Renormalization methods for higher order differential equationscaginalp/pub116.pdf · 2015-04-30 · Keywords: renormalization group, higher order differential equations, phase ﬁeld,

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 47 (2014) 315004 (33pp) doi:10.1088/1751-8113/47/31/315004

Renormalization methods for higher orderdifferential equations

Gunduz Caginalp1 and Emre Esenturk2

1 Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, USA2 Pohang Mathematics Institute, Pohang University of Science and Technology,Pohang, Korea

E-mail: [email protected] and [email protected]

Received 16 November 2013, revised 28 May 2014Accepted for publication 11 June 2014Published 23 July 2014

AbstractWe adapt methodology of statistical mechanics and quantum field theoryto approximate solutions to an arbitrary order ordinary differential equationboundary value problem by a second-order equation. In particular, we studyequations involving the derivative of a double-well potential such as u − u3 or− u + 2u3. Using momentum (Fourier) space variables we average over shortlength scales and demonstrate that the higher order derivatives can be neglectedwithin the first cumulant approximation, once length is properly renormalized,yielding an approximation to solutions of the higher order equation fromthe second order. The results are confirmed using numerical computations.Additional numerics confirm that the main role of the higher order derivativesis in rescaling the length.

Keywords: renormalization group, higher order differential equations, phasefield, transition layer, scalingPACS numbers: 05.10.Cc, 02.30.Hq, 02.30.Jr

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1. Introduction

Higher order ordinary and partial differential equations (DEs) arise in the context of someimportant issues in materials science and statistical mechanics. For example, in the phase fieldapproach to free boundaries, retaining wave modes up through k2, in the Fourier expansion,leads to the usual second-order phase field equations while retaining higher wave modes, leadsto higher order equations. It was shown [4, 8] in some early papers that higher order equationscould to be useful in understanding the detailed features of an interface such as anisotropy.

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J. Phys. A: Math. Theor. 47 (2014) 315004 G Caginalp and E Esenturk

For instance, in order to retain the 6-fold anisotropy, one needs to retain terms up to k6,leading to sixth-order DEs, since the anisotropy is largely ‘averaged out’ in the second-orderDEs [7].

Second-order DEs with a term arising from the derivative of a double-well have beenstudied extensively in terms of existence, uniqueness and monotonicity. In higher orders fewrigorous results are available such as [21] in which existence of traveling waves was established.In general, the addition of the higher order terms can dramatically change the nature of thesolution. Even if they are small they could represent a singular perturbation. Neverthelessunder some conditions, it was shown that this was not the case for small coefficients. Fourthand sixth-order equations have been studied in other contexts in [9, 22, 29, 30, 32, 35, 36].

Overall, theoretical methods for higher order DEs are not nearly as well-developed as theyare for the second-order equations. Computations that are routine for second-order equationsare also more difficult. Hence, it would be useful to develop tools to approximate solutions tosuch higher order DEs with those of second order.

The general ideas of renormalization group (RG) methods, which are closely related tomulti-scaling methods, have revolutionized our understanding of critical phenomena and otherareas of statistical mechanics, as well as quantum field theory [1, 10, 11, 16]. In a philosophicalsense there are a number of issues involving DEs that are related. One example is the blow-upor decay of solutions to DEs [3, 13, 17, 18, 26, 33]. Here, one has an ‘asymptotic self-similarity’that is reminiscent of the divergence of key thermodynamic quantities in statistical mechanics.Other recent applications of RG involve dynamical systems [27], difference systems [28],boundary flows [23] and other related problems [2, 12, 31].

In this paper we present a formal derivation, using RG and scaling methods, that showshigher order DEs can be approximated by suitably rescaled second-order equations. Thisformal derivation is supported by numerical computations that quantify the difference betweenan accurate solution to the sixth order computed directly, and the approximate solution derivedusing RG and scaling arguments. These comparisons are presented in section 5.

We develop an idea of using momentum (Fourier) space RG in order to approximatesolutions of a DE that is higher order than second and involves a nonlinear term. The nonlinearterm is the derivative of a double-well potential, so that solutions of the second order exhibittransition layers. The strategy is to write the solution of the DE as the minimizer of a freeenergy which is then expressed in Fourier space. We first truncate the Fourier space integralsto eliminate the very short and long space scales, and then express the nonlinear terms interms of the first cumulant (see, e.g., [10]). Analyzing these terms further we can rewrite thefree energy in the same form as the original with rescaled Fourier variables. By adjustingthe truncation, we can obtain coefficients that modify the nonlinearity. Transforming theseequations by rescaling, we can regain the original nonlinear term but with arbitrarily smallcoefficients for the higher order derivative terms. Finally, the space scale that has been modifiedin this process must be renormalized. We have shown that this can be done in two ways. One isthat, since we know that the rescaled solution to the second order is an approximate solution,we can minimize over functions of the same type. This is just a minimization over a singleparameter (which represents the length scale in the second-order DE), so that we need onlyto solve a polynomial to determine the scale. The other perspective is by using the surfacetension as an empirical quantity that rescales the length.

The key results of our RG analysis are (i) solutions of the higher order DE’s areclose (in L2 and L∞) to those of the second-order DE with rescaled length; (ii) solutionsof the sixth-order equation do not change significantly (as measured by the L2 and L∞

norms) as the coefficient of the higher order derivatives varies, provided space is rescaledappropriately.

2


The methodology can be illustrated with the prototype equation

c6φ(6) + c2φ

(2) + φ − φ3 = 0 (1)

with c6, c2 ∈ R+ (and not necessarily small) subject to boundary conditions

φ(k) (±∞) = 0 for k = 1, 2, 3 and φ (0) = 0. (2)

One can consider the effect of changing, say c6 = 10 to c6 = 0 (and appropriately droppingthe boundary conditions φ(k) (±∞) = 0 for k = 2, 3). In most nonlinear ordinary differentialequations (ODEs) and partial differential equations (PDEs) the highest order derivative is ofcrucial importance, and there would be a profound difference between the sixth-order andsecond-order equations. Even if the coefficient of the highest order is small, the perturbationfrom zero to a small value, ε, is generally singular and there is no guarantee that the differencein solutions will be small. Thus, it is surprising to see that the solutions of (1), (2) are close tothose of the second order, particularly for large values of c6.

The formal derivations are complemented by numerical computations (section 5).We compare the solutions of the ODEs obtained from the numerical solution and RGapproximation scheme. For the numerical solutions a multi-dimensional version of Newton’smethod is implemented. We find that, as the sixth-order coefficient is varied from c6 = 0to c6 = 10, not only do the two solutions (numerical and approximate) of (1), (2) stayqualitatively similar, as both exhibit single transition layer solutions, but also they are veryclose in the L2 and L∞ norms. We also show, in this section, that the effect of the sixth-ordercoefficient is weak in the sense that, as c6 is increased to large values, the true (numericallycomputed) solutions do not change appreciably. For example, solutions to (1), (2) for c6 = 8and c6 = 10 differ by 0.002 in the L2 norm, and 0.006 in the L∞ norm. In fact, furthernumerical experiments show that the main effect of changing the sixth-order coefficient is justrescaling the solution. In short, the renormalization analysis that we perform indicates that themain contribution of the higher order derivatives involves rescaling the length.

The rest of this paper is organized as follows. In section 2 we define the 2Nth orderDE and the associated functional that solutions must minimize. In section 3, we describe ourmethodology for the first stage of renormalization, namely rewriting the functional in Fourierspace and truncating in order to eliminate the small length scales. We analyze a key integral,and show how the DE can be modified as a result. In section 4 we perform the second stagein the RG process, namely, rescaling the length. Finally, in section 5 we perform numericalcomputations on the sixth-order equations and compare with our approximate solutions. In theconclusion (section 6) we summarize the results, and discuss additional research problems.The appendix contains much of the RG calculation and proofs of some technical lemmas.

2. Differential equations and statistical mechanics

2.1. Minimizers/differential equations

We consider, for N ∈ N, the 2Nth order DEN∑

j=1c2 jφ

(2 j) − 2aφ − 4bφ3 = 0, (3)

together with the boundary conditions

φ( j) (±∞) = 0 for j = 1, . . . N. (4)

Solutions to (3), (4) can be cast as a minimization problem. Let φ ∈ C2N (R) ∩ L∞ (R)

and satisfy

3


(i) boundary conditions (4) and∫

R|φ( j)|2 < ∞,

(ii) sgn(ab) = −1 and∫

R|a| φ2 + |b| φ4 < ∞.

We consider potentials for which solutions, φ (x), approach ±√−a/b or 0 as x → ±∞.

Define

L [φ] :=∫

R

N∑j=1

{(−1) j+1 c2 j

2[φ( j)]2

}+ aφ2 + bφ4. (5)

The following lemma is standard.

Lemma 1. Let φ ∈ C2N (R) ∩ L∞ (R) and satisfy conditions (i) and (ii). If φ is a minimizer of(5) then it satisfies (3).

Next we convert the sums into Fourier space by defining

φ̂(k) := (2π)−1/2∫ ∞

−∞φ (x) e−ikx dx, φ (x) = (2π)−1/2

∫ ∞

−∞φ̂(k)eikx dk. (6)

Lemma 2. The functional L [φ] can be written as:

L[φ] = L̂[φ̂] :=∫ ∞

−∞dk

N∑j=1

((−1) j+1 c2 j

2k2 j + a

)|φ̂(k)|2

+ b

2π

∫ ∞

−∞dk1 . . .

