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IL NUOVO CIMENTO VOL. 112 B, N. 5 Maggio 1997 NOTE BREVI

Hyperbolic secant-tangent method, coupled quantum-field equations

and static solitons

YI-TIANGAOand BOTIAN

Department of Applied Mathematics and Physics

Beijing University of Aeronautics and Astronautics - Beijing 100083, PRC

Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics P.O. Box 8009, Beijing 100088, PRC

(ricevuto il 28 Ottobre 1996; approvato il 4 Marzo 1997)

Summary. — In this note, a hyperbolic secant-tangent method is proposed, based on

the computer symbolic computation. We thus obtain some analytical sets of the static soliton solutions for a pair of equations relevant to the quantum field theory of charged solitons.

PACS 02.30.Jr – Partial differential equations. PACS 02.70 – Computational techniques.

PACS 03.40.Kf – Waves and wave propagation: general mathematical aspects.

Relevant to the quantum field theory of charged solitons, a pair of coupled differential equations [1, 2] reads as

sxx4 2 s 1 s31 dr2s , (1)

rxx4 fr 1 lr31 dr(s22 1 ) , (2)

where l , d and f are parameters.

In this note, to investigate eqs. (1) and (2), we try to generalize the tanh method ([3-6] and references therein) which originally works on some non-linear evolution equations only. We now introduce the hyperbolic secant function, and assume that for both the real scalar fields s (x) and r(x), some static soliton solutions be expressed as

s (x) 4

!

n 40 N VnQ tanhn(Gx 1L)1

!

m 40 M JmQ sech2 m 11(Gx 1L) , (3) r(x) 4

!

k 40 K UkQ tanhk(Gx 1L)1

!

l 40 L ClQ sech2 l 11(Gx 1L) , (4)

where N, K, M and L are integers, while Vn’s, Jm’s, Uk’s, Cl’s, G and L are constants. We note that G c 0 and that at least one of VNand JMand at least one of UKand CLare non-zero. The terms with even powers of sech and tanh have already been combined.

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YI-TIAN GAOandBO TIAN

820

With the computer symbolic computation, we substitute ansatz 3 and 4 back into eqs. (1) and (2). The leading-order analysis yields N 4K41 and M4L40, via the balance of the highest-order linear and non-linear contributions. Then, equating to zero the coefficients of the like powers of sech and tanh, we get a set of algebraic equations from which the values of Vn’s, Jm’s, Uk’s, Cl’s G and L are determined.

Thus we end up with two sets of static soliton solutions, one of which is

s (x) 46

o

2(d 2l) 2 d 2d2 2 l Q sech

y

o

l 2d2 2 d 2d2 2 lx 1a

z

, (5) r(x) 46

o

2( 1 2d) 2 d 2d22 l Q tanh

y

o

l 2d2 2 d 2d22 lx 1a

z

, (6)

with the constraints on the parameters as

f 4 (d 22)(l2d 2 ) 2 d 2d2 2 l , (7) with d D0: 2 d 2d2 D l D d2 and _{d Dl} _{if d E1 ,} (8) 2 d 2d2 E l E d2 and d El if d D1 , (9)

where a is a constant. Similarly, the other set can be expressed as

s (x) 46 tanh

y

o

d 2 2 l 2(d 2l)x 1b

z

, (10) r(x) 46

o

1 2d d 2l Q sech

y

o

d2_{2 l} 2(d 2l)x 1b

z

, (11) with f 4 d 2 2 l 2(d 2l) , (12) d2 D l , if l E d E 1 , (13) d2_{E l , if l D d D 1 ,} (14)

where b is a constant. We call the attention to the fact that there exist four solutions in each set in fact, since the 6 signs can be selected independently.

In comparison, the first set is recognized in [7] within certain ranges in the parameter space not completely correct, to our knowledge; the second set is the same as the one presented in [7] while ref. [1] provides the d D0 subcase. The hyperbolic secant-tangent method presented in this note has the advantage of conciseness and briefness.

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HYPERBOLIC SECANT-TANGENT METHOD,COUPLED QUANTUM-FIELD EQUATIONS ETC. 821

* * *

We thank the Director of Il Nuovo Cimento for the help with the publication expenses. This work has been supported by the Outstanding Young Faculty Fellowship and the Research Grants for the Scholars Returning from Abroad, State Education Commission of China. B. TIANis in charge of the text.

R E F E R E N C E S

[1] RAJARAMANR., Phys. Rev. Lett., 42 (1979) 200.

[2] BOYAL. and CASAHORRANJ., Phys. Rev. A, 39 (1989) 4298. [3] LUB., PANZ., QUB. and JIANGX., Phys. Lett. A, 180 (1993) 61. [4] MAW., Phys. Lett. A, 180 (1993) 221.

[5] PORUBOVA., J. Phys. A, 26 (1993) L797. [6] PARKESE., J. Phys. A, 27 (1994) L497.