Dispersion relation of the nonlinear Klein-Gordon equation through a variational method

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Dispersion relation of the nonlinear Klein-Gordon equation

through a variational method

Paolo Amorea兲

Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, México

Alfredo Rayab兲

Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, México and Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México. Circuito Exterior, C. U., A. Postal 70-543, México, D. F., 04510

共Received 26 May 2005; accepted 23 January 2006; published online 30 March 2006兲

We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equa-tion in the case of strong nonlinearities using a method based on the linear delta expansion. All the results obtained in this article are fully analytical, never involve the use of special functions, and can be used to obtain systematic approximations to the exact results to any desired degree of accuracy. We compare our findings with similar results in the literature and show that our approach leads to better and simpler results. ©2006 American Institute of Physics.

关DOI:10.1063/1.2176393兴

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Jo-sephson junctions. The simple sinusoidal solutions to the linear wave equation, which provide a dispersion relation independent of the amplitude, are lost when nonlinear terms are considered. As a matter of fact the exact solu-tions cannot be expressed in a simple form in terms of their linear counterparts, although they may still be os-cillatory. Moreover the dispersion relation obtained in this case turns out to depend upon the amplitude. The solution of the nonlinear wave equation poses an interest-ing challenge, especially in the case of strong nonlineari-ties, where perturbation theory by itself is not applicable: indeed in such cases the perturbative series does not con-verge and no sensible information can be extracted di-rectly from it. Here we present a variational method based on the linear delta expansion to find fully analytical approximate dispersion relations for the nonlinear Klein-Gordon and the Sine-Klein-Gordon equations for weak and strong nonlinearities. Our method can be easily general-ized to other cases and provides a systematic way to achieve the desired degree of accuracy. The solutions ob-tained in this paper are fully analytical and never involve the use of special functions.

I. INTRODUCTION

In this article we study the problem of describing the propagation of traveling waves obeying the nonlinear Klein-Gordon and the Sine-Klein-Gordon equations. Under certain con-ditions the effect of the nonlinearity is to preserve the oscil-latory behavior of the solutions and, at the same time, modify the dispersion relation for the traveling waves which

turns out to depend on the amplitude of oscillation. As a matter of fact we are considering conservative systems, for which the dynamics can be mapped to the nonlinear oscilla-tion of a point mass in a one-dimensional potential. The main goal of this article is to explore the effects of the nonlinearity on the solutions, providing simple and efficient approxima-tions. Although for weak nonlinearities, this task can be ac-complished by applying perturbative methods 共 correspond-ing to performcorrespond-ing an expansion in a small parameter which governs the strength of the nonlinearity itself兲, the situation is more complicated in the presence of strong nonlinearities. In such a regime perturbation theory cannot be applied, since the perturbative series do not converge.

Such a problem was studied in Ref. 1, where nonpertur-bative formulas for the dispersion relations of the traveling wave in the Klein-Gordon and the Sine-Gordon equations were derived. The formulas obtained by Lim et al. provide an accurate approximation to the exact results even when the nonlinearity is very strong.

In this article we consider the same problems of Ref. 1 and apply to them an approach which has been developed recently.2–6 Our approach is fully nonperturbative in the sense that it does not correspond to a polynomial in the non-linear driving parameter and, when applied to a given order, allows us to obtain analytical expressions for the dispersion relations, which never involve special functions, to any de-sired level of accuracy. It is worth mentioning that in the case of weak nonlinearities, an expansion of the nonperturbative results in powers of the nonlinear parameter is sufficient to recover the perturbative results.