∫ ∞

−∞dk4{φ̂ (k1) . . . φ̂ (k4) δ (k1 + · · · k4)}. (7)

Proof. See appendix A.�

2.2. Renormalization group method/statistical mechanics

Having noted that solutions to (1), (2) are minimizers of (7), we use RG ideas to obtain anapproximation to (7) that involve only terms associated with the second derivatives. The mainansatz is that there is a parallel between minimizing the Landau free energy and the concept ofthe partition function in statistical mechanics. The minimizers of the free energy describe thebehavior of the system (as solutions of a DE). The partition function Z (defined more preciselybelow) which is obtained by taking the Landau free energy L̂[φ̂] as a weight, and summingover all possible φ̂ in the Fourier space, also contains all essential information with regardsto the physical system at hand. Hence, we can use RG methods to eliminate the short lengthscales and write Z in terms of sum of an effective free energy. The minimizers of this new freeenergy, L̂new, should also well describe the behavior of the same system. Indeed, it will be seenthat L̂new has the same form as the original but with new parameters. By making particularchoices, the parameters can be adjusted to obtain a new functional in which the coefficientof the higher (than two) order derivatives are much smaller than that of the second-orderderivative. Iterating this procedure, one can make the coefficients arbitrarily small. Using theformal asymptotics and existing rigorous results [21] in the case of the sixth-order DEs (as anexample), these can be set to zero, leading to an approximation of the solution to the higherorder equation by that of the second order.

We start with (7) and first truncate the variables k, ki at some large � corresponding to asmall length scale. In statistical mechanics this could be an atomic length for example. Thus

4


we write

L̂[φ̂] =∫ �

−�

dk

{N∑

j=1

((−1) j+1 c2 j

2k2 j + a

)|φ̂(k)|2

}

+ b

2π

∫ �

−�

dk1 . . .

∫ �

−�

dk4{φ̂ (k1) φ̂ (k2) φ̂ (k3) φ̂ (k4) δ (k1 + · · · + k4)}. (8)

Note that all integrals are over both positive and negative k since they arise from Fouriertransforms.

Using this notation, we can define the partition function over the momentum space by themagnitude of such integrals (we omit the absolute value symbols below)

Z =∣∣∣∣∣∫ ∏

0<|k|<�

dφ̂(k) e−L̂[φ̂(k)]

∣∣∣∣∣ . (9)

Note that if k were discrete then we would have a large product of such integrals. Since we areconsidering a continuum of k values, we regard the integrals over k in the functional integralsense [15].

Given any l > 0, we define

φ̂>(k) := φ̂(k) if �l < |k| < �, φ̂<(k) := 0 if �

l < |k| < �

:= 0 if 0 < |k| < �l := φ̂(k) if 0 < |k| < �

l .(10)

Now, by the coarse-graining procedure, one can show (see appendix C) that the partitionfunction can be (approximately) written as

Z =∣∣∣∣∣C

∫ ∏0<|k|<�/l

dφ̂(k) e−L̂new[φ̂(k)]

∣∣∣∣∣ (11)

where L̂new is given below.

Result 1. The functional L̂ in (8) can be effectively replaced by

L̂new[φ̂] :=∫ �/l

−�/ldk

{β(k) + 3bC

π

}|φ<(k)|2

+ b

2π

∫ �/l

−�/ldk1 . . . dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4) (12)

with

β(k) :=N∑

j=1(−1) j+1 c2 j

2k2 j + a, C(�, l) :=

∫ �

�/l

1

β(k)dk. (13)

The derivation of result 1, involving field theoretic methods of RG, is presented inappendix C.

Thus we obtain an effective free energy L̂new in a form close to (8), except that the dkintegration does not extend to �, but only to �/l. By rescaling k we can write (12) in a formthat is identical to the original.

2.3. Renormalization and justification in this context

(The subsequent analysis can be read independently of this subsection.) The success of RG instatistical mechanics [14, 34] can be attributed to the idea that one can repeatedly average oversmaller length scales provided one rescales the interactions in a way that yields a fixed pointin the transformation. This fixed point must correspond to the exponent that characterizes the

5


divergence of the thermodynamic quantity near the critical point where the system exhibits aloss of scale due to the infinite correlation length.

A basic idea is that RG leads to successful calculation of critical exponents in statisticalmechanics due to the principle of universality. This principle states that the exponents are‘universal’ such that the changes in the details of interactions cannot alter the exponents. Forexample, altering the interactions from nearest to next-nearest neighbor interactions in a latticemodel produces not change in the exponents. However, dimension of the space or varying thesymmetry group of the spins can alter the exponents.

This universality arises from the fact that the correlation length (roughly speaking,the maximum distance at which spins influence one another). Consequently, the process ofaveraging over short length scales should not alter the basic properties of the system providedone adjusts, or renormalizes the interaction strengths to compensate for the reduction in size.Thus, the heart of the RG process and universality is the feature that one is interested in lengthscales that are much larger than the scales on which the averaging is taking place.

In the present framework one can distinguish the universal and non-universalquantities/features. What is universal in our setting is the form of the DE (as being abyproduct of the free energy). As a result of this, the general features of the solutions toDE are also universal. However, the coefficients of the DE are non-universal and depend onthe particular microscopic information which cannot be obtained through RG. Moreover, onecan implement an averaging process over small length scales (in Fourier space as describedbelow), provided one renormalizes the length scale in order to preserve the surface tension.In other words, a transition layer is associated with a surface tension (see section 4) whichis the crucial quantity in terms of the physical characteristics of the system (e.g., solution ofthe equation, the stability of the interface, etc). There is a class of solutions that will have thesame surface tension. Hence, if we perform an averaging process in which the higher orderderivatives are eliminated, then we can obtain the same behavior in terms of these universalquantities provided we renormalize and endow the new, simpler system with the same surfacetension.

There are several ways of performing the averaging and rescaling process [10]. Real spaceRG involves averaging the spins in a geometric region that can be used to render a smallerreplica of the original system. Momentum or Fourier space RG involves transforming thefree energy into Fourier space, and then integrating out, in the partition function, the fieldscorresponding to larger wavelengths or shorter physical lengths in the problem. This latterapproach is closest to the methodology we pursue for our problem. The idea is that if we havean integral over dk on the range [0,�] then we would like to divide this integral into [0,�/l]and [�/l,�] for some l > 0 to be determined. We then integrate over the larger k portion,namely [�/l,�] , leaving the free energy as an integral over just the smaller k part, namely[0,�/l] .

Historically, the justification of RG methodology involved proximity of the system toa critical temperature at which point correlation length diverges. In the original Landauenergy, temperature appears in the coefficient of the quadratic term of the functional(aφ2 ∼ (T − Te)φ

2). Near the critical point this coefficient diminishes toward zero. Althoughwe start with a coefficient (of φ2) which has a fixed value, one can show by simple scalingof the fields followed by changing the length scale, as done explicitly in section 3, that thefunctional can be cast into a form where this coefficient is small (in fact the coefficient can bemade to assume arbitrary values). Upon performing this rescaling, we can regard the systemas close to the critical point (since the coefficient of the φ2 term will be small), and henceall the RG arguments are justified. Of course, once the averaging process is over, we need torenormalize in order to restore the functional to its original form with new parameters. This

6


process, however does not destroy the essential information contained in the functional whichwill exhibit itself in the minimizers, i.e., the solution of the original problem.

In general, the justification of this process depends on the particular system. For a statisticalmechanics system consisting of spins (±1) on a triangular lattice, for example, it involves theidea that taking a ‘majority rules’ procedure to replace the three spins with just one, therebyreducing the size of the system by three. Of course, if one does not simultaneously adjust theinteraction strengths, one will obtain a trivial result. The ansatz here is that the length scaleof the smallest triangle is very small compared with the correlation length, so that averagingcannot alter a fundamental structure of the divergent quantity. Creswick et al [10] describe anumber of mathematical problems (fractals, random walk, etc) in which the justification of thetransformations is based upon exact or approximate self-similarity. In the case of a randomwalk of N (�1) steps, one can divide the walk into smaller subwalks of N/l steps that areaveraged out and rescaled so that the distribution has the same form as the original, but withdifferent parameters. In this way, one recovers the classical central limit theorems.

The philosophy of RG has been adapted to DEs in several works primarily in calculatingsome key features. For example in Goldenfeld et al [17], and Caginalp [5] RG was usedto calculate the nonclassical decay exponent of DEs. The former considered the Barenblattequation of porous media flow. The latter considered the heat equation with a nonlinearity.These were along the lines of real-space RG with the justification of ‘asymptotic self-similarity.’ In particular, if one has an equation in which there is decay of solutions forlarge x (space) and t (time), then one does not have exact self-similarity as in the fractals (forany finite x and t), but rather one approaches self-similarity as x and t approach ∞ within anappropriate scaling. DEs that exhibit a ‘blow-up’ of solutions [5] at a particular point have asimilar self-similarity, but as x approaches the blow-up point rather than infinity. In many casesthe results of the RG analysis of these equations have been confirmed by exact calculationsand theorems [6, 25].

The RG philosophy can also be used to derive simpler or alternative equations whilepreserving the essential physical information. For example, in [19], a set of ‘amplitudeequations’ were derived from the phase field crystal equation that work on a much largerlength scale thereby significantly reducing the computational time. Another example of thesame spirit was given in [24] where RG was used to derive Navier–Stokes equations fromthe Boltzman equation. These examples all show that RG approach can be used in a broadercontext and has far reaching application areas outside the realm of equilibrium statisticalmechanics.

The DEs problems we consider in this paper have four main features: (i) they arise ascritical points of a free energy functional, (ii) they have large order derivatives, (iii) solutionsexhibit a transition layer, (iii) one can define a ‘surface free energy’ related to the transitionlayer and the free energy functional. Like the decay and blow-up problems, the transitionlayer in which the solution moves from −1 to +1 can be viewed as asymptotically self-similar. Hence, RG ideas can be implemented with the same justification that led to successfulcalculations for the DEs discussed above. In particular, small length scales and higher orderderivatives should not have a significant impact on the transition layer except through rescalinglength. A key property of DEs with transition layers is surface tension, which is a bona fidephysical quantity in the phase field equations, but can be defined more generally in a purelymathematical formulation in terms of L2 norms of solutions, for example. Studies involvinghigher order phase field equations [7] indicate that these higher order derivatives serve tomodify the surface tension, but otherwise have a smaller influence than the second-orderderivative. Thus, the ansatz is that we can utilize the higher order derivatives to match theempirical surface tension which is an intrinsic quantity of the physical system.

