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Generalized tanh Method and Four Families of Soliton-Like Solutions for a Generalized Shallow Water Wave Equation Bo Tian3 and Yi -T ian Gaob

a Dept . of C o m p u t e r Sciences, and Inst, for Sei. & Eng. Computa t ions , Lanzhou Universi ty, Lanzhou 730000, Ch ina

b Inst, for Sei. & Eng. Computa t ions , Lanzhou University, Lanzhou 730000, China

Z. Na tu r fo r sch . 51a, 1 7 1 - 1 7 4 (1996); received November 13, 1995

We report tha t a generalized tanh method with symbolic computa t ion leads to 4 families of soliton-like solut ions for the J imbo-Miwa ' s (3 -I- l ) -dimensional generalized shallow water wave equa t ion .

Key words: Ma themat i ca l me thods in physics. Generalized t anh method , (3 + l ) -d imensional gen-eralized shallow water wave equat ion , Symbolic computa t ion , Soliton-like solutions.

During the recent years, aiming at the construction of the travelling-wave or solitary-wave solutions of some nonlinear evolution equations, remarkable attention has been devoted to several versions of the hyperbolic tangent method, or the tanh method, as seen, e.g., in [1-11]. Briefly speaking, the tanh method conjectures a priori that a solitary-wave solution u(x, y, t) for a nonlinear evolution equation can be expressed as

N

u(x, y, t) = X am • tahnm(fex + cy + d t) (1) m = 0

and proceeds with the substitution of Ansatz 1 back into the original equation, where N is an integer deter-mined via the balancing act of the highest-order linear and nonlinear contributions, while the am's, b, e and d are the constants given by the set of algebraic equa-tions resulting from the equating of the coefficients of tanhm (b x + c y + d r)'s to zero, with m = 0 ,1 , . . . , N. In addition, an attractive variety of the tanh method [12] has been used as a perturbation technique to derive an approximate solution.

Having investigated the current status of the tanh study, one might ask: "Are the travelling waves the only product of the tanh method? Can we go be-yond?"

We next try to answer the above questions by re-porting that a generalized tanh method with symbolic

Reprint requests and correspondence to Prof. Yi-Tian Gao.

computation leads to new soliton-like solutions for a generalized shallow water wave equation, i.e.,

Uyt + Uxxxy - 3 Uxx Uy ~ 3 Ux Uxy ~ " x z = 0 . (2)

Of current interest in both physics and mathematics are certain higher-dimensional nonlinear evolution discussed, e.g., in [13-17], from which the (1 + 1 ̂ di-mensional shallow water wave equations arise as their reductions. In this paper, we study the (3 + l)-dimen-sional generalized shallow water wave equation (2) introduced by Jimbo and Miwa [13].

Though originated as the second equation in the Kadomtsev-Petviashvili hierarchy [13], (2) is not com-pletely integrable in the usual sense, as concluded by Dorizzi, Gammaticos, Ramani, and Winternitz [15]. The very recent discussion [17] on the (1 + ^-dimen-sional version of (2) arouses the current importance to further investigate (2) itself.

We consider a generalized tanh method [18, 19], beginning with the assumption that the soliton-like solutions for certain nonlinear evolution equations, such as (2), being physically localized, are of the x-lin-ear form inside tanh as follows:

u(x,y,z,t) = X s/m{y,z,t) (3) m = 0

• tanhm [Z(y, z, t)x + 0(y, z, r)],

where s>/m(y, z, t)"s, I(y, z, t) and 0(y, z, t) are differ-entiable functions of y, z and t only, while N is an integer to be determined. Ansatz (3) is obviously more sophisticated than Ansatz (1). The x-linear proposal is

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172 Bo Tian a n d Yi-Tian G a o • Genera l ized t anh M e t h o d and F o u r Famil ies of So l i ton-Like So lu t ions

based on the consideration that in (2), or similar equa-tions, the physical field u(x,y,z,t) has only the first derivative with respect to y, z and t, but higher-order derivatives, with respect to x.

On balancing the highest-order contributions from the linear term (i.e., uxxxy) with the highest order con-tributions from the nonlinear terms (i.e., uxuxy and uyuxx), we get N = 1, so that Ansatz (3) becomes

u (x, y, z, t) = sd (y , z, t) • tanh [a x + 0 (y, z, f)] + M (y, z, t), (4)

where (y, z,t) = (y, z, t), si (y, z, t) = s/1 (y, z, f) ̂ 0, while a = constant ^ 0 replaces Z(y, z, t) just for sim-plicity of the future work.