Let us briefly describe the problem that we are interested in. We consider the nonlinear Klein-Gordon equation:

utt−uxx+V

⬘

共u兲= 0, 共1兲

whereV

⬘

共u兲 is a function ofu, which we will assume to be odd, and the prime is the derivative with respect to u. To

a兲_{Electronic mail: [email protected]} b兲_{Electronic mail: [email protected]}

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determine the periodic traveling wave, we set

u=u共␪兲, ␪=kx−␻t. 共2兲

After substituting into Eq.共1兲we find

⍀2_u¨₊_V

_⬘

_共_u_兲_{= 0,} _共3兲

where⍀2=共␻2−k2兲andu˙⬅du/d␪.u共␪兲is periodic with pe-riod 2␲and fulfills the boundary conditions

u共0兲=A, u˙共0兲= 0, 共4兲

withAbeing the amplitude of the traveling wave. The solu-tion of Eq. 共3兲with the previous boundary conditions oscil-lates between −A and A. By integrating Eq. 共3兲 and taking into account Eq. 共4兲we obtain

1

2⍀2u˙2+V共u兲=V共A兲. 共5兲

Considering ⍀2_⬎_{0 we observe that}

⍀=

_冑

␲

2兰0

A_关

V共A兲−V共u兲兴−1/2du 共6兲

gives the exact expression for the dispersion relation of the nonlinear Klein-Gordon equation. We neglect the case ⍀2

艋0 since there is no traveling wave for this configuration. This article is organized as follows: In Sec. II we de-scribe the variational nonperturbative approach and apply it to derive approximate analytical formulas for the nonlinear Klein-Gordon equation; in Sec. III we apply our method to two further nonlinear equations; and, finally, in Sec. IV we draw our conclusions.

II. VARIATIONAL METHOD

An exact solution of Eq. 共1兲 can be accomplished in a limited number of cases, depending on the form of the po-tential V共u兲. However, when the nonlinearities due to the potential V共u兲 are small, it is still possible to find useful approximations using perturbation theory. The focus of this section will be on the opposite situation, when the nonlin-earities are not small and a perturbative expansion is not useful. In such a case one needs to resort to nonperturbative methods, capable of providing the solution even in the pres-ence of strong nonlinearities. One of such methods, which we will use in the present article, is the linear delta expansion 共LDE兲.7–10

The LDE is a powerful technique that has been applied to difficult problems arising in different branches of physics like field theory, classical, quantum and statistical mechanics. The idea behind the LDE is to interpolate a given problem

Pg with a solvable one Ps, which depends on one or more arbitrary parameters ␭. In symbolic form P=Ps共␭兲 +␦共Pg−Ps共␭兲兲.␦ is just a bookkeeping parameter such that for ␦= 1 we recover the original problem, and for␦→0 we can perform a perturbative expansion of the solutions ofPin

␦. The perturbative solution obtained in this way to a finite order shows an artificial dependence upon the arbitrary pa-rameter, ␭ and would cancel if the calculation were carried out to all orders. As such we must regard such dependence as

unnatural; in order to minimize the spurious effects of ␭we then require that any observable O, calculated to a finite order, be locally independent of ␭, i.e., that

⳵O

⳵␭ = 0. 共7兲

This condition is known as the “principle of minimal sensi-tivity” 共PMS兲.11We call ␭PMSthe solution to this equation.

共In the case where the PMS equation has multiple solutions, the solution with the smallest second derivative is chosen.兲 We emphasize that the results that we obtain by applying this method do not correspond to a polynomial in the parameters of the model as in the case of perturbative methods.

The procedure that we have illustrated is quite general and it will be possible to implement it in different ways depending on the problem that is being considered. In Refs. 2–4 the LDE was used in conjunction with the Lindstedt-Poincaré technique to solve the corresponding equations of motion. Our approach here is to apply the LDE directly to the integral of Eq.共6兲as in Refs. 5 and 6. We will consider the potential

V共u兲=u

2

2 +

␮u4

4 . 共8兲

The dispersion relation in this case can be obtained using Eq. 共6兲as

⍀= ␲

冑

1 −␮A

2

2兰0␲共1 −msin2␾兲−1/2d␾

, 共9兲

withm=␮A2_{/ 2}_共_{1 +}_␮_A2_兲_.

We consider the following approach to obtain the disper-sion relation of a periodic traveling wave. This comes from the equation for the period of oscillations,

T=

冕

−A

+A

冑

2

冑

E−V共u兲du, 共10兲

where the total energyEis conserved and ±A are the classi-cal turning points.