7


3. Renormalization of the free energy

The original free energy (8) has been transformed using the momentum space RG procedureinto (12). Our goal now is to rescale the length and the field in order that the form of the freeenergy is exactly the same as the original with the new parameters substituted for the old. Thelength will be scaled twice, to restore the integration limits and the form of the nonlinearity.

With C defined by (13), the new parameters are given by

a → a + 3bC

π=: anew, b → b =: bnew. (14)

In order to obtain the same form for the free energy, with the new parameters, we need torescale the integrals by setting k′ = lk and write∫ <

dk :=∫ �/l

−�/ldk =

∫ �

−�

dk′/l (15)

and substitute into (12) to yield

L̂new[φ̂] :=∫ �/l

−�/ldk

{β(k) + 3bC

π

}|φ(k)|2

+ b

2π

∫ �/l

−�/ldk1 . . . dk4φ (k1) . . . φ (k4) δ (k1 + · · · + k4)

=∫ �

−�

1

ldk′

{β(k′/l

) + 3bC

π

}|φ(k′/l)|2

+ b

2π

∫ �

−�

1

l4dk

′1 . . . dk

′4φ(k

′1/l) . . . φ(k

′4/l)δ

(k

′1 + · · · + k

′4

l

). (16)

The delta function satisfies

δ(k′/l) =: lδ(k′). (17)

For some z in R+ to be determined, we let

φ̂(k′/l) =: φ̂1(k′)z. (18)

Using again k = k′/l we have in place of (16) the free energy

L̂new [φ] =∫ �

−�

z2

ldk′

⎧⎨⎩

N∑j=1

(−1) j+1 c2 j

2

(k′

l

)2 j

+ (anew)

⎫⎬⎭ |φ̂1(k)|2

+ b

2π

z4

l3

∫ �

−�

dk′1 . . . dk′

4φ̂1(k′

1

). . . φ̂1

(k′

4

)δ(k′

1 + · · · + k′4

). (19)

Basic Scaling. We set z = l so that the integrand (coarse-grained free energy density) has thesame form as the original density up to a factor of l. Hence,

c′2 j = c2/l2 j, (anew)′ = anew, b′ = b. (20)

With the primed parameters, we can write the free energy, (19), in the form

L̂new [φ] = l∫ �

−�

dk′

⎧⎨⎩

N∑j=1

(−1) j+1c′

2 j

2(k′)2 j + anew

⎫⎬⎭ |φ̂1(k)|2

+ lb

2π

∫ �

−�

dk′1 . . . dk′

4φ̂1(k′

1

). . . φ̂1

(k′

4

)δ(k′

1 + · · · + k′4

). (21)

8


This corresponds to the DEN∑

j=1c′

2 jφ(2 j)1 − 2anewφ1 − 4bφ3

1 = 0. (22)

Note that we can rescale the space variable with x̃ = lx and φ̃ (x̃) = φ1 (x) and writeN∑

j=1c2 jφ̃

(2 j) − 2

(a + 3bC

π

)φ̃ − 4bφ̃3 = 0. (23)

We will show below that, under appropriate conditions, C can be chosen arbitrarily in(0,∞) by choosing � and l appropriately, so that a + 3bC/π is arbitrarily small. LettingA := −2 (a + 3bC/π ) and B := 4b, we use the simple scaling described below to rewrite (23)in a form in which the coefficient of the terms beyond the second order are small if A is small.

3.1. Analysis of C

Let N be odd and c2N > 0. Recall the definitions of β(k), C(�, l) in (13) and let ε := l−1

where a < 0, and note that the leading term, c2Nk2N/2 is positive. Thus, the polynomial β(k)

has a real and positive root. We define k+ as the largest real root of β(k) = 0. Hence we canwrite

β(k) = (k2 − k2

+)p

Q(k2) = (k − k+)p (k + k+)p Q(k2) for p ∈ {1, . . . , N} ,

where Q is a polynomial of order 2N − p and Q(k2+)

> c > 0. Thus, the integral in C (�, l)has the dominant term∫

ε�

dk

(k − k+)p

which diverges as �ε → k+ for p � 1.

Note that for any � ∈ R+ we have C (�, 1) = 0. Also, by first choosing a large fixed

�, and then adjusting ε sufficiently close to k+/�, we can make C(�, ε−1) arbitrarily large.Hence, for any q ∈ R

+ we can find a pair (�q, εq) with �q � 1 and 0 < ε < 1 such thatC(�q, εq) = q. We summarize our results below using either of the two conditions.

Condition A. Let N be odd, and assume c2N > 0, a < 0.

Condition B. Let N be odd, and let β(k), defined by (13), have at least one positive root.

Lemma 3. Under condition A or B, for sufficiently large �, given any positive real numberδ, we can find ε depending on q and � such that

C(�, ε−1) = δ. (24)

For a > 0 we can utilize a less restrictive condition. Since C(�, ε−1) can be madearbitrarily small (note C(�, 1) = 0), if we can show that for some ε and � one has that

a + 3bC

π= 0 (25)

then by continuity we can choose (�, ε) such that a + 3bC/π attains any value between0 and a.

Condition C. Let N be odd. For a > 0 and b < 0 we require the existence of (�, ε) suchthat

∞ > C(�, ε−1

) =∫ �

ε�

dk

β(k)� π

3

∣∣∣a

b

∣∣∣ . (26)

9


When condition C is satisfied, one can find (�, ε) such that A := a + 3bC/π is arbitrarilysmall.

In the RG analysis, we assume that condition A, B or C hold as indicated below.

Example. Let c6 > 0 and c2 > 0 with all other ci = 0. Set a = 1/2 and b = −1/2 so that theequation to be considered is

c6φ(6) + c2φ

(2) − φ + 2φ3 = 0. (27)

Since we can make ε arbitrarily small, and � arbitrarily large, it will suffice to show∫ ∞

0

2dk

c6k6 + c2k2 + a� π

3=̃1. 047 2. (28)

One can verify numerically that condition on C will be satisfied, for example, if (c6, c2) satisfyany of the conditions: c6 � 40, c2 � 1, or c6 � 4, c2 � 4.5, or c6 � 0.1, c2 � 7.

3.2. Rescaling of the field

Here, we complete the renormalization process by scaling the field and the length so that theoriginal form of the DE is preserved. The equation

N∑j=1

c2 jφ(2 j) + Aφ − Bφ3 = 0 (29)

can be written as

N∑j=2

c2 jAj−1�(2 j) + c2�

(2) + �(1 − �2) = 0, (30)

with �(x) := B1/2A−1/2φ(x/A1/2). The RG analysis showed that the original equation (3) canbe replaced by another in which the coefficients (a, b) are transformed into (a + 3bC/π, b)

with a change in length scale. The analysis above showed that with a suitable choice of (�, ε)

one can adjust C so that −2 (a + 3bC/π ) is arbitrarily small provided a and b are of differentsigns.

Now using the simple scaling with δ := A := −2 (a + 3bC/π ) and B := 1 in (10) wehave the equation

N∑j=2

c2 jδj−1�(2 j) + c2�

(2) + �(1 − �2) = 0. (31)

Thus, we have derived the following.

Result 2. Solutions of the DE (3) can be approximated by those of (31) for arbitrarily small δ,

subject to the same boundary conditions (4) for some choice of length scale.

Hence, the coefficients of all terms beyond the second order can be made arbitrarilysmall. Thus, the RG transformations are used to obtain a small coefficient for the φ term,which can then be transformed into small coefficients for the higher (than two) order terms inthe derivatives.

10


4. Rescaling the length

Recapitulating our results thus far, the original equation (3) for a ≶ 0 and b ≷ 0 can betransformed via the simple scaling above to one of the prototype equations (32) and (33)below. Moreover, the RG scheme above shows that solutions to the equations (for N odd)

N∑j=1

c2 jφ(2 j) + φ − φ3 = 0, subject to conditions A or B (32)

andN∑

j=1

c2 jφ(2 j) − φ + 2φ3 = 0, subject to condition C (33)

with suitable boundary conditions, can be approximated by the solutions of the second-orderequation. However, the transformations have altered the spatial scale. The next step in ouranalysis is to set the length scale. This can be done in two ways discussed below.

4.1. Minimization of the functional integral to set length scale

The original DE, by construction, is the minimizer of the functional L. Our approximatesolution class consists of all functions which solve the second-order DE with an altered spatialscale. So, the best approximation among all candidates is the one with a stretching factor thatminimizes the functional L.

Example 1. We consider first prototype equation for the case c6 > 0, c2 > 0 (and all otherci = 0) and a = −1/2 and b = 1/4, so that the potential is [φ2 − 1]2/4 yielding the equation

c6φ(6) + c2φ

(2) + φ − φ3 = 0. (34)

Since tanh[x/(21/2c1/2

2

)]is an exact solution for the second-order DE in this problem, the

RG procedure suggests that an approximate solution has the form

ψη (s) = tanh

(s

21/2η

)(35)

for some value η > 0. We minimize the functional

L[ψη] =∫

R

{c6

2

[ψ(3)

η

]2 + c2

2

[ψ(1)

η

]2 + 1

4

[ψ2

η − 1]2}

(36)

over all ψη. Differentiating with respect to η we obtain the condition for the minimum as

0.667η6 − 0.667c2η4 − 1.904c6 = 0. (37)

This has a unique positive root η∗ for all positive c2, and c6. The approximate solution isthen given by ψη∗ (s) . For example, if c6 := 10, c2 := 1 one has η∗ = 1. 852, while c6 = 1and c2 = 1 implies η∗ = 1. 357. Numerical results below will show that this yields a goodapproximation to the computed sixth-order solution as measured by the L2 norm.

