Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA

Computer Physics Communications 65 (1991) 143-150 North-Holland

143

Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401, USA

A direct series method to find exact travelling wave solutions of nonlinear PDEs is appfied to Hirota's system of coupled Korteweg-de Vries equations and to the sine-Gordon equation. The straightforward but lengthy algebraic computations to obtain single and multi-soliton solutions can be carried out with a symbolic manipulation package such as MACSYMA.

1. Introduction The search for exact solutions of nonlinear P D E s becomes more and more attractive due to the availability of symbolic manipulation programs ( M A C S Y M A , R E D U C E , M A T H E M A T ICA, S C R A T C H P A D II, and the like) which allow to perform the tedious algebra c o m m o n to direct methods on a main frame computer or on a PC. In this paper we generate particular solutions of systems of nonlinear evolution (or wave equations) by a direct series method established by H e r e m a n et al. [1-3]. This method allows to construct single and multi-solitary wave solutions and applies to single equations as well as to coupled systems. The knack of the method is to represent the solutions as infinite series in real exponentials that satisfy the linearized equations. The coefficients of these series must satisfy a highly nonlinear coupled system of recursion relations, which can be solved with any symbolic computer program. The series is then finally summed in closed form and an exact solution of the given system of nonlinear P D E s is obtained. In section 2 we present the algorithm for the construction of a single solitary wave solution. In section 3 we apply it to a system of coupled K o r t e w e g - d e Vries (cKdV) equations [4,5]. In section 4 we show how the method can be generalized

to account for N-soliton solutions using the sineG o r d o n (SG) equation [6-9] as an example.

2. The algorithm We outline the algorithm to construct single solitary wave solutions to systems of nonlinear evolution or wave equations. Space is lacking to give all the details which m a y be found in ref: [1-3]. Step 1: Given is a system of two coupled nonlinear PDEs, o ~ ( u , v, u,, ux, v,, vx, u, . . . . . . U,,x, v , ~ ) = O, ~ ( U , V, llt, Ux, Vt, Vx, Ut. . . . . . Upx' V q x ) = O '

m, n, p , q ~ l N ,

(1)

where o~- and f¢ are supposed to be polynomials in their arguments and where un~ = 3 n u / b x " . Seeking travelling wave solutions for u ( x , t) and v( x, t ), we introduce the variable ~ = x - ct, where c is the constant velocity. The system (1) then transforms into a coupled system of nonlinear O D E s for ~(~) = u ( x , t) and ~b(~) = v ( x , t). The resulting equations m a y be integrated with respect to ~ to reduce the order. F o r simplicity, we ignore integration constants, assuming that the solutions q~ and ~b and their derivatives vanish at

0010-4655/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

144

W. H e r e m a n / E x a c t solitary w a v e solutions using M A C S Y M A

Step 2: We expand q, and + in a power series

Step 4." To find the closed form for 0 and tp we substitute eqs. (3) into (2). Hence,

~= ~ a.g',

~ = ~ b.g',

n=l

(2)

n=l

where g ( ( ) = e x p [ - K ( c ) ~ ] solves the linear part of at least one of the equations in the system. Hence, the wave number K is related to the velocity c by the dispersion law of (one of) the linearized equations. We substitute the expansions (2) into the full nonlinear system (1), rearrange the sums by using Cauchy's rule for multiple series [2] and equate the coefficient of g ' . This leads to a nonlinear system of coupled recursion relations for the coefficients a , and b,. Quite often, the relation K(c) and appropriate scales on ~ and allow to simplify the recursion relations.

Step 3." Assuming that a , and b, are polynomials in n, we determine [2] their degrees 81 and Next, we substitute 81 an~

8 2.

82

E Af ,

(3) into the recursion relations. The sums are computed by using the (factored) expressions of the sums of powers of integers [2,3,10], tl

k=0,1,2

.....

]=0

82

(6)

&

n= 1 j =0

/=0

where

~(g)--

~. nJg ",

j=0.1.2

.....

(7)

1, 2 . . . . .

(8)

n=l Using the relation [3] F]+l(g

) =gFjt(~),

j=O,

one can calculate any ~ ( g ) starting from F°(g)-

1

_g

(9)

g"

F,( g) - (1 _gg ) 2 ,

F2(g)

g(1 + g ) (1 _g)3 . . . . . (10)

Finally, we return to the original variables x and t to obtain the desired travelling wave solution of eq. (1).

