jmp32_3031

A new perturbative approach to nonlinear partial differential equations Carl M. Bender and Stefan Boettcher Department of Physics, Washington University, St. Louis, Missouri 63130

Kimball A. Milton Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019

(Received 28 December 1990; accepted for publication 20 May 1991)

This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation u, + u2(, = u,, the general nonlinear equation u, + u%, = u, is considered and expanded in powers of 6. The coefficients of the S series to sixth order in powers of S is determined and Pad& summation is used to evaluate the perturbation series for large values of S. The numerical results are accurate and the method is very general; it applies to other well- studied partial differential equations such as the Korteweg-de Vries equation, u, + UU, = u,,.

I. INTRODUCTION

In this paper, we will show how to solve a class of nonlinear partial differential equations

u, + u%, = 24, (1.1) by expanding in powers of 6. To illustrate our perturbative procedure in the simplest way, we have chosen Gaussian initial conditions. For such initial conditions, we will find that the perturbative analysis can be carried out explicitly to any order in S. Note that for S = 0 ( 1.1) is linear and for S = 1 ( 1.1) becomes the Burgers equation, a well-known nonlinear wave equation.

Perturbative expansions of the type we use here have been studied intensively in recent years.’ The simplicity and usefulness of this method is demonstrated in a variety of fields ranging from ordinary differential equations to quan- tum field theory. The purpose of the present paper is to show that this method can be employed to solve nonlinearpartial differential equations with the same ease and success as ordinary differential equations. The central idea of this method is to generalize the original nonlinear problem to a family of problems by introducing a parameter 6 in the exponent such that for S = 0 the problem becomes linear and solvable whereas for 6 = 1 the original nonlinear problem is recov- ered. An expansion in powers of S will then convert the nonlinear problem into an infinite sequence of linear problems which are all formally solvable.

The underlying philosophy of this approach relies on the observation that there is a smooth transition in the be- havior of the solution as S -+ 0 for many nonlinear problems. That is, what we perceive as the special case of linearity re- veals itself as just another degree on the nonlinearity scale which 6 measures.

The following treatment will serve as a demonstration that this point of view is also valid for a number of known nonlinear partial differential equations. In Sec. II, we show how to obtain the zeroth- and first-order solutions in S for the generalization of Burgers’ equation in ( 1.1) using Gaus- sian initial conditions. We then prove by induction that it is straightforward to calculate any order in perturbation theory explicitly. In Sec. III, we compare the results of a sixth- order calculation in S with the exact solution obtained by

numerical integration of the differential equation ( 1.1). For the chosen times the (3,3)-Pad& of the series in S reaches a high degree of accuracy globally in the spatial coordinate. In Sec. IV, we conclude with some remarks concerning the ap- plication of the perturbative S expansion to the Korteweg-de Vries equation.

II. PERTURBATIVE &EXPANSION FOR THE BURGERS EQUATION

We seek a perturbative expansion in powers of S for the solution u (x,t) of the generalized form of Burgers’ equation in (1.1):

#(XJ) = dO’(x,t) + Su”‘(X,t) + 62U’2’(X,t) + - *. . (2.la)

For the initial conditions, we choose dO’(x,O) = 24(x,0); u(“)(x,O) = 0 for n>0.(2.lb)

Substituting the formal perturbation series in (2.la) into ( 1.1) and expanding the result in terms of 6 yields 24;“’ + Suj” + . ..+u~“‘+Su~“+Su~o’ln(u’o’)+...

= ug) + sug + * * - . (2.2) It is a general feature of this type of expansion that the problem of a nonlinear differential equation decomposes order by order in 6 into infinitely many inhomogeneous l inear problems. For each of these linear problems, the associated homogeneous problem remains the same in any order. The inhomogeneity of the equation of nth order in S will in general depend on the solution of all previous n - 1 orders. Of course, for autonomous equations like ( 1.1)) the zeroth-order equation is always homogeneous.