In the spirit of the LDE we interpolate the nonlinear potential V共u兲with a solvable potentialV0共u兲and define the

interpolated potential V_␦共u兲=V0共u兲+␦共V共u兲−V0共u兲兲. Notice

that for ␦= 1, V_␦共u兲=V共u兲 is just the original potential, whereas for␦= 0 it reduces toV0共u兲. Hence we can write Eq.

共10兲as5,6

T_␦=

冕

−A

+A

冑

2

冑

E0−V0共u兲

du

冑

1 +␦⌬共u兲, 共11兲

where

⌬共u兲=E−E0−V共u兲+V0共u兲 E0−V0共u兲

. 共12兲

Obviously E=V共A兲andE0=V0共A兲.

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T_␦=

兺

n=0

⬁

共2n− 1兲!! n!2n 共− 1兲

n_␦n

冕

−A

+A

冑

_2共_⌬_共 u兲兲n

冑

E0−V0共u兲

du. 共13兲

Observe that the integrals in each order of Eq.共13兲have integrable singularities at the turning points because⌬共±A兲is finite. Assume that 兩⌬共u兲兩艋⌬0⬍1 for u苸关−A,A兴, which

happens if ␭, the arbitrary variational parameter, is chosen appropriately. Then, the series共13兲converges uniformly for 兩␦兩⬍1 /⌬0, which includes the case ␦= 1.

For the potential given in Eq. 共8兲 we can chooseV0共u兲

= 1 +␭2_{/ 2}_u2 _{as the interpolating potential and, hence, we}

have

⌬共u兲= 2 1 +␭2

冋

␮

4共a

2₊_u2_兲₋␭ 2

2

册

. 共14兲

The parameter ␭ should be chosen to be ␭

⬎

冑

␮A2_{/ 2}

冑

_{1 + 1 /}_␮_A2_{which guarantees the uniform}

conver-gence of Eq.共13兲.

It is straightforward to check that at first order,

T_␦共0兲+␦T_␦共1兲=

_冑

2␲ 1 +␭2

再

1 −

␦

1 +␭2

冋

3 8␮A

2₋␭ 2

2

册

冎

. 共15兲 The PMS共7兲withO=Tyields

␭PMS=

冑

3␮A

2 . 共16兲

The period is found to be

TPMS=

4␲

冑

4 + 3␮A2. 共17兲

Correspondingly,

⍀LDE共1兲=

冑

1 + 3

4␮A2. 共18兲

In Ref. 6 it was found that with this value of␭PMSall the

remaining terms of odd order in Eq. 共13兲 vanish. Hence, retaining only nonvanishing contributions, the expression for the period at orderN is

T_␦共N兲=

_冑

4␲ 4 + 3␮A2

兺

n=0

N

共− 1兲n

冉

− 1/2 n

冊冉

− 1/2 2n

冊

冉

␮A2 4 + 3␮A2

冊

2n ,

共19兲 and, correspondingly,

⍀LDE共N_兲= 2␲

T_␦共N兲. 共20兲

At second order we have

⍀LDE共2兲=

冑

4 + 3A2␮ 2

冉

1 +3A

4_␮2_{共1024 +}_A2_␮_{共1536 + 611}_A2_␮_兲兲

1024共4 + 3A2␮兲4

冊

.

共21兲

At third order, the dispersion relation is given by

⍀LDE共3兲=

冑

4 + 3A2_␮

2

冉

1 +3A

4_␮2_共₃₈₅_A8_␮4_{+ 560}_A4_␮2_共_{4 + 3}_A2_␮_兲2_{+ 1024}_共_{4 + 3}_A2_␮_兲4_兲

16384共4 + 3A2_␮_兲6

冊

. 共22兲

We will compare the results obtained using our method, Eqs.共18兲,共21兲, and共22兲with the results obtained in Ref. 1, where the same problem has been solved using the harmonic balance technique in combination with the linearization of the nonlinear Klein-Gordon equation. The findings of Ref. 1 at first order, their expression for the dispersion relation co-incides with our Eq. 共18兲, whereas at the second order they find

⍀Lim共2兲=

冑

40 + 31␮A2+

冑

1024 + 1472␮A2+ 421␮2A4

72 .

