Exact soliton solution in ice-like structures

Volume 139, number 1,2

PHYSICS LETTERS A

24 July 1989

EXACT SOLITON SOLUTION IN ICE-LIKE STRUCTURES Huibin LAN and Kelin WANG Centerfor Fundamental Physics of the University of Science and Technology of China, Hefei, Anhui, PR China Received 2 January 1989; revised manuscript received 3 April 1989; accepted for publication 18 April 1989 Communicated by AR. Bishop

In this Letter, we present a kind of exact soliton solution for the nonlinear equations which describe proton motion in a system with ice-like structure.

As is well known, the transport of protons in hydrogen-bonded systems and particularly in ice is a long-standing problem which has recently attracted a renewed interest with the introduction of new ideas from the physics of nonlinear phenomena [1,2]. In order to give an appropriate physical interpretation of the high mobility of protons in water, Antonchenko et al. proposed a theoretical model in ref. [1 1. They for the first time took into account the cooperative and nonlinear aspects of the proton motion in hydrogen-bonded chains in water, and chose the proton potential energy curve in each hydrogen bond as a double well with two minima corresponding to the two equilibrium states of a proton between two OH ions (see fig. 1), and taking the proton—proton coupling in a harmonic form, they finally obtained the following two equations of motion (following the same formulae as used in ref. [1]): (1) (1 —s2)ço” +2cx(l —~2)~--2flw~=O,

Antonchenko et al. obtained the exact soliton-like solution of eqs. (1) and (2) in some cases to account for the high mobility of protons. Since then a lot of work [3—6]has been done to extend the model of Antonchenko et al. to include damping and external force effects, which is more practically done in the real system; eqs. (1) and (2) will include terms representing damping and external force effects, and the authors of refs. [3—6] also obtained exact soliton-like solutions in some cases. In general cases they could only obtain perturbative or numerical solitonlike solutions except for ref. [7] where a different model was proposed for the same problem, which is certainly not very convenient for further research. In this Letter we shall give a new kind of exact solitonlike solution in a function series form of the simple

Ayi” +~‘= I

with the perturbative solution in ref. [I], and we perform a simple numerical computation to show that the number of terms for convergence of the function series is very small. Thus the solution here may be used to discuss the physical problem.

-

(2)

~2

where ~ describes the motion of the proton displacement, and w describes the motion of the relative displacement of the equilibrium of the OH— ions. 15-2

ev

~

0-

•

15-4 t

vs 0

\jJ

0

0

~jJ

T*I 0 sO

\.,JJ

fl4•2 0

~

t,J_JJ

Fig. 1. Potential well for a proton in a hydrogen-bonded chain. •: proton position, 0: OH— ion position.

case: the model of Antonchenko et al., i.e. we will deal eqs. (1) conditions and (2) here. can show that underwith appropriate our We solution coincides

In refs. [11 and [2], Antonchenko et al. and Laedke et a!. could only find an exact solution forthe case A = 0. When A 0, they could only find the perturbative solution (I) or the numerical solution (2). Here we will give a kind of exact solution in the form of a series of functions. As to eqs. (1) and (2), we assume that

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61

Volume 139, number 1,2

PHYSICS LETTERS A

co= ~ a21sech~/zxth/Lx,

(3)

i~O

2’~x.

(4)

~b21sech

~i=

x thdux, we have the following two equations for the coefficients a and h: 4(1 —s2)i2~2a 2)i(2i— I )~i2a 7,—2(1 —s 2(,_

(14) (15)

6aa

+2aF3 [1,], k, 1]

2ja2(~l)

—

2fl(a0b2, + b0a21) 2/3F7 [i,j, k] = 0, 2h~,—A [2(1—1 )~2~ 4(/— 1 )2~2 ]b

4Ai

(5) 2(,, + b~,

—F1 [i,j, k] + 2aO~4a2(,I)+ F1 [i

—

1,j, k] }.