Substituting (4) into (2), using Mathematicia, and we get

2 a 0 . sech2 (ax + 0 ) tanh (ax + 0 ) - s/,osech2(ox + 0) - 2s/yo2, sech4 (ax + 0)

+ 4 s/yo3 sech2 (ax + (9) tanh2 (a x + 0 ) + 6,3fwaya2 sech2 (ax + 0 ) tanh2 (ax + 0 ) -3 s/ s/yo2 sech4 (ax + 0 ) + 2 o0, sech2 (crx + 0 ) tanh (ax + 0 ) + s/y 0 , sech2 (ax + 0 ) + a2 sech2 (ax + 0 ) tanh (ax 4- 0 ) + 16 s/a3 0y sech4 (ax + 0 ) tanh (ax + 0 ) + 12,c/2a2 0V sech4 (ax + 0 ) tanh (ax + 0 ) -8.5/a3 0y sech2 (ax + 0 ) tanh3 (ax + 0 ) + s/t 0V sech2 (ax + 0 ) - 2 s/ 0,0y sech2 (ax + 0 ) tanh (ax + 0 ) + s/yl tanh (ax + 0 ) + Myt

+ rf 0yt sech2 (ax + 0 ) = 0 . (5)

Equation to zero the coefficients of like powers of tanh (ax + 0 ) yields a set of equations, a couple of which comes first, after algebraic manipulations, as

tanh5 (ax + 0): {2o + s/)0y = O, (6)

tanh4(ax + 0): (2a + 3 srf)s/y = 0 . (7)

To concurrently satisfy both of them, there are a cou-ple of possibilities

• 0 = 0 and s/ = 0 , > >

• s/ = — 2a = constant, the second of which then splits into more possibilities. The analysis in the rest of the paper shows that four families of exact solutions result from those possibili-ties.

Family I: 0y = 0 and sJy = 0.

In this place,

tanh2 (ax 4- 0): = 0 ^ sJ = ,s/ (t) only, (8)

tanh0 (ax + 0): Myt = 0 z, r) = a(y, z) + ß{z, t), (9)

where a(y, z) and ß{z, t) are the differentiable func-tions to be determined. Then ... .

(10)

tanh3 (ax + 0 ) and tanh (a x + 0): 3aay + 0 2 = O,

which implies that

a,(y,z)= (11) 3a

a differentiable function of z only. A couple of the integrations of (11) with respect to

y and z respectively yields

a(y,z) =;.2(z)y + y(z), (12)

0 (z, r) = - 3 a / (z) + (t). (13)

Hence we end up with the first family

ua)(x, y, z, t) = s/(t) • tanh [ox - 3ak(z) + pi(f)] + L(z)y + K(z,t), (14)

where the constant a and the differentiable functions s4 (t), A (z), FI (t), K (z, t) = y (z) + ß (z, t) all remain arbi-trary.

The Case of = — 2 a = constant.

Calculations give rise to

tanh2 (ax + 0): 0yt = 0 0(y , z, t) = a (z,t) + ß(y,z), (15)

tanh0(ax + 0): £yt = 0 -+&(y,z,t) = y{z,t) + fji{y,z), (16)

where a (z, f), ß{y, z). y (z, t) and p(y, z) are the differen-tiable functions to be determined. Then

tanh3 (ax-1-0) and tanh(ax + 0): oßz + oaz - 4a3 ßy - at ßy + 3 a2 ny = 0 . (17)

Three families of exact solutions will come out from (17).

B o T ian a n d Yi -T ian G a o • Genera l i zed tanh M e t h o d and

Family II: = - 2 a and ßy = 0.

Equation (17) reduces to

ocz=-ßz-3 any = Xz{z), (18)

a differentiable function of z only, since the left-hand side is only a function of z, t but the right-hand side is only a function of y, z. Integrating (18) yields

a M ) = A(z) + 0(t) , (19)

n(y,z) = + co (z). (20)

3 a

Correspondingly,

u{II)(x, y,z,t)= -2a • tanh [ax + /(z) + </>(t)] V Xz (*)

3 a + K (z, t) , (21)

where the constant a as well as the differentiable func-tions (j)(t),

k (z, t) = y (z, t) + co (z), and X(z) = ß(z) + A(z), (22)

are all arbitrary.