3. Example 1: the coupled Korteweg-de Vries equations

S1

( n + 1)n 2 '

(5)

. ( n + a)(Zn + 1) $2=

n= 1 j=O

(4)

For example, S o = n,

8~

E Aj~(g),

n = 1,2 . . . . .

y=0

S k = • i k, i=l

8~

E AjnJg n -

For instance,

b. = E Bjn ,

j=O

~o

O= E

6

.....

The algebraic (nonlinear) equations for the constant coefficients Aj and Bj are obtained by setting to zero the coefficients of the different powers of n. The problem is now completely algebraic and the unknowns Aj and Bj are readily obtained by solving the nonlinear system with e.g. MACSYMA.

We consider [4,5],

u , - a(6uu x + u3x ) - 2fly G = O,

(lla)

vt + 3uvx + V3x = 0,

(llb)

where a, /3 ~ N. These equations describe the interaction of two long waves with different dispersion laws in a nonlinear medium; e.g. shallow water waves in a channel. One can show that both u(x, t) and v(x, t) travel with the same velocity c, hence we look for q~(~) =- u(x, t) and ~ ( ~ ) = v(x, t) with ~ = x - ct.

IV. Hereman / Exact solitary wave solutions using MACSYMA

After one integration with respect to 4, eq. (11) becomes c~p + 3aft z + aq,2~ + fl~k2 = 0,

(12a)

-c~p~ + 3q,~p~ + ~P3~= 0,

(12b)

ignoring integration constants. Substitution of g ( ~ ) = e x p [ - K ( c ) ~ ] into the linear parts of eq. (12) leads to two dispersion laws, c = - a K 2 and c = K 2. The solutions ¢p and ~p can only be built up from the same real exponential g(~) if a = - 1 , which turns out to be a special case. A detailed study of all the cases reveals that either one of the dispersion laws will lead to the same result. Therefore, we m a y proceed with K = ¢~-, where c > 0. U p o n substitution of the conveniently scaled expansions, c

oo

c

d/, = -~ n=l y'~ a . g " ,

~k = ~

for n = 1, 2 . . . . and with a 0 = - e b 2 / [ 2 4 ( 4 a + 1)] > 0. Hence, /3 and 4et + 1 must have opposite signs. Substituting eqs. (15) into (13) and using the formulae for F 0 and F 1, given in eqs. (9) and (10), respectively, we obtain oo

ff=8c E (-1)"+'n(aog2) n=l

" (1

8ca°g2 + a0g2) 2 ' (16a1

c ~P= ~

~ ( . x " - ,, 2.+1 n=0 - l ) olaog

cbag = ~3~f~(1 + aog2) "

(16b)

Returning to the original variables we finally have

u(x,t)=2csech2[C~(x-ct)+8],

o0

n~=lbng n,

145

(13)

v(x, t)= +e into eq. (12), we get the coupled recursion relations:

~

-2(4a

/3

(17a)

+ 1)

×sech[¢-c(x-ct)

+ 8],

(178)

n--1

(l+anZ)a.+

E

(aata.-t+eb,b.-t)=O,

1=1

(14a) n-I

n ( n 2 - 1)b. + Y'~ lbta._t=O,

(14b)

l=1

both for n > 2. Furthermore, (1 + a)a I = 0, b I is arbitrary, and e = +__1 if 1131 = +_/3. We have to distinguish two cases:

Case 1: a 4= - 1 F r o m a ) = 0 it follows that a2n_ 1 =-0 and b2n = 0 (n = 1, 2 . . . . ). We c o m p u t e [2] the degrees 61 = 1 and ~ 2 = 0 of aEn and b2._ a, respectively, and we solve for the coefficients Aa, A 0 and B 0 occurring in a 2 n = A l n + A o ; bEn_l=Bo, (n= 1, 2 . . . . ). This calculation can be done by hand or with a symbolic p r o g r a m such as M A C S Y M A . The result is a2n

=

24n(- 1)"+la~,

bzn_ 1 = ( - - l.)~ . - 1 ~ola o, - 1 ,

with 8 = ½In [ 1 / a 0 [ = ½1n124(4a + 1)/b 21. N o t e that the c K d V equations remain indeed invariant 1 for reversing the sign of o(x, t). F o r a = 4, clearly b . = 0 ( n = 0 , 1 . . . . ); so v = 0 and the c K d V equations reduce to the K d V equation. For the special case a = ½ the c K d V equations are known to pass the Painlev6 test, the system is completely integrable and an N-soliton solution can be constructed [4,5].