Given this approximate solution, ψη∗ , to the sixth-order equation, one can also computethe surface tension [7], σ, which is a crucial quantity in phase field models. Roughly, speakingit is the difference per unit area of the free energy with the transition, minus the average of thefree energies of the constant states (e.g., φ = ±1). For the free energies we consider, one has(see [7])

σ̃ (η, c2, c4, . . . , cN ) :=N∑

j=1

jc2 j

∥∥ψ( j)η

∥∥2

2= 3c6

∥∥ψ(3)η

∥∥2

2+ c2

∥∥ψ(1)η

∥∥2

2. (38)

11


The quantity σ̃ would be identical to the theoretical surface tension if ψη (s) were replacedby the true solution to the full 2Nth order DE. For c6 = 10, c2 = 1, σ is approximated byσ̃ = 1.231 (compared with the true value of 1.176). For c6 = 1, c2 = 1, σ is approximatedby 1.046 compared with the true value of 1.02. For c6 = 5 one has σ̃ = 1.173 comparedwith the true value of 1.116. By ‘true value’ we refer to the value of σ obtained by using thenumerically computed solution of the sixth-order equation (discussed below).

Example 2. We implement the same procedure on a different prototype for potentials witha > 0 and b < 0, namely, − 1

2

(φ2 − 1

2

)2 + 18 , yielding the sixth-order equation

c6φ(6) + c2φ

(2) − φ + 2φ3 = 0 (39)

subject to the same Neumann conditions. The second-order equation now has a solution thatis qualitatively different from the first prototype. It is given by � (x) := ψ1 (x) where

ψη (x) := 1

cosh (x/η).

We use again the general solution and determine the minimizing value of η. Defining m1 andm3 below, and setting analogously, m0 := ∫

R

12

[ψ2

η − ψ4η

]we compute using the new ψη

defined above, m0 = 1/3, m1 = 2/3 and m3 = 2.95. One has similarly,

5c6 (2.95)

(1

η

)6

+ 2

3c2

(1

η

)2

− 2

3= 0.

For example, for c6 = 10 and c2 = 1 we have the solution η∗ = 2. 531. For c6 = 5 and c2 = 1we have η∗ = 2. 271 and for c6 = 1 and c2 = 1 we have η∗ = 1. 786.

Using these values, we can approximate the surface tension, σ � σ̃ , using (38) togetherwith ψη∗ (x) with the results σ̃ = 1. 114 (c6 = 10), σ̃ = 1. 023 (c6 = 5) and σ̃ = 0.861(c6 = 1). These compare with the true values (i.e., those obtained from the numericallycomputed solution of the sixth-order equation) of 1.07, 0.97 and 0.82 respectively, so thatthe estimate is 4% to 5% higher than the true value. Hence, this methodology allows one tocompute the surface tension of the sixth-order model to this level of accuracy with informationfrom the second-order equation alone.

The conclusion for the general 2Nth order equation is summarized below.Scaling by minimizing functional. Consider the general 2Nth order equations (32) (or (33)

(for N odd) ψ(x) being the solution of the second-order equation. Let

m0 :=∫ ∞

−∞[aψ2(s) + bψ4(s)]ds, mj := ‖ψ( j)‖2

2, j = 1, 2, . . .

ψη (x) := ψ(x/η), L[ψη] =N∑

j=1

(−1) j+1 c2 j

2

(1

η

)2 j−1

mj + ηm0.

The derivative of this functional satisfies

0 = dL[ψη]

dη, or,

N∑j=1

(−1) j+1 (2 j − 1) c2 jm j

(1

η

)2 j

= 2m0.

For N odd, this will have a positive solution for all c2 j and mj ( j = 1, . . . , N). Then, η∗ isdefined as the unique global minimizer, i.e., the value of η for which L[ψη] is the minimumamong finitely many roots.

Result 3. Let η∗ be defined as the minimizing stretching factor for the functional L. Then thefunction ψη∗ (x) is an approximation to the solution of (32) (or (33)).

12


While we have focused on two equations that have exact solutions for the second-orderequation, the process is the same for any potential with similar qualitative solutions. Thegeneral theory asserts the existence of a solution with these boundary conditions, so that anumerical solution would be obtained in place of ψη to calculate the mi.

4.2. Using empirical surface tension to set length scale

As an alternative to minimizing the functional integral to obtain the value of η∗, we now usean idea that is based on statistical mechanics instead. We regard the surface tension (38) as anempirical macroscopic quantity that the solution of the 2Nth order DE must satisfy. Let thetrue solutions of (32) (or (33)) be denoted by φ (x; c2, . . . cN ) and recall that the exact solutionsto the second-order equation are given by φ (x; c2, 0, . . . , 0) = ψη (x) := tanh[x/(21/2η)] (orψη (x) := 1/ cosh[x/ (η)]) . As before, we can compute

∥∥ψ( j)η (s)

∥∥2

2= mj

η2 j−1; mj :=

∫ ∞

−∞

{d jψ(s)

ds j

}2

ds. (40)

Then, we can rewrite σ̃ as

σ̃ (η, c2, . . . , cN ) =N∑

j=1

(−1) j+1 jc2 jm j

η2 j−1. (41)

In phase field models and other physical applications, σ̃ defined by (38) represents the(theoretically computed) surface tension. In order to renormalize the length scale we setσ̃ := σ∗, where σ∗ is the empirical value of the surface tension. Now let c2, . . . , cN be fixed,and suppose that we assume the solution to (32) (or (33)) is of the form of ψη (s) for some η.

We then propose that, among all possible functions ψη (s) , the better approximate solution isthe one for which the surface tension σ̃ coincides with σ∗. This is to say that we are lookingfor a positive solution, η∗, of

σ∗ =N∑

j=1

(−1) j+1 jc2 jm j

η2 j−1∗

, or

Nc2NmN = σ∗η2N−1∗ −

N−1∑j=1

(−1) j+1 jc2 jm jη2(N− j)∗ . (42)

The procedure described above can be summarized as fixing the surface tension based onempirical considerations. Denoting this by σ∗, we solve for η∗ in (42) and use this value inψη (s) as the approximation to the solution to the 2Nth degree equation (3), (4). This addressesthe question: if one knows the surface tension of the 2Nth equation, how well is the solutionprofile approximated by the solution to the second-order DE with the same surface tension?What is left is then to find a reasonable value for σ∗ (aside from physical measurement). Wewill consider two ways.

Scaling (surface tension) 1. Use the surface tension obtained from the second-orderequation as σ∗.Scaling (surface tension) 2. Use the true surface tension that is obtained from the numericalsolution (see section 5) of the 2Nth order DE.

Example 3. We again consider the sixth-order first prototype equation for which we willperform numerical computations in the subsequent section:

c6φ(6) + c2φ

(2) + φ − φ3 = 0. (43)

13


Let c2 := 1 for which the surface tension for the second-order equation is given by (38) asm1 = 2

√2

3 =̃ 0.943. For scaling 1, in order to retain the same surface tension in the sixth-order

equation, we set the value σ∗ := m1. We also compute m3 = 8√

221 =̃ 0.539. Given any value,

σ∗, for surface tension, if the solution profile for this sixth-order equation were given exactlyby ψ (s, η) and had surface tension σ̃ (η, 1, c6) then we would have

σ∗ = σ̃ (η, 1, c6) = 3c6m3

η5+ c2

m1

η. (44)

In order to retain the same surface tension as in the second-order equation, we set σ∗ := σ1 =0.943. Thus, η∗, can be defined as the unique (real) positive solution to the fifth degreepolynomial

0.943η5∗ − 0.943η4

∗ − 3 (0.539) c6 = 0. (45)

One can readily see that (45) has a unique real solution, η∗ (which depends on c6), for anyc6 > 0, and that η∗ > 1. In order to match this surface tension, the approximate solution (35)must be given by

ψη∗ (x) = tanh

(x√2η∗

). (46)

For example, one has the following roots η∗ (c6) for c6 = 1, 2, 5 and 10: η∗ (1) = 1.42,

η∗ (2) = 1.57, η∗ (5) = 1.80 and η∗ (10) = 2.02 when we compute the roots using thesurface tension of the second-order DE (m1 = 0.943). In the next section we will computethe L2 difference between the computed solution of (43) and the approximate solution (46)for these values of c6. Also, once the solution is computed, we can use the computed surfacetension for the sixth-order equation as σ∗ in (44) to obtain a better η∗ value (see section 5).

Example 4. Let N = 5 with the ci = 1 in (32) and set σ∗ := m1 = 0.943, so (42) is then(upon computing the σi)

(0.943) η9 − (0.943) η8 + 2 (0.377) η6 − 3 (0.539) η4 + 4 (1.51) η2 − 5 (6. 857) = 0,

with unique real solution η∗ = 1. 559. The approximation to the 10th order equation is thengiven by tanh[x/(1.559

√2)].

Example 5. We consider the second prototype, namely equation (39). For c6 := 0 and c2 := 1one has the exact solution �(x) := ψη (x) where ψη (x) := 1/ cosh (x/η) as before. Onceagain we have

∥∥ψ( j)η

∥∥22 = η1−2 jm j and the mj are given as in (40). Then we have m1 = 2/3

and m3 = 2.95 so that

σ̃ (η; c2, c6) = 3c6

∥∥ψ(3)η

∥∥2

2+ c2

∥∥ψ(1)η

∥∥2

2

= 3c6m3

η5+ c2

m1

η.

Using the same idea as above, we define σ∗ := m1 = 2/3 so η∗ is the real root of2

3η5 − 2

3η4 − 3c6 (2.95) = 0. (47)

For example, if c6 = 10 then η∗ = 2.89. Then, the approximate solution is ψη∗ (x) =1/ cosh

(x

2.89

)which solves the equation η2

∗ψ(2) −ψ +2ψ3 = 0. We can also use the computed

(for the sixth-order equation) surface tension σ∗ = 1.03 and obtain η∗ = 2.56, so that theapproximate solution is [cosh (x/2.56)]−1. These will be compared with the numericallycomputed solutions below.

Result 4. With η∗ defined by either scaling 1 or 2, the function ψη∗ (x) is an approximation tothe solution of (32) (or (33)).

