Exact solutions of various Boussinesq systems

Math. Lett. Vol. 11, No. 5, pp. 45-49, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0893-9659/98 $19.00 + 0.00 PII: S 0 8 9 3 - 9 6 5 9 ( 9 8 ) 0 0 0 7 8 - 0 Appl.

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Exact Solutions of Various Boussinesq Systems M . CHEN D e p a r t m e n t of M a t h e m a t i c s , P e n n s y l v a n i a S t a t e U n i v e r s i t y

University Park, PA 16802, U.S.A. chen_mOmath,psu. edu ( R e c e i v e d S e p t e m b e r 1997; a c c e p t e d O c t o b e r 1 9 9 7 )

Abstract--It was shown in [1,2] t h a t surface water waves in a water tunnel can be described by systems of the form tit + u x + (u~)x + aux=x - boxer = O,

(1)

ut + ~z + u u z + crlxx= - du-~=t = O,

where a, b, c, and d are real constants. In this paper, we show t h a t to find an exact traveling-wave solution of t h e system, it is suffice to find a solution of an ordinary differential equation, and t h e solution of the ordinary differential equation in a prescribed form can be found by solving a system of nonlinear algebraic equation. T h e exact solutions for some of the systems are presented at the end of t h e paper. (~) 1998 Elsevier Science Ltd. All rights reserved.

Keywords--Exact

solution,Traveling-waves, Boussinesq system, Water waves.

1. I N T R O D U C T I O N To describe small amplitude and long waves in a water channel, systems in the form of (1), which include the classical Boussinesq system (cf. [3]), were derived by Bona, Sant and Toland in [1], where a, b, c, and d are real constants and determined by three parameters A, #, and 0 < 0 < 1 in the following way: a=~

02 -

c = ~ I (1 -t9 2) #,

A,

02 -

(l-A),

d = 7 1 (1 -/~2) (1 -/~).

The dimensionless variables x and t are sealed, respectively, by h and (h/g) 1/2 where h denotes the undisturbed water depth and g denotes the acceleration of gravity. The variable r/(x, t) is the nondimensional deviation of the water surface (scaled by h) from its undisturbed position and u(x, t) is the nondimensional horizontal velocity (scaled by v f ~ at a height Oh with 0 < 0 < 1 above the bottom of the channel. These three parameter family of systems are formally equivalent and correct through first order with regard to the small parameter e = sup{r/(x,t)}. In this paper, we concentrate on finding exact traveling-wave solutions of (1) which approach constants at infinities. The existence of these solutions is useful in the theoretical and numerical studies of This work was supported in part by NSF Grants DMS-9410188 and DMS-9622858.

Typeset by .a.~V-T~ 45

46

M. CHEN

the model systems. In fact, one of the exact solution we found here for the regularized Boussinesq system (a = c = 0, b = d = 1/6) has been used in [2] to demonstrate the convergence rate of a numerical algorithm. 2. M A I N

RESULTS

Denoting ~ = x + xo - Cst with x0 and Cs being constants, we first present the result on the existence of traveling-wave solution o ( x , t) =

u ( = , t) =

(2)

that ri(~) and u(~) are asymptotically small at large ~ and proportional to each other, so lira (O(~), u(~)) = 0, ~--.+oo

ri(x,t) = B u ( x , t),

(3)

with B being a constant. Substituting (2) and (3) into system (1) and using the fact that the resulting two equations are consistent, one can prove the following. THEOREM. For a given system ha the form of (1), ff the constants a, b, c, d satisfy one of the following conditions: (i) (ii) (iii) (iv) (v)

a - b + 2d ~ O, p = ( - b + c + 2d)/(a - b + 2d) > 0, and 09 - 1/2)((b - a)p - b) > O;

a=b=c>0, a=b=c<0, a-b+2d=0, a-b+2d=0,

d=0; d=0; a=c,d>0; a=c,d<0;

then the given system has solitary-wave solutions. Moreover, the exact solitary-wave solutions are of the form

ri(Z, t) -- ri0 seeh 2 ()k(x q- xO - Cat)), u(x, t) =

~

ri0 sech 2 (A(x + Xo - Cst)),

where

c,=

3 + 2rio -I-X/3(3 + ri0)'

= 5

3 ( a - b) +2b(rio + 3 ) '

and rio can be any constant satisfies • • • • •

ha ha ha in ha

Case Case Case Case Case

(0, ri0 = (3(1 - 2p))/2p; (ii), 0 < rio < +oo; (iii), - 3 <_ rio < O; (iv), ri0 > - 3 and 3/(ri0 + 3) is not ha the closed interval between 1 and b/d; (v), ri0 > - 3 and 3/(ri0 + 3) is in the closed interval between 1 and b/d.

With a more general approach, one can find exact solutions where u(~) and ri(~) are not proportional to each other and they do not approaches zero at infinity. Assuming that the traveling-wave solution (u(~), ri(~)) tends to (u~, ri¢¢) as ~ tends to +c¢. Substituting functions = ri( ) - ri ,

=

-

(4)

into (1) and integrating the system once, one obtains - C s h + v + vh + rioov + uooh + av" + bCah" = O, 1 2 - C a v + h + ~v + uoov + ch" + dCav" = O.