Case 2: a = - 1 A p p a r e n t l y a 1 is arbitrary and so is b 1. One can calculate all a . and b. recursively in the hope to obtain a general closed form, which is not known yet. All a t t e m p t s failed so far, it seems only possible to obtain a d o s e d form solution provided b 2 = ~a 1 1 2 and for fl > 0 (i.e. e---1). U n d e r these conditions one obtains a,,

=

12n( - 1 .)~ + 1 a0, .

(18a)

(15a)

b~ = 7a.12 = 72n2ao2.,

(15b)

for n = 1, 2 . . . . . and with a 0 = al/12. Substitution

(18b)

14( Hereman / E x a c t solitary wave solutions using M A C S Y M A

146

of eqs. (18) into (13) leads, upon return to the original variables, to u(x, t)=c

sech2[ ~1/ c ( x -

ct) + 8 ] ,

(19a)

proceeds as in the previous example. In summary• upon substitution of the scaled expansions• 1

~ ' ( x - a ) = 0(~) - ¢_,.

~ a,Y(~), n=]

3

,)

VOl/~l

3c

~sech

-

'3C

t) 2

[1

=~b(~)=

g'(x-ct) -

a)

+ a

]

y~ b,,g"(~),

(22)

with g ( ( ) = e x p [ - K ( c ) ~ ] into eqs. (21), one obtains the coupled recursion relations, (19b) n

with 6 = ½1n 112/al I- Observe that for v(x, t) = (3/~/61 fl I ) u ( x , t) both equations in (11) become identical to the KdV equation, u, + 3uu~ + u3.~. = O.

1

(n2-1)a,, - ~

a,b,,

,=0,

(23a)

/=1 ,,l

l

2b,,+ ~ ( b , b , , ~ , + l ( n

,)=0,

l)a/a,,

(23b)

l=l

4. Example

2: t h e s i n e - G o r d o n

equation

The generalization of our method to construct N-soliton solutions is straightforward but the calculations become rather cumbersome and are therefore very suitable for symbolic manipulators such as MACSYMA. In this section we will just treat an illuminating example. Details about the method and another example, namely the ubiquitous KdV equation for which we constructed the N-soliton solution, may be found in ref. [2,3]. The SG in light-cone coordinates, u,, = sin u,

(20)

describes e.g. the propagation of crystal dislocations, superconductivity in a Josephson junction, and ultra short optical pulse propagation in a resonant medium [6-8]. In mathematics, the SG is long known in the differential geometry of surfaces of constant negative Gaussian curvature [7,8]. To apply our series method we must remove the transcendental nonlinearity in eq. (20) and transform it into a coupled system with strictly polynomial terms [2]: O.

for n>_2, and where a~ is arbitrary and b~=0. We also used the dispersion law K 2 = - - c for c < 0, obtained from the linear part of eq. (21a), to simplify eqs. (23). The solution of eqs. (23) is a,,, _

=

a2n+l

0,

=

b2,, = 8 ( -

4(-1)"

ao

1)

2,+1 •

" n a o2,, ,

..... (24a)

n=l,2

b~,, =0, = + I

n=O, 1,

with a 0 = ¼ a I > 0 . Substituting eqs. (24) into (22) and using the formulae for F0 and F I, we obtain O-

4 (-c

y. ( _ l ) , , ( a o g ) 2 , , ~ l ,,=o

4 -

aog

~

(25a)

1 + ( a o g ) 2"

~b= - 8

(-1)

~tt+l

rt=]

ntaog

~2n

-8(aog) 2 =

[1 + (a,,g)-J (25b)

O g " = O,

(21a)

Returning to the original variables yields

2q" + q,2 + O2 = 0,

(21b)

cos[u(x, t)] - 1

-

•

-

where O = u ~ and q ' = c o s u - 1 . For the construction of the single solitary wave solution, one

....