(6)

D2(i)= [4(1 —s

0J

x {A [2(/— 1

4 ( i— 1 )2~2 +2aOö4a71~)+F1 [i— I,j, k]} )~2+

j)

—

F [i, k]

—2a0~3{[—6cta~ó2+2(1 —s

F1[ij,k]= F2[i,j, k] =

~

a2a~,

~a2Jb2k,

F3[i,j, k, /1 =

~

2(,)

(7) (8)

and for j~0,k~0,

+2c~F3[i,j,k,l]—2aF3[i—l,j,k,1] +2flF2[i,j, k]}.

a2}a2ka,,.

(9)

(2a—2aa~)a0—2fla0b0=0, b0+a~=l

(19) (20)

.

In the sum of eq. (9), only one subscript equals 0, among j, k and 1, and

The boundary conditions for ~ and

(5=1

~(x)},~1=0,

i~0 /=0,

(10)

(54 ~,

1,

/=1,

(11)

/0, i

(12)

~ 1

±1,

(21)

w(x)I~~=0. We can have

(22)

a0=±1, b0=0.

(23)

For i~>1, we have the following two cases: (A)Wheni=landD(i=l)0,i.e. 2+l)[4(l—s2)R2—4a]+4fl0, (4AR

(24)

/=1.

we have

(13)

Thus we have the following recurrence formulae for the coefficients a 21 and b7~: 62

~(x)j~~=

cu are as follows:

i~1,

(53z~l, i~0, ~,

(18)

When /=0, we have

j + k + 1=

=~,

~,

2)i(2i— I )/12Ja

where

(521,

(17)

2)i2F2+2a—6aa~ —2,8b

—2aa~,~ — F [i,j, k]

+2a~O4a2~l)+FI[i—l,j,k].

=~,

(16)

-

—2aF3[t— l,j, k, 1] +2flF7[i,j, kI 2ti2 ]b +2aofl{A [2 (i I )t~+4(/— 1 ) 2(,_)

+6cta~ó2a2~~ + 2aF3 [i— 1,j, k, ~ —

2+l)[4(1s2)i2/i2+2a D(i)zr(4A1JL —6aa~ó,—2/3b 0}+4flö1a~, 2ji2+ I ){ [2(1 —s2)i(2i— I )j~2 D1 (1) (4Ai —

+2aa,, —6cta~ä1a7, —2ctF3[i,j, k, 1]

=~,

D1(i) a21=—--, D(i) b 2~= ~

Inserting the (3) same and (4) intoofeqs. and (2) equating powers sech(I) i~xand sechand /L1

—

24 July 1989

a~=~a, b~=0.

(25)

From eq. (25) we can see that the solution in this case cannot

return

to the perturbative solution when

Volume 139, number 1,2

PHYSICS LETTERS A

A is small (see ref. [1]), so we do not deal with it here (B) When 1=1 and D(i=1)=0, i.e. (4A~+1 )[4(l —s2)j~—4aj+4fl=0, we ~2

have [8A(1—s2)]1

N o= i~O ~ a21 sech2’ux thux,

(32)

(1 —s2)~” (26)

(33)

2a(~2— l)çs+2flyi’ N

(4aA—(1—s2) 2) ] 2 + 1 6A( a—fl) (1 —s2 ) }

~=

Lb 2~sech~’~.

/2)

±{ [4aA—(1 —s a

24 July 1989

(34)

(27)

~i=1—to2—A~”.

(35)

(28)

The result is shown in fig. 2.

2+ 1) ]a 2 = ~[1 —b2(4A/J

0.

Now the solution in this case can return to the perturbative solution which is given in ref. [1]. We can show this as follows. When A is small, we can have from eq. (27) that 2) 16A(a—fl)(l —s x [8A(l—s2){4aA—(l—s2) —

—

[4aA

=

—

(1— ~2)

—

+

1 6A( a

fi) (1 s2) ]~ /2}] —

-

in eqs. (32) and (34). Thus Nfor convergence of the function series is about 40 for the parameters given by eq. (31); and we have carried out similar

(29)

numerical computations for some other parameters which satisfy eq. (26), with similar results. Thus we believe that at least the solution in case (B) is the exact solution of eqs. (1) and (2). Following the same procedure as in ref. [I], we can calculate the energy and the momentum of the soliton.