14


5. Numerical computations and comparisons

In this section we describe the numerical techniques for the solutions of two prototype (sixth-order) DEs. Both examples are in the form

0 = c6φ(6) + c(2)

2 φ − W ′i (φ), (48)

where W1(x) = (1 − x2)2/4 and W2(x) = (x2 − x4)/2. Although both forms appear to besimilar, they exhibit solutions with quite different behavior due to the differing signs. We solvethe above DEs in the interval [−10, 10] with the following respective boundary conditions

I : φ(±10) = ±1, φ(1)(±10) = 0, φ(2)(±10) = 0,

II : φ(±10) = 0, φ(1)(±10) = 0, φ(2)(±10) = 0. (49)

We use a finite difference scheme in order to solve these high order nonlinearDEs, and discretize the interval [−10, 10] into N equal pieces and enumerate the pointsx0 = −10, . . . , xN = 10. The second, fourth and sixth-order derivatives are numericallyapproximated by

d2φ(xi)

dx2∼= {φ(xi+1) − 2φ(xi) + φ(xi−1)} /�x2,

d4φ(xi)

dx4∼= {φ(xi+2) − 4φ(xi+1) + 6φ(xi) − 4φ(xi−1) + φ(xi−2)}/�x4,

d6φ(xi)

dx6∼= {φ(xi+3) − 6φ(xi+2) + 15φ(xi+1) − 20φ(xi)

+15φ(xi−1) − 6φ(xi−2) + φ(xi−3)}/�x6.

respectively.The discrete system then constitutes a system of nonlinear algebraic equations. If we

denote the value of φ at xi as φi, then the system of equations can be regarded as a nonlinearmap S : R

N → RN such that S(φ) = 0, where φ ∈ R

N . To solve it we use a multi-dimensionalversion of Newton’s method. Starting with a reasonable initial solution profile (for instanceφ(x) = tanh x for the first type of equation, 1/ cosh(x) for the second), one can perform theiteration scheme

φn+1 = φn − [DS(φn)]−1S(φn),

where [DS(φn)]−1 denotes the inverse of the gradient of S. Actually, in order to avoidconstruction of large inverse matrices and save computational power, we implement anequivalent procedure by solving an auxiliary linear equation, and inserting the temporarysolution as the second term in the above equation, i.e.,

DS(φn)V ntemp = S(φn),

φn+1 = φn − V ntemp.

Once the solution is obtained the surface tension integral is evaluated numerically on the samemesh. The mesh size needs to be chosen judiciously. For mesh sizes �x < 1

20 , it is observedthat surface tension values do not change appreciably. Also, the matrices constructed in thescheme become close to singular when �x < 1

60 . The codes have been written using theMATLAB software.

5.1. Numerical solutions for the first prototype equation

We consider equation (43) for c6 � 0, together with (49). We let ‖·‖2 and ‖·‖∞ denotethe usual L2and L∞ norms and denote by φ (c2, c6) the solutions to with those particular

15


Figure 1. The true solution (red) for the full sixthorder (first prototype) equation(c6 = 10) is displayed together with the solution of the second-order equation (blue)and modified second-order equations with scaling 1 (green), scaling 2 (yellow) andminimization (black).

Table 1. The L2 and L∞ differences between the true solution and approximations forc6 = 10.

c6 = 10 Pure Scaling 1 Scaling 2 Scale by min

η value 1 2.02 1.89 1.86L2 diff 0.45 0.31 0.25 0.23L∞ diff 0.25 0.12 0.10 0.09ST = 1.176 0.94 0.94 1.17 1.23

parameters. The objective is to examine the difference between the L2 and L∞ normsbetween solutions to the sixth-order equation and the approximations discussed above, namely,ψη∗(c6 ) (x) = tanh

(x

21/2η∗(c6 )

)with η∗ determined as described in each of the three procedures.

(i) For c6 = 10, the surface tension computed from the sixth order equation is σ = 1.176.

The polynomial equations to be solved for the two approaches that use the surface tension (toset the length scale) are

0.94(η5 − η4) − 16.2 = 0 (scaling 1),

if one uses the surface tension obtained from the second-order equation, and

1.176η5 − 0.94η4 − 16.2 = 0 (scaling 2),

if one uses the true surface tension. The latter is obtained from solving the sixth-order equation,but we can regard it as a proxy for the empirical surface tension. The results of these twomethods in determining η∗, together with the minimization method described above, areshown in the figure and tables for varying values of c6.

For c6 = 10, the results are depicted in figure 1 and table 1.As a measure of the differences, we can compute ‖φ (c2, 0) − φ (1, 0)‖2 where

φ (c2, c6) is the exact computed solution to the DE for a range of values of c2. In

16


Figure 2. The true solution (red) for the full sixth-order (first prototype) equation(c6 = 5) is displayed together with the solution of the second-order equation (blue)and modified second-order equations with scaling 1 (green), scaling 2 (yellow) andminimization (black).




this way we can see the relative effect of changing c6 as opposed to changing c2. Forexample, ‖φ (10, 0) − φ (1, 0)‖2 =̃1.63. One can also note the relative norm difference{∫ ∞

−∞ |φ (1, 0) − sgn (x)|2 dx}1/2 =̃ 1.48 as a measure of the numerical difference between

the two similarly behaving functions.(ii) For c6 = 5, the pertaining polynomial equations for scaling 1 and scaling 2 are

0.94(η5 − η4) − 8.1 = 0,

1.115η5 − 0.94η4 − 8.1 = 0

respectively, and the surface tension is 1.115. The results are shown in figure 2 and table 2.(iii) For c6 = 1, the respective polynomial equations for the two type of scalings are

0.94η5 − 0.94η4 − 1.62 = 0,

1.02η5 − 0.94η4 − 1.62 = 0.

The results are shown in table 3.

17


Figure 3. The true solution (red) for the full sixth-order (second prototype) equation(c6 = 10) is displayed together with the solution of the 2nd order equation (blue)and modified second-order equations with scaling 1 (green), scaling 2 (yellow) andminimization (black).




5.2. Numerical solutions for the second prototype equation

Next, we compute solutions to the equation (39) and compare the true (i.e., directly computedsolutions) for the sixth-order equation with the exact solution (pure) for the second-order andthe (rescaled) approximations.

(iv) For c6 = 10 one obtains the true surface tension σ = 1.07 by computing the solutionto the sixth-order DE. One also computes m1 = 2/3 and m3 = 2.95. The polynomial equationsto be solved are

0.66(η51 − η4

1) − 88.5 = 0,

1.07η52 − 0.66η4

2 − 88.5 = 0

for the two scalings, respectively.The results are displayed in figure 3 and table 4.(v) For c6 = 5 we note that the true surface tension is σ = 0.97 and obtain the pertaining

polynomials as

0.66(η51 − η4

1) − 88.5/2 = 0,

0.97η52 − 0.66η4

2 − 88.5/2 = 0

for the two scalings. The results are displayed in figure 4 and table 5.

18


Figure 4. The true solution (red) for the full sixth-order (second prototype) equation(c6 = 5) is displayed together with the solution of the second-order equation (blue)and modified second-order equations with scaling 1 (green), scaling 2 (yellow) andminimization (black).






η value 1 2.56 2.39 2.27L2 diff 0.71 0.72 0.61 0.52L∞ diff 0.35 0.24 0.20 0.17ST = 0.97. 0.66 0.66 0.97 1.016

(vi) For c6 = 1 we have the analogous polynomials

0.66(η51 − η4

1) − 8.85 = 0,

0.82η52 − 0.66η4

2 − 8.85 = 0.

The results are displayed in figure 5 and table 6.

19


Figure 5. The true solution (red) for the full sixth-order (second prototype) equation(c6 = 1) is displayed together with the solution of the second-order equation (blue)and modified second-order equations with scaling 1 (green), scaling 2 (yellow) andminimization (black).




Table 7. Magnitude of φ(c6, 1) (in norm) for the first prototype equation.

Norms φ(4, 1) φ(5, 1) φ(8, 1) φ(10, 1) φ(12, 1) φ(16, 1)

L1 77.58 77.52 77.38 77.31 77.25 77.15L2 8.753 8.749 8.739 8.734 8.730 8.723

5.3. Additional numerical analysis of prototype equations

We examine the accuracy of the RG approximation by presenting additional computations thatconfirm the ansatz. All computed tables of this section are given in appendix D.

Let φ(c2, c6) be the true solution for the sixth-order equation (i.e., (1), (2)), that wecompute directly (without RG). Let φapp(c2, c6) denote the approximation that we obtainfrom RG and the scaling via minimization. We investigate the L2 and L∞ norms of thedifferences between φ(c2, c6) and φapp(c2, c6). We have already presented, in the tables andthe graphs of the previous subsections, the norm differences and qualitative behavior ofφ(c2, c6), φapp(c2, c6) for c6 = 1, 5, 10. Now, we increase c6 to even higher values (for thefirst prototype equation) and investigate the difference in norm (with c2 := 1).

20


Implementing a linear regression of these values c6 � 12 (table D1) we obtain∥∥φ(1, c6) − φapp(1, c6)∥∥

1

‖φ(1, c6) − 0‖1=̃ 0.8 + 0.01c6

20=̃ 0.04 + c6

2000.

By this measure, the error is almost independent of c6 for large values of c6. This is a surprisingresult, since higher order derivatives are often regarded as dominant in the behavior of thesolution.

The RG analysis that is supported by the numerical computations suggests that exactsolutions of the sixth-order equation (i.e., (1), (2) with c2 := 1) are not sensitive to changes inc6 when c6 is large. We test this idea by computing the solutions directly and examining thequantity e(c6, n, p) := ‖φ(1, c6) − φ(1, c6 + n)‖p as shown in table D2. We see that there is asmall difference between the true solutions corresponding to largely different c6 e.g., c6 = 16and c6 = 26. The difference decreases as c6 increases. A similar observation can be made forthe second prototype DE. Table D3 shows that for larger values of n the differences e(c6, n, p)

decrease with increasing c6 for the second prototype equation.We can further investigate the nature of the small difference between the solutions of (32)

corresponding to different values of c6 (with c2 = 1 fixed). The RG ansatz suggests that themain difference should be a rescaling of space. In order to test this hypothesis, we can vary thelength scale of a solution for a given c6 and compare that with those of the equation in whichc6 is replaced by c6 + n. In other words, for φc6,α (x) := φc6 (αx) , we compute

minα∈(0.1,10)

∥∥φc6,α− φc6+n

∥∥p .

