We hereby try to m o d i f y the generalized t a n h m e t h o d by a s s u m i n g t h a t certain soliton-like solutions, physically localized, be of the f o r m L u (x, t

(1)

372 Notiz M o d i f i c a t i o n o f the G e n e r a l i z e d tanh M e t h o d with s e c h F u n c t i o n f o r G e n e r a l i z e d H a m i l t o n i a n E q u a t i o n s

Bo T i a n , K e Y i Z h a o , a n d Y i - T i a n G a o Department of Applied Mathematics and Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

and

Laboratory of Computational Physics,

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

Z. Naturforsch. 52 a, 372 (1997);

received January 9, 1997

We modify the generalized tanh method with the sech function, so as to obtain a new class of the soliton-like solutions to a coupled set of generalized Hamiltonian equations.

T h e generalized t a n h m e t h o d a n d its extensions [ 1 - 3 ] are a t o o l to directly c o n s t r u c t the t a n h - p r o f i l e d soliton-like solutions for certain e q u a t i o n s of m a t h e - matical physics.

T h e sech function, like t a n h , does n o t diverge at infinity either (for real arguments). We hereby try to m o d i f y the generalized t a n h m e t h o d by a s s u m i n g t h a t certain soliton-like solutions, physically localized, be of the f o r m

L

u (x, t) = X sJ, (x, t) • t a n h ' [ f (x, t)]

1 = 0

j

+ X t) • sech [ f ( x , t)] • t a n h ' ' [ f ( x , f)], (1)

7 = 0

where L a n d J are the integers determined via the b a l a n c e of the highest-order c o n t r i b u t i o n s f r o m b o t h the linear a n d n o n l i n e a r t e r m s of the original equa- tion, while i (x, tys, 0Sj(x, r)'s, a n d f (x, t) are differentiable f u n c t i o n s . f ( x , t) s h o u l d be real to m a k e sure t h a t the soliton profiles hold.

It is n o t e d t h a t in the generalized t a n h m e t h o d , the soliton-like s o l u t i o n s only c o m e f r o m the first s u m m a - tion in A n s a t z (1). W h a t we d o is to m a k e the p o w e r

series m o r e c o m p l e t e via the i n t r o d u c t i o n of the sec- o n d s u m m a t i o n .

We n o w a p p l y the a b o v e m o d i f i c a t i o n to a coupled set of generalized H a m i l t o n i a n equations,

ut = ux + 2 v , (2)

vt = 2 e u v , where e = ± 1 , (3)

which has an infinite n u m b e r of conserved densities, the n o n d e g e n e r a t e H a m i l t o n i a n structure, certain B ä c k l u n d t r a n s f o r m a t i o n s a n d exact solutions [ 4 - 8 ] .

We use A n s a t z (1) for b o t h u(x, t) a n d Ü(X, t), a n d t h e n e q u a t e to zero the coefficients of like p o w e r s of sech a n d t a n h t o get the explicit expressions for s/t(x, r)'s a n d ^7 (x, r)'s. F o r simplicity, we consider a

3

trial f ( x , f ) = X {Fj(x)tj, a n d after c o m p u t e r i z e d j=o

symbolic c o m p u t a t i o n end u p with Y (x, r) = ß t -I- y (x), w h e r e ß is a real, n o n - z e r o c o n s t a n t , while y(x) is a real, differentiable f u n c t i o n w i t h yx(x)-/> + oo when x —* ± oo. T h u s , we o b t a i n a n e w class of the soliton- like s o l u t i o n s as

u(x, t)= ~{±i\ß\ sech [ßt + y (x)]

— eßtanh[ßt + y(x)]} , (4) 7 fx) — ß

V(x, t) = >x ' 4 f -{±i\ß\sech[ßt + y(x)]

• t a n h [ß t + y (x)] + £ ß sech2 [ßt + y (x)]} . (5) T h i s class is different f r o m t h a t in [2, 7, 8]. It c a n also be reduced to solitary waves w h e n y(x) = a x + (5, w h e r e a a n d <5 a r e real c o n s t a n t s .

Acknowledgements

T h i s w o r k h a s been s u p p o r t e d by the O u t s t a n d i n g Young F a c u l t y F e l l o w s h i p & t h e Research G r a n t s for the Scholars R e t u r n i n g f r o m A b r o a d , State E d u c a t i o n C o m m i s s i o n of C h i n a .

[1] B. Tian and Y. T. Gao, Computer Phys. Comm. 95, 139 (1996).

[2] B. Tian and Y. T. Gao, to appear in Appl. Math. Lett.

(1997).

Reprint requests and correspondence to Prof. Dr. Y.-T. Gao.

[3] B. Tian and Y T. Gao, to appear in Chaos, Solitons &

Fractals (1997).

[4] G. Tu, Phys. Lett. A 94, 340 (1983).

[5] W. Ma, Kexue Tongbao 32, 1003 (1987).

[6] Z. Chen, Comm. on Appl. Math. & Comput. 4, 71 (1990).

[7] W. Ma, J. Fudan Univ. (Natural Sei.) 33, 319 (1994).

[8] Y. Li and C. Tian, Kexue Tongbao (Lett.) 29,1556 (1984).

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