(24b)

=l-2sech2[~@c

(~v - ct ) + S ] ,

(26a)

147

W. Hereman / Exact solitary wave solutions using M A C S Y M A

u(x, t)= + ~ -2c f s e c h [ ~ -1c (X-Ct) + 3] d x .I

in eq. (21b). Thus, using the dispersion law (28), we find

= +4arctan{exp[-~_c(X-Ct)+3]}, (26b) with 3 = In 14/al l- This is the well-known kinktype solution [6-9] of the SG equation. Let us now turn to the construction of the N-soliton solution of eq. (20), for which various techniques are available [7,8]. Neither one of these methods is very transparent in explaining how N-solitons are built up from N real exponentials. In the spirit of our earlier work [2], we focus on this aspect of soliton generation and show how multi-solitons evolve from the mixing of real exponential solutions of the underlying linear equation. As a generalization of eqs. (22) we substitute N ~(l) =

E

d i j = _ ½cicj~~i~'~j =

i=1

(27)

Note that any term in the expansion of • (ko, respectively) will only have an odd (even, respectively) number of g's, which is in agreement with eq. (25a) and (25b), respectively. The analogue of the first term in eq. (23a), i.e. (n 2 - 1)a,, will result from the action of the linear operator 22°

L. =-~-~

- 1.

(32)

on the (2n + 1)-th term in the expansion of ~, namely, N

~(2n+l) =

1

Ki'

i = 1 , 2 . . . . . N.

(28)

The amplitudes a i could be absorbed into the exponentials to account for the arbitrary phase factors; the constants ci(Ki) will be chosen later. Observe that the starting term in the expansion of • , say ~(2), must be of the form N

~//(2)= E E i=1 j = l N N

=

1 Oi+Iaj "2CiCj-Ki + Kj" (31)

into the linear part of eq. (21a), implying that

N

-

N

cigi(x' t) = E ciai exp(Kix - ~2it),

i=l

Iai=

1 cicj 2 KiK j

N

N

£E---E

i=1 j = l

Cij . . . . g i g j

" " " gs,

s=l

2n + 1 summations n =0, 1.....

(33)

The analogue of the second term in eq. (23a) will, in its most symmetric form, look like

dijgigj 1

52 d , j a , a j

E

~(2t+ l)

(K,, K, ..... Ko)

l=0

i=1 j = l

×exp[(K,+K,)x-(&+gaj)t],

n-I

2l + 1 arguments

(29)

X a/.t(2n-2/, ( K p ,

Kq ..... Ks)

so that -2~/A2) balances with N

2(n - l) arguments

N

~}?(1)= E E c i g j ~ - ~ i ~ g i g j i=1 j = l N N

+ x/,(2,-2,) (K,, Kj . . . . . K,) 2(n - l) arguments

= E E cicjaiajl2iI2j i=1 j = l

Xexp[(Ki+Kj)x-(~i+~2j)t]

X~(2I+I)(Km, K. ..... Ks). (30) 2l + 1 arguments

(34)

I~ Hereman / Exact solitary wave solutions using M A C S Y M A

148

The analogue of eq. (23b) reads N

N

q.(2.)= ~

thus

N

-1

~_ ... ~ d,j .... g i g j ' ' ' g r

Ci lk =

2n summations 1 n] -

2 E

tb(20(K,,

Kj .....

provided we set c~ = 1. After some lengthy calculations, carried out with M A C S Y M A , we obtain

K.)

l=l

(-1)" C,g...... = 4 " ( K , + K,)(Kj + K,) " " ( K r + K~) '

2l arguments X

~(2n-2/) (Ko, g p . . . . . K~)

n=0,1

2(n - l) arguments n

.....

-2(-11"($2,+ ~2j+ ... +~2r) d,,...r= 4"(K, + Kj)(K,+ K k ) . . ' ( K q + Kr)'

1

+ E *t(2'+l)(K,, Kj . . . . .

(38)

Ki)(K/+ K , ) '

4(K/+

r=l

i=1 j=l

Ko)

/=0

21 + 1 arguments X(~t(2n

2/-1)

(Kp,

Kq .....