A. Zeroth-order calculation Keeping only terms to zeroth order in S, we get, from

(2.21, ui”’ + u;o’ - ugt”,’ = 0. (2.3)

The general solution of (2.3) is

u’“‘(-w) = & _ m -Jm d<qb([) exp( - r(x-~t-‘12),

(2.4)

3031 J. Math. Phys. 32 (1 I), November 1991 0022-2488191 /I 13031-08$03.00 @ 1991 American Institute of Physics 3031

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wherefrom (2.lb) #(x) = u’~‘(x,O) = u(x,O) is the initial condition. For the special case of Gaussian initial conditions,

4(x) = a@Ze- b*, (2.52 we obtain from (2.4)

dO’(x,t) = aJGZ3-‘exp{ - (x - t)“/[4f(t) ]I, (2.6)

where, for convenience, we define j-(t) = t + l/(46).

Note that b lim 4(x) = a&x). - m

(2.7)

B. First-order calculation

Choosing only terms of first order in S, (2.2) yields u:*) + ukl) - ~2) = - ui”) ln(u(O)), (2.8)

Because this equation is linear, it is easy to obtain a general solution: t P(x t) = dr

9 s s m dc1”‘(&,7)

f+Gqz7J --m

xexp{- [(x-t) - (~-f)12/Pw--7)lh (2.9)

where the inhomogeneity is given by I(“(x,t) = - U~“‘(x,t)ln[u’o’(x,t)]. (2.10)

Note that in accordance with (2.lb), u”‘(x,O) ~0. For the case of the Gaussian initial conditions in (2.5)) (2.10) reads explicitly

I”‘(x,t) = - --L (x - tj3 V-(t)*

+ ln[4rrft)/a2] 4f(G

(x - t) 1 X [a/$WXlexpC - (x - t)2/[4f(t)lI.

(2.11) If we insert (2.11) into (2.9) and make the change of vari- able

5-4 + T + [fc~,/fW 1 (x - 0, (2.12) we obtain a simple integral of a Gaussian multiplying a poly- nomial in f ‘. It is remarkable that the residual exponent after the shift in (2.12) does not depend on r, but rather repro- duces U(O) (x,t). Thus, after evaluating the 6 ’ integral, we are left with trivial integrals of the form

with k,i,m& such that the integrals converge. The final result is

- - To prove this observation, we merely need to point out that, if (2.18) is true, then all (u”‘/#‘))aCin (2.16b) and conse- quently Pn [u] are polynomials in (x - f) with coefficients depending on t, Since

(2.13) and

C, Calculation in nth order

It is quite clear now that to nth order in S we will have to solve the linear, inhomogeneous partial differential equation

tp + up - ug:’ = I(n)(x,t). (2.14) To obtain I(“) (x,t) we observe that we can rewrite ( 1 rl 1) as

ut+u,-uu,= -u,(u”-1). (2.15) Next we expand U’ - 1,

u&-l= c w c(lnU)“, n=l n!

and substitute (2. la) for U. Reexpanding this series in powers of 6 gives

u6 - 1 = j, 6” y; ;$I yyJ/,- f

l#n

x c F, [u]b’-FI[u]b’

(2.16a) X:f= ,b,= m b,!.**LQ! ’ Zi= ,ib, = f

where the functional F, [u] is given by

F,,[u] = 2 ( - l)k-l(k- l)! k-1

x c (U(l)/U(0))“l,..(U(~)/u~o))~~

. L:= ,a? = k a, ! =*+a,! z:= ,ro, = n

(2.16b) Next, wemultiply (2.16a) by u,, where u isgiven by (2.la). Expanding in powers of S gives the right side of (2.15) :

- u,(d- 1) = 2 ljnl(n)(x,t), (2.1’7a) n=l

with [m/21 m--l [ln(u’O’)]“-k-I

I(n)(XJ)= - i uy-m) ,go & (m-k-l)f m=l

kfm

x c

F, [i,bl’**t-k[tt]bk (2.17b)

Z:, ,bi= I b,!*‘*bk! *

Bf= ,ib, = k

For the case of Gaussian initial conditions (2.5), we observe now that if, for all i < n, u(‘)(x,t) has the form

I

3i tP(x,t) = x a:“(t) (x - t)k

1 d”)(x,t), (2.18)

k=O then

I(n)(x,t) = i

2 H:"'(t)(x - igk I

d”‘(x,t). (2.19) k=O

In(P) = --J- (x - tp 4f(f)

(2.20)

