomes proportional to , where is the dimension of the manifold . A detailed analysis of all cases shows that the terms of the highest order in in the integral correspond to terms of the form
in the sum , where . Denote
This is a manifold of dimension , smooth outside the origin. The latter lies outside if , and is a singular point of if .
As shown in KM13KZ (), in the limit the sum can be approximated by the integral
where is a certain function on , smooth outside zero, such that
where is the volume of the -ball in .
Finally, we define
(so that and as goes to infinity), choose