If this is small for a broad range of c6 (particularly large values of c6), then it provides goodevidence for the assertion that varying c6 essentially rescales the solution. In appendix D,we list the norm differences for the (c6, n) for various α values. The results summarized intables D4– D7 confirm the ansatz that rescaling the solutions with the right parameter leadsto approximate solutions with errors (measured by the Lp norms) that are five to ten timessmaller than without the scaling. As a measure of the size of this norm difference, we computethe norms of the true solutions as shown below.

We compare them with the ‘relative errors’ where a standard measure of the ‘relativeerror’ can be given as

re(c6, n, p) :=minα∈(1,1.5)

∥∥φc6,α− φc6+n

∥∥p

avg(∥∥φc6

∥∥p ,

∥∥φc6+n

∥∥p)

.

Computing this expression for c6 = 4, we obtain

re = minα∈(1,1.5)

∥∥φ4,α− φ4+1

∥∥1

avg(‖φ4‖1 , ‖φ5‖1)= 0.036/77.55 � 5 · 10−4,

re = minα∈(1,1.5)

∥∥φ4,α− φ4+1

∥∥2

avg(‖φ4‖2 , ‖φ5‖2)2= 0.01/8.75 � 0.001.

Hence, the relative error in the rescaled solution is very small.In a similar way we can form tables for the second prototype equation as shown in tables

D8–D11. Again, we observe that not only are the norm difference of the solutions at largec6 values small, but also, solutions corresponding to different c6 coefficients, are extremelyclose to one another upon changing length-scale. Summarizing these computations we havethe following:

Result 5. For both prototype equations, the computations show that the relative error arisingfrom a significant change in the coefficient of the sixth-order derivative leads to a very smallchange in the solution as measured by L1, L2 and L∞ differences.

21


6. Conclusion

Motivated by statistical mechanics and quantum field theory, we have presented a formalRG methodology that can be used to approximate solutions of a class of higher (than two)order differential equations (DEs) of boundary value type by second order. The central ideasinvolve writing the DE as a minimizer of a functional which is then treated as a Hamiltonian instatistical mechanics. By writing the free energy arising from this functional, one can averageover the large Fourier modes, i.e., the short length scales. Going through this process, one canrender the coefficients of the higher order derivatives arbitrarily small, so that the equation canbe approximated by one of second order. The length scale, however, has been modified in theprocess. Thus, the second step in the RG procedure is to rescale length. We showed that thiscan be done in two ways. One of these involves determining the length scale by minimizingthe functional. This is easily done since there is only one parameter, and it amounts to solvinga polynomial.

An alternative procedure involves using the surface tension as an empirical constant thatcan be used to rescale the equations. One can use the surface tension obtained from the secondorder in order to set the scale (scaling 1) or utilize the surface tension of the sixth-ordersolution (scaling 2). The latter is a proxy for using the measured surface tension in obtainingthe solution.

The surface tension that is an inherent feature of equations exhibiting a transition layer isparticularly important in phase field models (see references in [7])). We find that the surfacetension of the computed solution to the sixth order differs from the approximation (usingminimization) by 2%–5% in the equations studied. This range holds even when the coefficientof the sixth-order term is as large as 10. Mathematically, the surface tension can be defined interms of Sobolev norms such as

∥∥φ′∥∥L2 .

With either approach for rescaling length, the solutions provide an approximation thatshows a qualitative agreement (in terms of exhibiting a similar transition layer), and goodquantitative agreement in terms of the L2 and L∞ norms. The minimization approach providesslightly better accuracy.

The RG analysis suggests that the chief role of the higher derivatives in a sixth-orderequation involves rescaling length. Using computed solutions to the two basic prototypes ofsixth-order equations, we have shown that the main effect of varying the coefficient of thesixth-order derivative is to rescale the length scale.

The overall philosophy of our approach entails examining the underlying physics for onepossible derivation of a DE. Using approximation methods particular to that application, weobtain simplified equations. In many physics and engineering problems one has a system ofmany equations. In situations of high symmetry one has a system of ODEs which can bewritten as a single higher order ODE. Thus our methodology is a first step in significantlysimplifying large systems.

Numerical computations on DEs of high order pose many challenges such as stability.While our numerical examples have focused on sixth-order equations, the same RGmethodology can be expected to provide a good approximation for DEs of much higherorder whose solutions are much more difficult to compute using typical methods.

In many problems involving transition layers, one can utilize a local coordinate system inwhich one coordinate is orthogonal to the interface. Thus, while our methods have been appliedto ordinary DEs, there is a good reason to believe that it will be an integral part of a study ofpartial DEs in which there are transition layers, for example. Moreover, whenever one has aproblem in which there is a large space scale (e.g., the transition) and complex behavior (e.g.,the intricacies of interfacial pattern) one should be able to use this approach, since the complex

22


behavior is governed properties of the larger scale (e.g., surface tension) that are preservedduring this RG process. Clearly a next step in this direction of research would be to find amethod for successive approximations that converge to the true solution in a particular norm.The cumulant expansion can be carried out to higher terms, thereby eliminating the error in thisapproximation. Another direction is to obtain rigorous error bounds for the approximationsobtained from the RG approach.

Acknowledgments

Both authors thank the two anonymous referees and the associate editor for their helpfulcomments. The second author was supported by the Pohang Mathemtatics Institute throughthe grant National Research Foundation of Korea (NRF) funded by the Ministry of Education(2013053914)

Appendix A. Interaction terms in Fourier space

We define the Fourier transform

φ̂(k) := (2π)−1/2∫ ∞

−∞φ (x) e−ikx dx, φ (x) = (2π)−1/2

∫ ∞

−∞φ̂(k) eikx dk (A.1)

and calculate the interaction terms in the Hamiltonian. One way to do this is by consideringthe discrete version and taking the continuum limit as discussed in [4, 7]. We can also computethese quantities directly in the continuum:

A1 :=∫ ∞

−∞dxφ2

x =∫ ∞

−∞dx (2π)−1

(∂

∂x

∫ ∞

−∞φ̂ (q) e−iqx dq

)2

= −∫ ∞

−∞dqφ̂ (q) φ̂ (−q) q (−q) =

∫ ∞

−∞dq|φ̂(q)|2q2. (A.2)

A4 :=∫ ∞

−∞dx [φ (x)]4 = (2π)−2

∫ ∞

−∞dx

∫ ∞

−∞φ̂ (q1) e−iq1x dq1 . . .

∫ ∞

−∞φ̂ (q4) e−iq4x dq4

= (2π)−2∫ ∞

−∞dq1 . . .

∫ ∞

−∞dq4

∫ ∞

−∞dx e−ix(q1+···+q4 )φ̂ (q1) . . . φ̂ (q4)

= (2π)−1∫ ∞

−∞dq1 . . .

∫ ∞

−∞dq4δ (q1 + · · · + q4) φ̂ (q1) . . . φ̂ (q4) (A.3)

A3 :=∫ ∞

−∞dx [φxxx (x)]2 =

∫ ∞

−∞dx

{(2π)−1/2 ∂3

∂x3

∫ ∞

−∞dqφ̂ (q) e−iqx

}2

=∫ ∞

−∞dx

{(2π)−1/2 ∂3

∂x3

∫ ∞

−∞dqφ̂ (q) e−iqx

}2

= −∫ ∞

−∞dqδ(δ + δ̃)φ̂ (q) φ̂ (−q̃) q3 (−q)3 =

∫ ∞

−∞dq|φ̂(q)|2q6. (A.4)

Appendix B. The cumulant expansion

The cumulant expansion (see p 260 of [16])

〈e−V 〉0 =⟨1 − V + 1

2V 2 + · · ·

⟩= 1 − 〈V 〉0 + 1

2〈V 2〉0 + · · ·

23


log〈e−V 〉0 = log

(1 − 〈V 〉0 + 1

2〈V 2〉0 + · · ·

)

=̃ − 〈V 〉0 + 1

2

[〈V 2〉0 − 〈V 〉20

] + · · · (B.1)

Hence, one has the approximation

〈e−V 〉0=̃ e−〈V 〉0+ 12 (〈V 2〉0

−〈V 〉20)

so that the first cumulant approximation is

〈e−V 〉0=̃ e−〈V 〉0 . (B.2)

Appendix C. Derivation of result 1

The partition function is defined by

Z :=∫ ∏

0<|k|<�

dφ̂(k) e−L̂[φ̂(k)].

We note some identities related to complex integration before we delve into actualcomputations. Let z̄ denote the complex conjugate of z, and write∫

dz dz̄ :=∫

dz ∧ dz̄ =∫

d (x + iy) d (x − iy)

=∫

dx ∧ dx − 2i∫

dx ∧ dy +∫

dy ∧ dy = −2i∫

dx ∧ dy. (C.1)

Here we are using the standard ideas of area integration in complex variables (see, for example,[20]). For any fixed k, we can define the integral

∫dφ̂(k) analogously since the integrals will

occur in complex conjugate pairs as defined by (C.1). With dz denoting dφ̂(k), the productintegral, and noting that the definition implies φ̂(k) and φ̂ (−k) are complex conjugates wewrite

∫dφ̂(k) dφ̂ (−k) =:

∫dz dz̄ . Also note that the integrals in the definition of Z involve

pairs (k,−k) yielding a grouping in the form of (C.1). Thus, we have the identity∣∣∣∣∣∣∫ ∏

0<|k|<�

dφ̂(k)

∣∣∣∣∣∣ =∣∣∣∣∣∏

0<k<�

∫dφ̂(k) dφ̂ (−k)

∣∣∣∣∣=

∣∣∣∣∣∏

0<k<�

2∫

d(Re φ̂(k))d(Im φ̂(k))

∣∣∣∣∣ , (C.2)

and note that the factor of two is a constant factor that will not be relevant to our results thoughwe retain it.