共23兲

In the left-hand panel of Fig. 1 we make a comparison of the ratios of the dispersion relations obtained from Eqs.共18兲 and 共21兲–共23兲 to the exact dispersion relations for␮A2_⬍_0,

and in the right-hand panel of Fig. 1 we display the relative error

⌬= log10

冏

⍀−⍀exact

⍀exact

冏

共24兲

for␮A2_Ⰷ_{0. We can appreciate that our variational method at}

second order provides a smaller error than the method of Ref. 1 applied to the same order. The error is further reduced by using the LDE to the third order and can then be system-atically reduced using the general formula共19兲.

III. FURTHER EXAMPLES A. Sine-Gordon model

We now consider the Sine-Gordon model, which is gov-erned by the potential

V共u兲= − cosu 共25兲

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⍀2_u¨_{+ sin}_u_{= 0.} _共26兲

The exact dispersion relation in this case can be obtained from

⍀= ␲

2

冕

0

␲/2

共1 −m2sin2t兲−1/2dt

共27兲

withm= sin共A/ 2兲. Observe that in this case

T= 4

冕

0

␲/2

共1 −m2sin2t兲−1/2dt⬅4K共m2兲, 共28兲

withK共m兲being the elliptic integral of the first kind. We take advantage of this fact and make use of the nonperturbative series for the elliptic integral which was derived using the LDE technique.12At orderN, setting␭= −m/ 2 and␦= 1, it is given by the following expression:

KN共m,␭兲=␲ 2

兺

k=0

N

兺

_j

=0

k

⫻ ⌫共j+ 1/2兲 j!2共k−j兲!⌫共1/2 −k兲

共−m兲k

2k−j

冉

1 −m 2

冊

k+1/2. 共29兲

This expression provides a nonperturbative series for the

el-liptic integral of the first kind since it does not correspond to a simple polynomial in m.

To further improve this series we can use the Landen transformation13

K共m兲= 1 1 +

冑

mK

冉

4

冑

m

共1 +

冑

m兲2

冊

共30兲 and the inverse relation

K共m兲=2共1 −

冑

1 −m兲

m K

冉

共− 2 + 2

冑

1 −m+m兲2

m2

冊

. 共31兲

Notice that f共m兲= 4

冑

m/共1 +

冑

m兲2 _{maps a value 0}_⬍_m_⬍₁

into a new value m

⬘

=f共m兲⬎m. The inverse transformation f−1共m兲=共−2 + 2

冑

1 −m+m兲2/m2 maps a value m into a smaller one. Using this transformation we obtain more accu-rate approximations for the elliptic integrals. For example, at order 1 we find

KLDE共1兲共m兲=

␲

冑

1 −m

2 + 3

冑

1 −m

共32兲

and, correspondingly

⍀LDE共1兲=

1

4

冑

cos共A兲+ 12

冏

cos

冉

A

2

冊

冏

+ 3. 共33兲

At second order we find

⍀LDE共2兲=

16 cos2

冉

A

4

冊冉

3 + 12 cos

冉

A

2

冊

+ cos共A兲

冊

2

冑

2 + 2 cos

冉

A 2

冊

sec

4

冉

A

4

冊

2713 + 2520 cos

冉

A

2

冊

+ 2580 cos共A兲+ 360 cos

冉

3A

2

冊

+ 19 cos共2A兲

. 共34兲

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It is noticeable that⍀LDE共3兲=⍀LDE共2兲. In fact, for the

follow-ing consecutive orders, the same statement holds, i.e.,

⍀LDE共5兲=⍀LDE共4兲,⍀LDE共7兲=⍀LDE共6兲, and so on. The same

pat-tern of equal value of the observables for consecutive orders of approximation was found in Ref. 3 for the Duffing poten-tial at largen.