The Case of ßy * 0 with sJ = - la

Equation (17) implies that

a, ßy - a a, = a ßz-4 a3 ßy + 3 er2 ny = Ay(y, z), (23)

where /.(y,z) is a differentiable function of y and z only. Integrating (23) over y yields

(y, z) = a.,(z, t)ß(y, z) — aa2(z, t)y-K(z, t), (24)

/ . (y , z) 4 a „ 3 a2

- -1- \ ßz(y, z)dy + (f)(z), 3 o

(25)

where K (z, t) and cf) (z) are a couple of differentiable functions.

In addition, the left side of (23) does include certain functions of t also. Its first derivative with respect to t results in another constraint,

a« (z, 0 ßy (y> z) — a txzt (z, f) = 0 , (26)

from which the remaining two families appear.

Four Families of Soliton-Like Solutions

Family III: si = - 2 <7, ßy # 0 but a„ = 0.

173

Corresponding to an = 0, one has also a., = 0, from (26). Thus,

ct{z, t) = \J/(z) + £ t + rj, (27)

so as to make the third family,

u(III){x,y,z, t)= - 2 a • tanh [ax + i^t + ß{y, z) + \jj [z) + q]

+ ß(y, z) + F(z, t) - - - I ßz{y, z)dy 3 3a

+ z) ayij/z(z)], 3 a

(28)

where the constants a, >/, £ as well as the differentiable functions t//(z), z) and

F(z, r) = y(z, t) + 0 ( z ) - K ( z , t)

are all arbitrary.

(29)

Family IV: si = -2a, ßy^0 and a„ ^ 0.

The separation of the variables in (26) comes out with

oc , (z, r) ßy(y,z) = a-^-^-=co(z)*0,

a„(z,t)

or equivalently,

z) = co(z)y f](z),

o" a.t (z, t) = co (z) aft (z, f),

(30)

(31)

(32)

where co (z) and t] (z) are differentiable functions. With the introduction of an auxiliary variable

v(z, t) = ar(z, t), (32) becomes a first-order linear par-tial differential equation

(33)

(34)

a v. (z, t) = co (z) vt (z, t).

Solving for it, we get £ f

V (z, t) = — CO (z) dz + £ t + T

J or

<? f f a(z, t)= — t \co(z)dz + - t 2 + xt + x, (35) J 2

where / and r are constants, with £ ^ 0 since a„ ^ 0.

174 Bo Tian and Yi-Tian G a o • Generalized tanh Method and F o u r Families of Sol i ton-Like Solut ions

Putting everything together,

uUV)(x,y,z, t)= — 2a- tanh a x H f | c o ( z ) d z a

+ - t2 + xt + co(z)y + >1(z) + y

4 a zco(z)y + r(z, t) H—— co(z)y +

3 3 a

(36)

+ 3 a~

oj(z)y c o ( z ) d z -coz(z)y2 r\z (z) y

6a 3a

To sum up, with symbolic computation, our gener-alized tanh method hereby leads to 4 families of new analytic solutions for (2), the Jimbo-Miwa's (3 4- ^-di-mensional generalized shallow water wave equation.

Those families, represented by (14), (21), (28) and (36), are potentially interesting in physics since they are soliton-like solutions, although they involve cer-tain arbitrary functions and so are not necessarily bounded everytime.

Al l the results have been verified using Mathematica with respect to (2).

where the constants y, t , c ^ 0, a ^ 0 as well as the differentiable functions rj(z), co(z) ^ 0,

Ctl(z) T ( z , t) = y (z, D + W

3 a c o ( z ) d z +

+ T rj(i 4a

+ —— r] (z) —

all remain arbitrary.

Ctri(z) 3 a2

K(Z, t) + 0 ( z ) (37)

Acknowledgements

Th i s work has been supported by the Outstanding Young Faculty Fellowship & the Research Grants for the Scholars Returning from Abroad, State Education Commission of China; the Youth Sci.-Tech. Founda-tion & Natural Science Foundation, Gansu Prov., China; and the Youth Science Foundation, Lanzhou Univ., China. The text is written by Y. T. Gao, to whom correspondence should be addressed.

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