272 Notiz Exact Solitary-wave Solutions for the General Fifth-order Shallow Water-wave Models Woo-Pyo Hong and Young-Dae Junga
Department of Physics, Catholic University of Taegu-Hyosung, Hayang, Kyongsan, Kyungbuk 712-702, South Korea
a Department of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea
Z. Naturforsch. 54a, 272-273 (1999);
received December 15, 1998
We perform a computerized symbolic computation to find some general solitonic solutions for the general fifth-order shal- low water-wave models. Applying the tanh-typed method, we have found certain new exact solitary wave solutions. The pre- viously published solutions turn out to be special cases with restricted model parameters.
We investigate the generalized fifth-order shallow water-wave model which describes certain physically- interesting (1 + l)-dimensional waves but is not integra- t e by the conventional methods [1]:
Vt+ avxxxxx* PVxxx + ydxi2wxx + v2]
+ 2qvvx + 3rv2vx = 0 , (1)
where a, p, y q, and r are model parameters, v is a real scalar function of the two independent variables, x and t, and the subscripts denote partial derivatives.
We apply the tanh-typed method [ 2 - 4 ] with symbolic computation to (1) and assume that the physical field v (JC, t) has the form
N
v(x,t)= X ^ „ ( 0 - t a n h " [ 0 ( f ) * + a m (2) n=0
where N is an integer determined via the balance of the highest-order contributions from the linear and non- linear terms of (1) as N= 2, w h i l e/ fN( t ) , 0 ( 0 , and t) are the non-trivial differentiable functions to be deter- mined.
With the symbolic computation package Maple we introduce Ansatz (2) together with the above conditions into (1) and collect the coefficients of like powers of tanh:
(tanh7): - 6 / *2( O 0 ( O
• [rst2{t)2+ 16y0(O2/*2(O + 1 2 O a 0 ( O4] (3) (tanh0): - 5 ^ , ( 0 0 ( 0
[ 2 4 a ^ ( f )4 + 2 0 7 0 ( O V2( O + 3r s f2( t )2] (4) Reprint requests to Prof. W. Hong;
E-mail: wphong@cuth.cataegu.ac.kr
(tanh5): - 2 0 ( 0 [-U0a/f2(t) 0 ( O4
+ l O 0 ( r )2y ^ , ( O2+ 1 2 0 ( O W2( O + 240(O27/*o(O/*2(O - 96 7 / *2( O20 ( O2
- 3 r ^2( 03 + 6 r ^ , ( 0 V2( 0
+ 6 r ^o( r ) ^2( 02 + 2 ^2( 02] (5) (tanh4): ~ / fx{ f ) 0 ( 0 [ 3r *x( t )2 - 2 4 O a 0 ( O4
+ 6 0 ( O V ~ 1 8 4 / 0 ( 0 V2( 0 + 1 2 0 ( O27 ^ o ( O + 1 8 r ^o( O / ^2( O
+ 6 ^2( 0 - 1 5 r ^2( 02] (6)
(tanh3): -2/t2{i) ^ '»(t) - \232arf2(t) 0 ( O5
+ 1 2 ^ , ( 0 ^ 2 ( 0 0 ( 0
- 2/42(t) £ 0 ( 0 x + 1 2 r ^o( O ^2( O20 ( O - 6rst0(t)*x (O20(O - 6 / v *o( 0 V2( 0 0 ( 0 - 2 q *x (O20(O - 4 q * o ( t ) *2( t ) 0 ( 0 + SOy/40(t)/42(t) 0(O3 + 40psf2(t) 0(O3 - 1 1 2 r ^2( O20 ( O3 + 327/^1 (O20(O3
+ 4 ^2( O20 ( O (7)
(tanh2): ~/tx (0 ^ ( 0 + 3 rAx (03 0 ( 0
+ A ^2(O + 8 iu ^1( O 0 ( O3
+ 1 6 7 / * o ( O / * i ( O 0 ( O3- / * , ( 0 ^ 0 ( 0 * - \36asfx(0 0(O5
+ 1 8 ^ ( 0 / * , ( 0 ^ ( 0 0 ( 0 (8) - 3 r ^o( 02^ , (0 0 ( 0 + ( f ) / *2( 0 0 ( 0
- 2qsf0(t)/4x (0 0 ( 0 - 9 2 7 ^ , ( 0 0 ( O V2( O (tanh1): 2/*2(f) ^ ^ ( 0 + 212 a / f2( t ) 0 ( O5
- 32y*0{t)*2{t) 0(O3 + 2/42(t) ± 0 ( 0 x + 6 r ^o( 0 ^ i (O20(O + 6 r ^o( 0 V2( 0 0 ( 0 + (0 - 1 6 ^ 2 ( 0 0 ( O3 + (O20(O + 1 6 7 ^2( O20 ( O3 + 4q/40{t)/f2{t) 0 ( 0
- 1 2 7 ^ , ( O20 ( O3 (9)
(tanh0): ( 0 £ 0( 0 * + (0 £ » ( f ) df dr - 2ps4x(t) 0 ( O3 + 1 6 a ^ , ( O 0 ( O5
+ 3 r ^o( o V , ( O 0 ( O + 2q/40(t)s4x (t) 0 ( 0
+ 8 7 ^ , ( 0 0 ( 0 V 2( 0 - 4 7 ^ O ( 0 ^ I ( 0 0( O3
+ (10) dr
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Notes 273 We find the conditions f o rj 4N{ t ) , $ (t), and t) which
simultaneously cause the above terms to become zero:
(tanh • x): s f2 (r) — 0(t)x = 0
dt (11) - » g ( t ) = = non - zero constant,
2ry , i 4\ , . 38y ^ + r f i (tanh4): / fQ( t ) = - y
(tanh7 and tanh6): (f) = - 6
100
(tanh5): = 1 2 / c s g n ( ^ ) ^ "
-500 -5 0
- 1 0 0 0 -100
(14) Pig- 1- A kink-typed solitary wave with the parameters yU=l, 7=1, <7 = 1, r=-1, and + 12/30.
where csgn(x) is defined by c s g n ( x ) = 1 if R e ( x ) > 0
- 1 if Re(x) < 0 ' (15) (16) (tanh0): '»(t)
_ (3#/u2 r2 + 300^py2r +6924 $5y4)t 4
5 y2 r + 5
where ^ is an arbitrary constant. In addition to the above conditions, the following auxiliary condition is required for v(x, t) to be the solution of (1):
1 = 3 r/x + 5 0 /2^2
(17)
-100
500 X ^ 5 0 1000 100
Fig. 2. A kink-typed solitary wave with the parameters f i = \ , 2 y
Thus, computerized symbolic computation helps us to y = \ , q = \, <2=1, r=-1000, and 18r + 12/30.
obtain a new family of exact solitary-wave solutions of (1), i e.
»2
••tanh v(x, t) = —
(3#H2r2 + 300^HY2r + 6924#5y*)t 4 CfX + 2 + C
5y r 5
- 6 y c s g n ( g ) ?2 2
| ( 3 ^2r2 + 3 0 0 ^ V y2r + 6 9 2 4 ^5y4) r | 4 ^ 5 y r
38y2 + r n 2 ry
Lastly we present figures with the some selected param- eters . For simplicity we set p. = 1, y= 1, the auxiliary con- dition q = jr + 15#2 = l, and In Figs. 1 and 2 we plot the kink-typed solitary waves with parameters:
r = - 1 , - 1 0 0 0 , -V-18 r + 12 / 3 0 , respectively.
Acknowledgement
This work was supported by the special research Grant in 1991 of Catholic University of Taegu-Hyosung.
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