For comparison, Limet al.1 have found the dispersion relation to be given at first order as

⍀Lim共1兲=

冑

2J1共A兲

A , 共35兲

and at second order as

⍀Lim共2兲=

冑

g共A兲+

冑

g2共A兲−h共A兲, 共36兲

where

g共A兲=共b0−b2−b4+b6兲A+ 18a1+ 2a3

36A ,

共37兲

h共A兲=a1共b0−b2−b4+b6兲

18A ,

and

a1= 2J1共A兲, a3= − 2J3共A兲,

共38兲 b2i= 2共− 1兲iJ2i共A兲, i= 0,1,2,3.

Jn共A兲 being the Bessel function of the first kind.

In the left-hand panel of Fig. 2 we display the ratio of the dispersion relations from Eqs.共33兲–共36兲to the exact and in the right-hand panel the corresponding relative errors. From the graphs we see that the LDE curve calculated to second order display much smaller errors than the curves obtained with the method of Limet al.even close toA=␲. A second observation is that our formulas can be systematically

improved simply by going to a higher order and that they do not involve any special function, as in the case of Eq.共35兲.

B. Pure quartic potential

Our final example is the Klein-Gordon equation in a pure quartic potential

V共u兲=u

4

4 , 共39兲

which leads to the equation of motion

u¨+u3= 0. 共40兲

This is a particular case of the first example where the contribution of the quadratic term in potential 共8兲 is ne-glected. As such, the corresponding dispersion relation can be derived from the expression of the period of oscillations, Eq.共19兲, since the quadratic term contributes with the l in the square root in the front of the double sum and in the argu-ment in the sum, and is simply given by

T_LDE_共N兲= 4␲

冑

3␮A2

兺

n=0

N

共− 1兲n

冉

− 1/2 n

冊冉

− 1/2 2n

冊

1

32n 共41兲

and correspondingly⍀LDE共N_兲 can be obtained as in Eq.共20兲. Results for the first three orders are the following:

⍀LDE共1兲=

24

冑

3A

49 , 共42兲

⍀LDE共2兲=

13 824

冑

3A

28 259 , 共43兲

⍀LDE共3兲=

1 990 656

冑

3A

4 069 681 . 共44兲

Limet al.found, at first and second order of approxima-tion, respectively,

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⍀Lim共1兲=

冑

3 2 A,

共45兲

⍀Lim共2兲=

1

12

冑

62 + 2

冑

421A.

In the left-hand panel of Fig. 3 we display the ratio of the approximate to the exact dispersion relation and in the right-hand panel the relative error from our findings at first, second, and third order and those of Ref. 1 given previously. At first order, our findings perform just as the second order of Lim et al.,1 and at second and third orders, the perfor-mance of the variational results is excellent.

IV. CONCLUSIONS

We have derived analytical expressions for the disper-sion relations of the nonlinear Klein-Gordon equation for different potentials by means of the Linear Delta Expansion. This technique is implemented by computing the period of oscillations in the given potential. In the particular example of the Sine-Gordon potential, where the dispersion relation is given in terms of elliptic integrals, we have implemented the LDE to compute such an integral and, by means of the Landen transformation, we have obtained an improved series for the elliptic integral. We have observed that the expression obtained by using the first few terms in this series performs remarkably well even close to theA=␲. We believe that our results are appealing in two respects: first in that they

pro-vide a systematic way to approximate the exact result with the desired accuracy, and second in that the expressions that we obtain never involve special functions, as in the case of Ref. 1. An aspect that needs to be underlined is that the method described in Sec. II provides aconvergentseries rep-resentation for the dispersion relation, provided that the ar-bitrary parameter fulfills a simple condition.

ACKNOWLEDGMENTS

One of the authors 共P.A.兲 acknowledges the support of Conacyt Grant No. C01-40633/A-1. The authors also ac-knowledge the support of the Fondo Ramón Alvarez Buylla of Colima University.

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