(5)

Exact Solutions

47

Eliminating one of the dependent variables, one can find that v(~) (or h(~)) satisfies a forth-order ordinary differential equations (cf. [4]). For instance, in the case that c ~ 0, one can eliminate h(~) and obtain an ordinary differential equation on v(~) as follows. Notice from (5) that h and h" can be expressed as a function of v(~), h = gl(V) f(v)

and

h"

g2(v) = f(v)'

(6)

where f(v) = c(-C, + v + u~) - be,,

gi(v) = c ( - v - av" - ~loov) - bC, ( C , v - l v 2 - dC, v" - uoov) , g2(v) = v + a v " + nooV + ( - c a + v + u ~ )

(

12

c,v - ~v

- d C , v" - uoov

)

.

Differentiating the first equation in (6) twice with respect to ~ and using the second equation, one finds f2g2 = gi' f 2 -- g l f " f -- 2 g l f f ' + 2gi ( f ) 2 , (7) which is an ordinary differential equation with dependent variable v(~). One can therefore established the fact that in order to find a traveling-wave solution of (1), it is suffice to find a solution v(~) satisfying the ordinary differential equation. Notice again that the ordinary differential equation (7) involves only

v"", ¢¢", (¢,)2, ¢,, (¢)2 terms, the Ansatz equation

(v') 2 = pv 2 + a v 3,

(8)

p > o,

can be used to find solutions in the form of

(cf. [6]). Substituting (8) into (9) yields a polynomial equation on v(~) where the coefficients depend on p, a, C8, u¢~, and ~/~. By requiring the coefficients to be zero, one obtains a system of algebraic equations and the solution p, a, C,, u ~ , and ~/~ provides the solution of ordinary differential equation in the form of (9), which in turn yields the exact traveling-wave solution of the system with the help of (6) and (4). The method described above is used on a large class of the systems in (1) which includes the system in [5] (formula (13.101)), the systems in [6], regularized Boussinesq system in [1], Boussinesq's original system (cf. [3]), and the integrable version of the Boussinesq system (cf. [7]). The exact traveling-wave solutions founded are listed in next section. The method presented in this paper is quite general and it recovered the solutions founded in [8,9], where a homogeneous balance method was used. Other Ansatz equations can be used to find solutions in different forms [10].

3. E X A C T SOLUTIONS

TRAVELING-WAVE FOR SYSTEMS IN

(1)

Denote ~ = x + xo - C,t, where x0 and C, are arbitrary constants, one can find the exact traveling-wave solutions for the following systems (p >_ 0 is an arbitrary constant). • a=0:

v(e) = - 1 .

48

M. CHEN • a=O,

c#O:

u(~) = ~c (-b + 2c + 2 d - bcp) + C, bpsech 2

~/"p~ ,

~,

,1(~) = - 1 + - ~ (b 2 - 4 ~ + , ~ 2 _ b2cp + 2b~dp) + -2-g-c b (b -

2d)p ~ h 2

(~v ~ )

• a=c=O:

u(~) = --~Ca(3 - 5bp) + 5Csbpsech 2 ( 2 ~/'-p~) b (lOb - 6d)'p 2

n(~) = - 1 +

5 2

(2 sech2

I

* b-c=O,d#O:

u(~) = - a + 2dC~- 2d2C~P (2) 2dC~ + 3dCspsech 2 "~"I~ ,

y(¢) = - 1 + ~

(a-

a

2d2C2p)

+ 5apsech

~(~Vr-p~)

• c=d=O,b#O:

_o

~

( ~~/'p~) , 02 ~ ~ ( ~ ) -1 "4- 16b2------~s 2 ~-~o,p-'l- C2b2p2 "4- ~a,p -{- C28b2p2

u(~) = 4bC, + C, - -~C, bp + 5C, bpsech 2 r/(~) =

sech2

? b~k2p2 sech ' ( l ~-p~) . • b=c=d=O,a>O:

~1(~) = -1 + gapsech

~"p~ .

• b=c=d=O,a
1

1

TI(~) = -1 - ~ap + ~apsech

2(~ ~/'-I~) •

• b=d=O,a=c: u(~) =

=}=v~(l+cp)+ 2Ca 3cp sech2 ( 1 ) 2

,7(~) =

±~

¢-~

l+cP ~-~sech~(1 2

) J-~"

'

( ~~-I~)

Exact Solutions

49

REFERENCES 1. J.L. Bona, J.-C. Saut and J.F. "roland, Boussineeq equations for small-amplitude long wavelength water waves (preprint), (1997). 2. J. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physiea D., (submitted). 3. J. Boussinesq, Thdorie de l'intumescence liquide appelde onde solitaire ou de translation se propageant dans un canal rectangulalre, Comptes Rendus de l 'A cadmie de Sciences 72, 755-759, (1871). 4. M. Chen, Exact tavelling-wave solutions to bidirectional wave equations, (in preparation). 5. G.B. Whitman, Linear and Nonlinear Waves, John Wiley, New York, (1974). 6. J.L. Bona and R. Smith, A model for the two-way propagation of water waves in a channel, Math. Proc. Cambridge Phil. Soc. 79, 167-182, (1976). 7. E.V. Krishnan, An exact solution of the classical Boussinesq equation, J. Phys. Soc. Japan 51, 2391-2392, (1982). 8. M. Wang, Solitary wave solutions for variant Bonssineeq equations, Physics Letter8 A 199, 169-172, (1995). 9. M. Wang, Y. Zhou and Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematics physics, Phy.¢ics Letters A 216, 67-75, (1996). I0. Z.J. Yang, R.A. Dunlap and D.J.W. Geldart, Exact traveling wave solutions to nonlinear diffusion and wave equations, Internat. J. Theoret. Phlls. 33, 2057-2065, (1994). 11. S. Kichenassamy and P.J. Olver, Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, S I A M J. Math. Anal. 23, 1141-1166, (1992).