3032 J. Math. Phys., Vol. 32, No. 11, November 1991 Bender, Boettcher, and Milton 3032


*~rn)=((~),+(~)(~))u(O), (2.21a)

with u(o)

x -= -- p)

l (x-t) VI0

(2.21b)

are also polynomials having this form, then (2.19) follows from (2.17b). Furthermore, by construction we can prove that, if (2.19) holds for I(“), then u(‘) will be of the form (2.18). Therefore, the generalization of (2.9) from n = 1 to arbitrary n > 0 reads

t P(x,t) = dr

Jwfw -“edI

2 H:n)(7-) (c - ~)%~“~(~,r)] k=O

[(x-t) - (g-7)1* w(t) --f(dl ’

(2.22)

or

447$(r) [ 1 -fc~vY-ct) 1

(2.23)

Note that u(“)(x,O) ~0 in accordance with (2.lb). Again, shifting g-5 + r + Lf(r)/f(t>] (x - t) as in

(2.12), we obtain

U(n)w) = u’0’kt) k$o ,i, (;) [Jgq,

s I X drH:“‘(r)f(r)’

0 447$(T) [ 1 --fc~)mt> 1

X

Xexp 1 Cl2 - 4f(7)[1 -f(r,/f(t,] I *

Finally, we use

if r odd

= [Ctcr+ “‘21-(+!-), if r even

and rearrange the summation to get

d”)(x,t) = dO’(x,t) a:“‘(t)(x - t)k )

with

3033 J. Math. Phys., Vol. 32, No. 11, November 1991

(2.24)

(2.25)

(2.26)

l(3n- kV21 k

x ( - l)ifift)j-i s rd?-H;;:&)f(r)2i-J. 0

(2.27)

This completes the proof.

III. NUMERICAL RESULTS

Before we proceed with the presentation of some numerical results, it is instructive to conduct a dimensional analysis of ( 1.1) . Related to this is the subject of minimal sensitivity that we address later. The subsequent numerical results show that with such a high-order approximation in the 6 expansion a high degree of accuracy can be reached at S = 1, even without imposing the condition of minimal sensitivity.

A. Scaling properties of the Burgers equation

To provide ( 1.1) with the physically correct dimensions, we note that since u has dimensions of velocity, we must introduce a velocity M and a diffusion coefficient Y such that

24, + M( u/M)%, = vu,. (3.1) By making the scale transformations

x= (v/M)X, t= (v/M2)T, u=MU, (3.2) we obtain dimensionless quantities X, T, and U and, with those, we reproduce ( 1.1). Note also that the initial conditions need to be resealed accordingly. For instance, for the Gaussian initial conditions in (2.5), we need to scale

a = VA, b = (M2/&B, 4 = Ma. (3.3) We then obtain

a(X) = A(m)e- Bxz, (3.4)

where Q(X) c U(X,O). (3.5)

Because of (3.3), for any choice of a and b, the perturbative solution of ( 1.1) will always be implicitly dependent on M and Y. This dependence can be made explicit by using the transformations (3.2).

B. Minimal sensitivity

Note that at S = 1, (3.1) is independent of M. However, the velocity parameter M appears in every finite order of perturbation theory. Apparently then, as the order of perturbation theory increases, the results should become less and less sensitive to the value of M when 6 = 1. Thus, if we are interested in the solution to ( 1.1) at S = 1, in every order in perturbation theory, it seems reasonable to choose the value ofM to be the point at which the perturbative solution is least sensitive to changes in M. Thus, one could compute the de- rivative with respect to M of the perturbation expansion in nth order and set the result equal to 0. This gives an algebraic equation which can, in principle, be solved for M. The resulting solution for Mean then be substituted back into the nth-

Bender, Boettcher, and Milton 3033


I5 ., I1 I,.II., * nc i : o=u.a

n $ ’ _I n v \ ~

,,i/. ! , ,!-7 -Yj LO’

10’

100

lo-’

10-z

10-3

lo-’

ool 102

IO’

100

10-l

10-2

lo-’

IO-’

lo-’ Ill,,r,ll*l -5 -5 0 0 5 5

x x

15

10

05

0.0

IO0

lo-’

lo-’

10-1

10-’

-5 0 5 x

10-S - -5 0 5

x

IS., II.I, . ,%

R,fl4 t - ~“.Y !&I) n 1 1 I \ , P3,

0.5

_./

1

00

10’ ~,““I”’ II

6=0.5 (3.3)-Pade 10'

10'