For subsequent calculations we may write the integrals in (C.2) more concisely as∫ ∏0<|k|<�

dφ̂(k) =∏

0<k<�

2∫

d(Re φ̂(k)) d(Im φ̂(k))

=∏

0<k<�/l

2∫

d(Re φ̂<(k)) d(Im φ̂<(k))∏

�l <k<�

2∫

d(Re φ̂>(k)) d(Im φ̂>(k))

=∫ ∏

0<k<�/l

2 dφ̂<(k)

∫ ∏�l <k<�

2 dφ̂>(k) =:∫

�< dφ̂(k)

∫�> dφ̂(k).

(C.3)

24


Recall the definitions (10) and abbreviate the second line of (8), the quartic nonlinearity,by V (see below). To simplify the notation, let∫ <

dk :=∫

0<|k|< �l

dk and∫ >

dk :=∫

�l <|k|<�

dk

β(k) :=N∑

j=1

(−1) j+1 c2 j

2k2 j + a (C.4)

Z :=∫ ∏

0<|k|<�

dφ̂(k) e−L̂[φ̂(k)]

L̂[φ̂(k)] =∫

0<|k|< �l

dk∫

�l <|k|<�

dkβ(k)

∣∣∣φ̂(k)

∣∣∣2 + V [φ̂(k)]

V [φ̂(k)] = b

2π

∫ �

−�

dk1 . . .

∫ �

−�

dk4

{φ̂ (k1) φ̂ (k2) φ̂ (k3) φ̂ (k4) δ (k1 + · · · + k4)

}. (C.5)

Using (C.3) we write (omitting the absolute values)

Z = e−βBF :=∫ ∏

0<|k|<�

dφ(k)e−L̂[φ̂(k)]

=∫

�< dφ(k) e− ∫ < dkβ(k)|φ(k)|2∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 e−V [φ(k)]. (C.6)

We would like to perform the integration over the large wavelengths, namely∫ > dk, rendering

the exp {−V [φ(k)]} , namely the φ4 terms into a form that is essentially an average over theφ2 terms. By doing this integration we are averaging over short length scales. We define theaverage

〈A [φ(k)]〉> :=∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 A [φ(k)]∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2

⟨e−V [φ(k)]

⟩>

:=∫

�> dφ(k)e− ∫ > dkβ(k)|φ(k)|2 e−V [φ(k)]∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 =:

⟨e−V [φ(k)]

⟩num>

〈1〉num>

(C.7)

and use it to rewrite (C.6) as

e−βBF =∫

�< dφ(k) e− ∫ < dkβ(k)|φ(k)|2 ⟨e−V [φ(k)]⟩>

〈1〉num> . (C.8)

We use the first cumulant expansion (see appendix B) and approximate⟨e−V [φ(k)]

⟩>

=̃ e−〈V [φ(k)]〉> (C.9)

so that (C.8) can be written, within this approximation, as

e−βBF =∫

�< dφ(k) e− ∫ < dkβ(k)|φ(k)|2 e〈−V [φ(k)]〉> 〈1〉num>

=∫

�< dφ(k) e− ∫ < dkβ(k)|φ(k)|2∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 e〈−V [φ(k)]〉> . (C.10)

Note that 〈V [φ(k)]〉> does not depend on∫

�> dφ(k) integrals since they have already beenintegrated out. Now we can view (C.10) as a partition function in terms of a new free energy,and write

e−βBF =∫ ∏

0<|k|<�

e−L̂new[φ̂(k)] where

L̂new =∫

0<|k|< �l

dk∫

�l <|k|<�

dkβ(k)|φ̂(k)|2 + 〈V [φ(k)]〉> (C.11)

25


Our goal now is to evaluate 〈V [φ(k)]〉> and rescale variables so that we can obtain a freeenergy in the same form as the original, namely, (8). First we compute

〈V [φ(k)]〉> =∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2V [φ(k)]∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 (C.12)

= B∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 b

2π

∫ �

−�

dk1 . . .∫ �

−�

dk4

{φ̂ (k1) . . . φ̂ (k4) δ (k1 + · · · + k4)

}(C.13)

and let B := ∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 . We write (C.13) by splitting up the∫ �

−�dk1 . . .

∫ �

−�dk4 integrals in terms of the long and short wave modes:∫dki =

∫ <

dki +∫ >

dki.

resulting in 16 terms. Using (10) we also write

φ (ki) = φ< (ki) + φ> (ki) .

One can then write (C.13) as

B 〈V [φ(k)]〉> =:b

2π

∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2 {I1 + 4I2 + 6I3 + 4I4 + I5}

=: Q1 + 4Q2 + 6Q3 + 4Q4 + Q5 (C.14)

I1 :=∫ <

dk1

∫ <

dk2

∫ <

dk3

∫ <

dk4φ< (k1) φ< (k2) φ< (k3) φ< (k4) δ (k1 + · · · + k4)

I2 :=∫ <

dk1

∫ <

dk2

∫ <

dk3

∫ >

dk4φ< (k1) φ< (k2) φ< (k3) φ> (k4) δ (k1 + · · · + k4)

I3 :=∫ <

dk1

∫ <

dk2

∫ >

dk3

∫ >

dk4φ< (k1) φ< (k2) φ> (k3) φ> (k4) δ (k1 + · · · + k4)

I4 :=∫ <

dk1

∫ >

dk2

∫ >

dk3

∫ >

dk4φ< (k1) φ> (k2) φ> (k3) φ> (k4) δ (k1 + · · · + k4)

I5 :=∫ >

dk1

∫ >

dk2

∫ >

dk3

∫ >

dk4φ> (k1) φ> (k2) φ> (k3) φ> (k4) δ (k1 + · · · + k4)

(C.15)

Lemma 4. Q2 = Q4 = 0.

Proof. We prove Q2 = 0; the proof of Q4 = 0 is similar. We compute (see (C.3) for thedefinition of

∫�> dφ(k) )

2π

bQ2 =

∫�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2

∫ <

dk1

∫ <

dk2

∫ <

dk3

∫ >

dk4

· φ< (k1) φ< (k2) φ< (k3) φ> (k4) δ (k1 + · · · + k4)

Now use the fact that φ< (k1) is independent of the∫ > dki integrals and evenness of β) to

write this as2π

bQ2 =

∫ <

dk1

∫ <

dk2

∫ <

dk3φ< (k1) φ< (k2) φ< (k3)

∫ >

dk4

·∫

�> dφ(k) e− ∫ > dkβ(k)|φ(k)|2φ> (k4) δ (k1 + · · · + k4)

26


=∫ <

dk1

∫ <

dk2

∫ <

dk3φ< (k1) φ< (k2) φ< (k3)

·∫ >

dk4

∫�>

k �=k4dφ(k) e− ∫ >

k �=k4dkβ(k)|φ(k)|2

δ (k1 + · · · + k4)

·∫

dφ (k4) dφ (−k4) e−2β(k4)|φ>(k)|2φ> (k4) (C.16)

Note that in the last integration, one has φ = φ> in the domain of integration. Hence, one has∫dφ (k4) dφ (−k4) e−2β(k4)|φ(k)|2φ (k4) =

∫dz dz̄ e−2β|z|2 z

= 2∫

dx∫

dy e−2β(k4)(x2+y2) (x + iy) = 0 (C.17)

so that the integral vanishes and Q2 = 0. �

Lemma 5. Q1 = bB2π

∫ < dk1 . . . dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4).

Proof. We compute

2π

bQ1 :=

∫ <

dk1 . . .

∫ <

dk4

∫�> dφ>(k) e− ∫ >

β(k)|φ(k)|2

· φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4)

=∫ <

dk1 . . .

∫ <

dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4)

·∫

�> dφ>(k) e− ∫ >β(k)|φ(k)|2

= B∫ <

dk1 . . .

∫ <

dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4) . (C.18)

Note that B also decouples, but we will not need to evaluate it. Note that the factors of B cancelin the exponent for 〈V [φ(k)]〉> in (C.14). �

Lemma 6. One has the following identity for Q3 :2π

bQ3 = BC

∫ <

dk1 |φ< (k1)|2

C :=∫ >

dk31

β (k3). (C.19)

Proof. From the definition we write2π

bQ3 =

∫�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2

∫ <

dk1

∫ <

dk2

∫ >

dk3

∫ >

dk4

· φ< (k1) φ< (k2) φ> (k3) φ> (k4) δ (k1 + · · · + k4)

=∫ <

dk1

∫ <

dk2φ< (k1) φ< (k2)

∫ >

dk3

∫ >

dk4

· δ (k1 + · · · + k4)

∫�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2φ> (k3) φ> (k4) (C.20)

and let

T3 :=∫

�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2φ> (k3) φ> (k4) . (C.21)

27


We now divide the evaluation of (C.20) into three parts. In the sum over k3 and k4 we evaluateit for (i) k3 �= k4, (ii) k3 = k4, and (iii) k3 = −k4.