K~)

n = 1, 2 . . . . . The final objective is then to write

2( n - l) - 1 arguments n = 1,

2. . . . .

(35)

and allows to subsequently determine the coefficients du. ..r. To make this less obscure, let us give an example. • (~) and qp(2) being c o m p u t e d in eq. (27), and eq. (29) with (31), we equate N

N

n=O

N

i=1 j=l

i=1

N

i=1 j=l

-}-

X ( K k + K , ) K~K/K c'~'k k &gjgk'

X

k=l

+ ...

s=]

g, g j ' ' " g; (K, + K j ) ( K , + K , ) ' ' -

(K r + K,)

(39) and a similar expression for '/', in forms. Various authors [2,7,9] have this can be done by introducing the tity matrix I and the N × N matrix ments

½{ ~ ( 1 ) ( K , ) ~ ( 2 ' ( K j , K , )

Kj)~(I)(Kk)}

i=1 j=l

.

(361

to

N

.

i=1 j=l

k=l

4 Z E

.

N

E (K,+K,I(K,+K~)

i=1 j=l

1 N

N

k=l

= E E

-

N

gigjgk (K,+K,)(K,+K~)

N

X ( K , + K~ + K,) - 1]c,,kg, g,g k

+ ~(2'(K/,

N

= x , + x x x(41

La,'"= E E E

N

(38)

,

their closed shown that N × N idenB with ele-

N

E (Ki+Kj)(Kj+Kk)

1

~dia/

B,, - 2 ( Ki + K, )

k=l

C i Cy Ck

X (K k + Ki) KiKjKI' g ig jg k ,

(37)

×exp{½[(K,+Ki)x-(~d,+~/)t]}.

(40)

W. Hereman / Exact solitary wave solutions using MA CSYMA

149

k. t

h

i

,--2..0

Fig. 1. The two-soliton solution of the sine-Gordon equation.

The N-soliton solution to the SG equation (21) is then found to be q~(x, t ) = 4[Tr(arctan B)] x,

(41a)

~/'(x, t ) = - 2 ( l n [ d e t ( I + B2)]}xt,

(41b)

that for N = 1 eq. (42) simplifies to eq. (26b). The two-soliton solution,

u(x, t) =4arctan

K1--K2

- e x p [ K2x - ~2t + 32]} {1 + exp[(K1 + K2)x where Tr stands for trace. Finally, with • = u x we have

u(x, t ) = + 4 Tr(arctan B) =

+_(2) Trtln[ , r,+i 7--Z-~ ] ),

(42)

which satisfies eq. (20). This solution is in agreement with the one obtained in e.g. ref. [9]. It is left as an exercise to the reader to verify

-(~1 +~2)t+31 + 3 2 ] } - 1 ] } ,

(43)

is found by setting N = 2 in eq. (42). This solution is shown in fig. 1 or K 1 = 1, K 2 = f'2-, thus ~1 = 1, ~2 = - ½ ~ , and with 31 -- 32 = 0. A further discussion of other kink and anti-kink solutions of the 8G equation may be found in ref. [6-8].

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W. Hereman / Exact solitary wave solutions using MA CS YMA

References [l] W. Hereman, A. Korpel and P.P. Banerjee, Wave Motion 7 (1985) 283. [2] W. Hereman, P.P. Banerjee, A. Korpel, G. Assanto, A. Van lmmerzeele and A. Meerpoel, J. Phys. A: Math. Gen. 19 (1986) 607. [3] W. Hereman and M. Takaoka, J. Phys. A: Math. Gen. 23 (1990) 4805. [4] R. Hirota and J. Satsuma, Phys. Lett. 85 A (1981) 407.

[5] J. Satsuma and R. Hirota, J. Phys. Soc. Jpn. 51 (1982) 3390. [6] G.L. Lamb, Rev. Mod. Phys. 43 (1971) 99. [7] A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985). [8] P.G. Drazin and R.S. Johnson, Solitons: An Introduction (Cambridge Univ. Press, Cambridge, 1989). [9] W. Zheng, J. Phys. A: Math. Gen. 19 (1986) L485. [10] M.R. Spiegel, Mathematical Handbook of Formulas and Tables, Shaum's Outline Series (McGraw-Hill, New York, 1980).