10-1

10-Z

10-a

lo-

FIG. 1. A compar ison of the exact solution to the general ized Burgers equat ion in ( 1.1) for the Gaussian initial condit ions in (2.5) with D = 5 and b = 1 and the first six Pad6 approximants in the diagonal sequence, P y , P I , Ph , P: , P: , Pi, constructed from the 6 expansion at 6 = 1. Both graphs in each of the upper diagrams display u(x,t) at t = 1. Below each compar ison is a plot of the relative error \erC, 1 between the exact solution and Padi approximation. Observe that except in the vicinity of the poles of the Pad& approximants, the numerical accuracy is good. Poles appear as upward spikes in the relative error plots. Note that as the order of the 6 expansion increases, the widths of the neighborhoodsof the poles in which the Pad& approximation is inaccurate shrink rapidly to zero. The downward spikes on the error plots are points where the Pad& approximation happens to cross the exact solution.



order perturbation series. It is interesting that the value of A4 tally when the principle of minimal sensitivity is used.’ obtained in this way is actually not a constant as we have However, the resulting equations are complicated and, fur- assumed all along but rather has a dependence on both space thermore, our high-order perturbation calculations are so and time. Thus, the principle of minimal sensitivity elevates accurate that we see no reason at this point to discuss the what was initially a constant to a space-time dependent clas- principle of minimal sensitivity further in this paper. Our sical field. Some initial experimentation shows that the accu- approach here is simply to take A4 = 1 and proceed with the racy of a given perturbation calculation increases dramati- calculation.

*O-6- -s 0 5

x -s 0 5

x

1.5 1 ( 3 I , 3

6=1.0 i

-5 0 5 x

(2.3)-Pade :

1

I

---T v- I.” = I m -f -! Pl 1 II i

0.5 t /\-II ,,px.;, , ,ii

102 I , I , I I I , I

6=1.0 (3,3)-Pade

100 100

10-l 10-I

10-a 10-a

10-a 10-s

lo-’ lo-’

10-a lo-’ -5 0 5 -5 0 5

x Y

FIG. 2. Same as in Fig. 1 except that S = 1. As S increases, the accuracy of the Pad.6 approximation gradually decreases, as expected.



C. Numerical results

The results of the S-expansion calculation for the Gaus- sian initial conditions in (2.5) with parameters a = 5 and b = 1 are given in Figs. 1-4. On these figures, we consider four values of 6: S = l/2, 1 (Burgers’ equation), 2, and 4. (Only the case S = 1 is analytically solvable by means of the

Hopf-Cole transformation.3 ) On each of these figures, we compare the exact solution at t = 1 found numerically with the first six elements of the diagonal Pad.5 sequence, Py , P i, P l, P s, P :, and Pi, constructed from the S expansion. We have used a Pad6 summation here to improve the rate of convergence of the S perturbation series. We do not know what the radius of convergence of this series is. However, our

1.5 I ’ * 11 l- : ’

0.5

6=2 0 IO'

i l.2)-Pade

, 4 .F &=2.0 (3.3)-Pade 1

10-J- -5 0 5

x

.I 4=2.0 (l,l)-Pade 1 ‘6=2.0 10’

(2,2)-Pade q

7

FIG. 3. Same as in Fig. I except that 6 = 2

3036 J. Math. Phys., Vol. 32, No. 11, November 1991 Bender. Boettcher. and Milton 3036


experience’ has been that the accuracy of the S series is sig- IV. PERTURBATIVE 6 EXPANSION OF THE nificantly enhanced by Pad6 summation. We also plot the KORTEWEG-DE VRIES EQUATION relative error between the exact solution at t = 1 and the Padi approximation. Observe that, as one might expect, the To demonstrate the generality of the perturbative S ex- accuracy of the Pad6 decreases slightly as S increases away pansion, we use it in this section to obtain at least formally from 0. However, except in the vicinity of poles (zeros of the the solution to a two-parameter class of nonlinear partial denominator) the Pad6 approximations are good. differential wave equations:

FIG. 4. Same as in Fig. 1 except that 6 = 4.

IO’ ,

6=4.0 Il2kPade 10’

.