2 (a fi) 8Afl—4Aa— (1 —s2) +4Aa— (1 —s2) —

a fi (1 _~2) [1 —4Afl/ (I _s2)] a /3 ~ [I+4A/3/(l—s2)]+O(A2) —

=

—

=

.

In eqs. (32) and (34) N indicates the terms we preserve in the summation. In fig. 2, we only draw the case N= 40. The relevant error is about 1%. When N increases, the accuracy will arbitrary accuracy increase, if one preserves can in principle sufficient reach terms an

In order to do this, we first ask for a first integral When a 2=

for eqs. (1) and (2). Multiplying eq. (1) by a~ and

~ö, we have from eq. (28) 2+1) —~o=~--~b2(4A# That is —

9*

b

I +ô 2 2= l+4Aj~ =(l+(5)[1—4Aj~2+O(A2)] =1+(5—4AJL+O(A2)

.

6 -

(30)

~

Eqs. (29) and (30) are consistent with the pertur-

~ 3

flandA: bative Now results taking in certain ref. [1]. valuescomputation ofA= the—0.2, parameters s,fola, we s=0.9, carry a=0.6, out the numerical /3=0.082, for the(31) lowing four quantities:

o~4.21 0.2

02

0.4

06

03

i.O

~2

Fig. 2.

63

Volume 139, number 1,2

PHYSICS LETTERS A

eq. (2) by a,çui and integrating overxwe get two integro-differential equations: 2 (I —s~)q~’~—a( I —p2)—2 jC flw—~---dx=0, (36) d~ th

f

(37)

.

Then we can have the formula for the energy from eq. (33) above and eq. (2.20) in ref. [1] as

J

2+fl(a2+a~—A)w’2] (39) 1 dx [2~’ We have also the formula for the momentum from eq. (2.21) in ref. [1] as P=

.

J

dx (mu~çt’2+Mp~i’2) (40) a Then after performing the integration in eqs. (34) and (35), we can finally get the energy and the momentum of the soliton as -~

F[B,2/]=

~

i+k=1

B21B2~,

.

(43)

B FI[C, 2/]=

~

c=

—

2i1jb21.

(44)

C 21C2~,

Eliminating the common integral term we find the following first integral: 2)2+flw2—2flcu’(l—~2) a(l—~ = (1 —s2)~’2—-flAy’2 (38)

E=A

where

2=~[(2i—l)a1111—2ia,,],

2~+2~2—2 ~ A~’+y—

24 July 1989

(45) (46)

as We wellgive as the hereenergy the exact andsolution the momentum of eqs. (I)ofand the (2) soliton, which we believe will be useful for future research on the high efficiency of energy and electron charge transfer in quasi-one-dimensional molecular systems. Finally, one can see from the above that in principle the function series solution of eqs. (3) and (4) can exist for appropriate parameters and for some kinds of nonlinear equations like those of eqs. (1) and (2); thus we can deal with the more complex case of eqs. (1) and (2) with terms indicating damping and external field effects and other similar nonlinear coupling equations. This will be our future work. We are grateful for the editors’ kind and helpful

E=4A

+

1 ~ FTB,2i] (i—1)(i—2)...2xl (1—5)(l—,)...2X2 2+a~—A) ~Afl(a

advice.

References

z=1

xF 1 [C, 2i]

(41)

~

V mu02 ~ 2F[B, 21] —i a .

+

V —~

a

.

mp~~ F1 [C. 2i]

(j ~.

l)(i2).2X ~. ~ —

2 )

—

2 /2

5

~

2

~

2 /

2 /2

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36(1987)903.

[41St. Pnevmatikos, Phys. Lett. A 122 (1987)249. [51L.N. Kristoforov et al., Phys. Stat. Sol. (b) 146 (1988) 487.

‘

2

‘42 ‘ /

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