Case (i) k3 �= k4 : The integrals in (C.21) decouple, and noting that we have pairs ofintegrals including∫

dφ (k3) dφ (−k3) e−2β(k3)|φ(k3)|2φ> (k3)

=∫

dz dz̄ e−2β|z|2 z = 2∫

dx∫

dy e−2β(k4)(x2+y2) (x + iy) = 0

(identical to the calculation of (C.17). Hence T3 = 0 and so these terms vanish.Case (ii) k3 = k4 : We evaluate T3 as

T3 :=∫

�>k �=±k3

dφ(k) e− ∫ >

k �=±k3dkβ(k)|φ(k)|2 T31

T31 :=∫

dφ (k3) dφ (−k3) e−2β(k3)|φ(k3)|2φ (k3)2

=∫

dz dz̄ e−2β(k3)|z|2 z2 (converting to polar coordinates)

= 2∫ ∞

0r dr

∫ 2π

0dθ e−2β(k3)r2

r2 e2iθ = 0,

since the θ integral vanishes.Case (iii) k3 = −k4 : In this case T3 is expressed as

T3 =∫

�>k �=±k3

dφ(k) e− ∫ >

k �=±k3dkβ(k)|φ(k)|2 T32

T32 :=∫

dφ (k3) dφ (−k3) e−2β(k3)|φ(k3)|2φ (k3) φ (−k3) . (C.22)

We perform the complex area integration for T32 in polar coordinates:

T32 =∫

dz dz̄ e−2β(k3)|z|2 |z|2

= 2∫ ∞

0r dr

∫ 2π

0dθ e−2β(k3)r2

r2

= 4π

β2

∫ ∞

0

(β1/2r

)3e−2(β1/2r)

2

d(β1/2r)

= π

2β (k3)2 . (C.23)

Now we evaluate the remaining integrals in (C.22):∫dφ(k) dφ (−k) e−2β(k)|φ(k)|2 = 2

∫ 2π

0dθ

∫ ∞

0r dr e−2β(k)r2

= 4π1

−4β(k)

∫ ∞

0(−4βr) e−2β(k)r2

dr = π

β(k). (C.24)

Using (C.22)–(C.24) to evaluate T3, we obtain the identity

T3 =⎛⎝ ∏

k �=k3,k>0

π

β(k)

⎞⎠ π

2β (k3)2 = 1

2

1

β (k3)

∏>

k>0

π

β (k3). (C.25)

28


Now recalling the definition of Q3 and noting the results of the three cases so that onlyk3 = −k4 contributes, we see that the dk4 integral leads to a delta function and yields

2π

bQ3 =

∫ <

dk1

∫ <

dk2φ< (k1) φ< (k2)

∫ >

dk3

∫ >

dk4

· δ (k1 + · · · + k4)

∫�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2φ> (k3) φ> (k4)

and observe that the dk4 integration introduces a delta function δ (k3 + k4) . Thus, we have2π

bQ3 =

∫ <

dk1

∫ <

dk2φ< (k1) φ< (k2)

∫ >

dk3

∫ >

dk4

· δ (k1 + · · · + k4) δ (k3 + k4)1

2

1

β (k3)

∏>

k>0

π

β(k)

=∫ <

dk1

∫ <

dk2φ< (k1) φ< (k2)

∫ >

dk3δ (k1 + k2)1

2

1

β (k3)

∏>

k>0

π

β(k)

=∫ <

dk1φ< (k1) φ< (−k1)

∫ >

dk31

β (k3)

∏>

k>0

π

β(k). (C.26)

Recall that B := ∫�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2 so that by the calculation of (C.24) we have

B =∏>

k>0

π

β(k), (C.27)

and we can write Q3 as2π

bQ3 = B

∫ <

dk1φ< (k1) φ< (−k1)

∫ >

dk31

β (k3)

= BC∫ <

dk1 |φ< (k1)|2

C :=∫ >

dk31

β (k3). (C.28)

The calculation of C is in section 3.1. �

Lemma 7. Q5 = Const.

Proof. We note that2π

bQ5 = b

2π

∫ >

dk1 . . .

∫ >

dk4δ (k1 + · · · + k4)

·∫

�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2φ> (k1) . . . φ> (k4)

= Const (C.29)

since each of the terms integrates out. �

Lemma 8. One has the result

〈V [φ(k)]〉> =:1

B{Q1 + 4Q2 + 6Q3 + 4Q4 + Q5}

= b

2π

∫ <

dk1 . . . dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4)

+ 6Cb

2π

∫ <

dk1 |φ< (k1)|2 + Q5 (C.30)

where Q5 is a constant (independent of φ).

29


Recall (C.10), namely

e−βBF =∫

�<dφ(k) e− ∫ < dkβ(k)|φ(k)|2∫

�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2 e〈−V [φ(k)]〉>.

We note that e〈−V [φ(k)]〉> has already been integrated over∫ > and so does not depend on

|k| > �/l. Hence we can integrate the term∫�>dφ(k) e− ∫ > dkβ(k)|φ(k)|2 =: B

by itself. Using (C.30) in (C.10) then we can write (C.10) as

e−βBF = B∫

�<dφ(k) e− ∫ < dkβ(k)|φ(k)|2 (C.31)

· exp

[−

{6C

b

2π

∫ <

dk1 |φ< (k1)|2

+ b

2π

∫ <

dk1 . . . dk4φ< (k1) . . . φ< (k4) δ (k1 + · · · + k4) + Q5

}]

We can combine the |φ< (k1)|2 terms in the exponential and write the partition function in theform of (C.11), namely,

e−βBF = B e−Q5

∫ ∏0<|k|<�/l

e−L̂new[φ̂(k)]. (C.32)

Appendix D. Tables

Here we list the tables used in section 5.3

Table D1. The L2 and L∞ differences between the true solution and approximations forc6 = 10, . . . , 35.

c6 10 12 15 18 20 23 25 27 30 35

η 1.856 1.906 1.965 2.020 2.056 2.098 2.123 2.149 2.182 2.238σ 1.176 1.194 1.217 1.237 1.246 1.266 1.277 1.286 1.300 1.317L2 0.232 0.241 0.250 0.0257 0.264 0.270 0.273 0.276 0.28 0.289L∞ 0.94 0.954 0.097 0.100 0.10 0.103 0.102 0.103 0.104 0.105

Table D2. List of e(c6, n, ∞) values for the first prototype equation.

e-Table (L∞) n = 2 n = 4 n = 6 n = 8 n = 10

c6 = 2 0.042 0.06 0.09 0.1 0.11c6 = 4 0.026 0.045 0.06 0.07 0.08c6 = 6 0.019 0.035 0.047 0.058 0.069c6 = 8 0.015 0.028 0.039 0.050 0.061c6 = 10 0.013 0.024 0.035 0.046 0.051c6 = 12 0.011 0.022 0.033 0.038 0.044c6 = 16 0.012 0.017 0.024 0.031 0.037

30


Table D3. List of e(c6, n, 2) values for the second prototype equation.

e-Table (L2) n = 2 n = 4 n = 6 n = 8 n = 10

c6 = 2 0.084 0.14 0.18 0.21 0.24c6 = 4 0.055 0.097 0.13 0.16 0.18c6 = 6 0.042 0.076 0.101 0.130 0.151c6 = 8 0.034 0.045 0.088 0.110 0.131c6 = 10 0.029 0.054 0.076 0.097 0.114c6 = 12 0.025 0.036 0.056 0.086 0.103c6 = 16 0.022 0.034 0.055 0.071 0.085

Table D4. minα∈(1,1.5)

∥∥φ4,α − φ4+n

∥∥Lp is reached is for α = αmin = 1.03.

α 1 1.01 1.02 1.03 1.04 1.05∥∥φ4 − φ4+1,α

∥∥1

0.103 0.068 0.038 0.036 0.056 0.086∥∥φ4 − φ4+1,α

∥∥2

0.031 0.020 0.011 0.01 0.018 0.028∥∥φ4 − φ4+1,α

∥∥∞ 0.017 0.011 0.006 0.004 0.009 0.014


∥∥φ8,α − φ8+n


α 1 1.01 1.02 1.03 1.04 1.05∥∥φ8 − φ8+2,α

∥∥1

0.124 0.083 0.044 0.022 0.051 0.087∥∥φ8 − φ8+2,α

∥∥2

0.034 0.022 0.011 0.006 0.015 0.026∥∥φ8 − φ8+2,α

∥∥∞ 0.015 0.09 0.005 0.002 0.006 0.011


∥∥φ12,α − φ12+n


α 1 1.01 1.03 1.04 1.05∥∥φ12 − φ12+4,α

∥∥1

0.18 0.137 0.054 0.032 0.053∥∥φ12 − φ12+4,α

∥∥2

0.048 0.036 0.013 0.007 0.015∥∥φ12 − φ12+4,α

∥∥∞ 0.022 0.17 0.006 0.002 0.007


∥∥φ20,α − φ20+n


α 1 1.02 1.04 1.05 1.06∥∥φ20 − φ20+8,α

∥∥1

0.324 0.199 0.089 0.079 0.098∥∥φ20 − φ20+8,α

∥∥2

0.072 0.044 0.020 0.017 0.025∥∥φ20 − φ20+8,α

∥∥1

0.029 0.019 0.009 0.006 0.010

31



∥∥φ4,α − φ4+n


α 1 1.01 1.02 1.03 1.04 1.05∥∥φ4 − φ4+1,α

∥∥1

0.172 0.112 0.064 0.025 0.071 0.123∥∥φ4 − φ4+1,α

∥∥2

0.049 0.033 0.017 0.007 0.018 0.034∥∥φ4 − φ4+1,α

∥∥∞ 0.022 0.015 0.007 0.002 0.007 0.013


∥∥φ8,α − φ8+n


α 1 1.01 1.02 1.03 1.04 1.05∥∥φ8 − φ8+2,α

∥∥1

0.203 0.141 0.082 0.022 0.071 0.128∥∥φ8 − φ8+2,α

∥∥2

0.055 0.038 0.020 0.007 0.017 0.033∥∥φ8 − φ8+2,α

∥∥∞ 0.023 0.016 0.008 0.002 0.006 0.013


∥∥φ12,α − φ12+n


α 1 1.01 1.02 1.03 1.04 1.05 1.06∥∥φ12 − φ12+4,α

∥∥1

0.294 0.229 0.163 0.100 0.046 0.083 0.137∥∥φ12 − φ12+4,α

∥∥2

0.076 0.058 0.040 0.0023 0.009 0.017 0.034∥∥φ12 − φ12+4,α

∥∥∞ 0.030 0.23 0.016 0.008 0.003 0.005 0.012


∥∥φ20,α − φ20+n


α 1 1.02 1.04 1.05 1.06 1.07∥∥φ20 − φ20+8,α

∥∥1

0.381 0.237 0.096 0.043 0.094 0.152∥∥φ20 − φ20+8,α

∥∥2

0.095 0.058 0.021 0.009 0.020 0.038∥∥φ20 − φ20+8,α

∥∥1

0.036 0.022 0.008 0.003 0.006 0.013

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