I

100 ------I

10-I

Ad/ ,

10“

10-3

.‘_; 1

if A

lo-’

10-J - -5 0 5

x

1.5 1 , I I I . 1 ,-s

1 o=‘t.v -i A r ! 1

i lOI- P’zly h, -

IO’ IO’

10’ 10’

100 100

lo-’ lo-’

10“ 10“

10-5 10-5

IO-’ IO-’

10-S -5 -5 0 0 5 5

x x

I”“! 6=4.0

, A i i

i

lo-’ r

10-J 3 ’ j ’ ’ ‘. a ’ j ’ -5 0 5

x

1.5,., ll,!l.‘.,,,

E a=‘4.0 1 4 1 P’3 i

10

05 I

I

~ u(x.1) -j

1

i 1

00 ,l,,,,I,[

102 1 , I I I 1 I 1 , ,

6=4.0 (3,3)-Pade

10-5

LO-’ 1 !

lo-st ’ ’ ’ ’ * 4 ’ 1 ’ ’ ’ ’ 1 -5 0 5

x

3037 J. Math. Phys., Vol. 32, No. I I, November 1991 Bender, Boettcher, and Milton 3037


u, fdu, =a;,, m= 1,2,3 ,.... (4.1) This class of wave equations contains not only Burgers’ equation (S = 1, m = 2), but also the Kortewegde Vries equation (8 = 1, m = 3). When m = 3, for a Gaussian initial condition it is easy to get the leading-order term in the perturbative S expansion explicitly in the form of product of an Airy function and an exponential.

A. Perturbative S expansion for (4.1)

If we substitute the series (2. la) into (4.1)) the problem in (4.1) decomposes to

ui”’ + u:‘“’ - d~rd”) = 0, m = 1,2,3 ,..., (4.2a) uy + up) - d,“u’“’ = Icn)(x,t), m = 1,2,3 ,...,

(4.2b) with the appropriate initial conditions given in (2. lb). We point out that I ‘“‘(x,t) will in general be some complicated function depending on all previous uCi) (x, t), i < n. The solution of (4.2a) for general initial conditions q+(x) = u(x,O) = u’“‘(x,O) is u(O) (x,t) = s 1, g 5%) J;, $

XexpC( -i)“tp”+ip[~- (7x--t)lI, m = 1,2,3 ,... . (4.3)

The solution of (4.2b) with u(“)(x,t) =O for n > 0 is

P(x,t) = Id?- I I - Lx. I’“‘(fJ,T) 0 --Jz;r

X s

m 4 -exp{( - i)“(t - r)pm -mfi

+&[(c-T)--(x-t)]), m=1,2,3 ,.... (4.4)

An interesting special case is obtained by imposing the Gaus- sian initial conditions in (2.5). With these initial conditions, we get for (4.3)

ucO)(x,t) = a - exp{( - i)mtpm

- i(x - t>p) I

co d< - -mJ21;:

Xexp{ - bc2 + ip{), m = 1,2,3 ,... . (4.5)

Evaluating the Gaussian integral gives

dO’(x,t) = es_” dp exp{( - i)‘“tpm m

- ( 1/4b)pz - i(x - t)p}. (4.6)

B. Korteweg-de Vries equation

For the particular case of the Korteweg-de Vries equation, we choose rn = 3 in (4.6) and obtain

u”‘(x,t) =$~~_dpexp[itp’~$p”-i(x-tf)p].

(4.7) A shiftp +p’ - i/( 12bt) eliminates the quadratic term in the exponent with the result that

- (X - r>/12fir

s m dp’

0

X cos tp’3 + i ( [

-$& - (x - t)]P’} . (4.8)

We easily identify the above integral as a representation of the Airy function Ai. Thus,

XAi I

1/(48b*t) - (x - t)

(3t)‘/3 1 (4.9)

is the leading term in the solution of the Korteweg-de Vries equation for Gaussian initial conditions in the perturbative S expansion.

ACKNOWLEDGMENT

We thank the U.S. Department of Energy for financial support.

’ See C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, Jr., J. Math. Phys. 30, 1447 ( 1989) and references therein. See also B. J. Laur- enzi, J. Math. Phys. 31,2535 ( 1990). After the completion of this work, it was pointed out to us by H. P. Greenspan that W. L. Kath and D. S. Cohen in Adv. App. Math. 67,79 ( 1982) applied techniques similar to those used in this paper to a nonlinear diffusion equation in which the diffusion constant is a power of the concentration.

‘H. F. Jones and M. Monoyios, Int, I. Mod. Phys. A 4, 1735 (1989). 3 G. B. Whitham, Linear and Nonlitzear Waws (Wiley, New York, 1974),

p. 97.



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