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Exact solutions to the Korteweg-de Vries-Burgers equation
WAVE M O T I O N 11 (1989) 559-564 NORTH-HOLLAND
EXACT
SOLUTIONS
559
TO THE
KORTEWEG-DE
VRIES-BURGERS
EQUATION
Alan J E F F R E Y and Siqing XU Department of Engineering Mathematics, University of Newcastle Upon Tyne, England, NE1 7R U Received 7 October 1988
We introduce a transormation which reduces the Korteweg-de Vries-Burgers (KdVB) equation, u, + 2auux + 5bux,. + cu~x~ = 0, to a quadratic form involving a new dependent variable and its partial derivatives. Exact solutions o f the KdVB equation can be obtained by solving this equation. The exact form of the travelling wave solution to the KdVB equation is obtained in this paper, and its nature depends on the direction o f propagation of the wave.
1. Introduction
The KdVB equation for u ( x , t) has the form ut + 2 auux + 5 buxx + cuxxx = 0,
(1.1)
with a, b and c constant coefficients. It arises in many different physical contexts as a model equation incorporating the effects of dispersion, dissipation and nonlinearity. Johnson [7] derived (1.1) as the governing equation for waves propagating in a liquid-filled elastic tube in which the weak effects of dispersion, dissipation and nonlinearity are present. Grad and Hu [3] used a steady state version of (1.1) to describe a weak shock profile in plasmas. In the limits b ~ 0 or c-* 0, that is when the weak effect of dissipation or dispersion is comparatively insignificant and so can be neglected, the KdVB equation can be approximated by the KdV equation u, + 2auu,, + cUxxx = 0,
(1.2)
or the Burgers' equation ut + 2auux + 5buxx = 0,
(1.3)
respectively. The KdV equation (1.2) and the Burgers equation (1.3) are both exactly soluble and they each have a wide range o f applications in physical problems. A number of theoretical issues relating to the KdVB equation have received considerable attention. In particular, the travelling wave solution to the KdVB equation has been studied extensively. Johnson [7] examined the travelling wave solution to the KdVB equation in the phase plane by means of a perturbation method in the regimes where b ,~ c and c ,t: b, and developed formal asymptotic expansions for the solution. Grad and Hu [3] studied the same problem using a method similar to that used by Johnson [7] and a related problem was considered by Jeffrey [8]. A numerical investigation of the problem was carried out by Canosa and Gazdag [2]. More recently, Bona and Schonbeck [1] studied the existence and uniqueness of bounded travelling wave solution to (1.1) which tend to constant states at plus and minus infinity. They also considered the limiting behaviour of the travelling wave solution of (1.1) as b tends to zero while c 0165-8641/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
A. Jeffrey, S. Xu / Solutions to the KdVB equation
560
is O(1), and as c tends to zero while b is o(1). The case where both b and c tend to zero with the ratio b/c held fixed was also examined. Qualitative results concerning the travelling wave solutions to the KdVB equation were also obtained by the above mentioned authors and others, but they did not find the exact functional form of the travelling wave solution, or any other exact solutions. The KdVB equation was examined by G i b b o n et al. [4] within the framework of soliton theory. A comprehensive account of the travelling wave solution to the KdVB equation can also be found in the review p a p e r by Jeffrey and Kakutani [5]. For other theoretical issues concerning the KdVB equation, we refer the reader to Bona and Schonbek [1] and to the reference cited there. In the present study we solve (1.1) foi" exact solutions by means of a direct method. In Section 2, we will introduce a transformation of the dependent variable u which reduces (1.1) to a quadratic form in a new dependent variable and its partial derivatives. Exact solutions are then obtained by solving this equation in terms of a series of exponentials. Two travelling wave solutions to (1.1) are obtained in this fashion. The situation is similar to that of the KdV equation in bilinear form in terms of Hirota's bilinear operator. The KdV equation (1.3) can be written in its bilinear form by making a suitable change of the dependent variable, and an infinite sequence of exact solutions, the N-soliton solutions, can be then obtained by solving the bilinear equation in terms of a series of exponentials. For the KdVB equation, however, we will show that the quadratic form yields only the two travelling wave solutions, and not the infinite n u m b e r found for the KdV equation. A similar situation may occur with equations which can be put into a bilinear form in terms of Hirota's bilinear operator. In such cases, at least two solutions can be obtained by solving the bilinear equations in terms of a series of exponentials; see, for example, Newell [9].
2. The transformation and exact solutions
To introduce our transformation of the dependent variable u, it proves convenient to use a variable representing an integral over the pulse profile that satisfies equation (1.1). Let
= I x u(X, t) dX,
(2.1)
then integrating (1.1) with respect to x once gives, in terms of ~3,
~t+a~E + 5b~xx+c~xxx+~(t)=O,
(2.2)
where ~ ( t ) is an arbitrary function of t arising from the integration. The function if(t) can be eliminated by introducing a new variable v(x, t) defined as
v(x,t)=~-f~
~b(X) dX,
(2.3)
which reduces (2.2) to the form
v , + a v 2 + 5bvxx+cVxxx=O. Equation (2.4) is equivalent to (1.1), and hereafter we will work with (2.4). The main result of the present study is stated in the following theorem.
(2.4)
A. Jeffrey, S. X u / Solutions to the K d V B equation
561
Theorem 1. The K d V B equation (2.4) is transformed into the following quadratic f o r m f o r F ( x , t) 6 b F ( Ft + 5bFxx + cF~x) + 6cF( F, + 5 b F ~ + c F ~ ) x - 6 c F x ( F , + 5bFxx + dFxxx)
+ 18c2(FEx - F~Fxxx) + 6 b ( b F 2 - cFxF~x) = 0
(2.5)
by the transformation v(x, t) = 6cFx+ 6b In F. aF a
(2.6)
The proof of Theorem 1 is straightforward. By replacing the t and x derivatives of v in (2.4) by the corresponding t and x derivatives of F using the transformation (2.6), it is readily established that F satisfies (2,5). We omit the detail here. Notice that (2.5) cannot be expressed in terms of Hirota's bilinear operator. Equation (2.5) looks even more complicated than (2.4) and it is not necessarily easier to solve. However, the advantage of dealing with (2.5) rather than (2.4) is that (2.5) admits particular solutions in terms of a series of exponentials. We now show how this can be accomplished. First, we look for a solution of (2.5) of the form F = 1 + e o,
(2.7)
O = ioc + cot + fl,
(2.8)
where
with to, k, B constants to be determined. On substitution of (2.7) into (2.5), we find that a function F of the form (2.7) will be a solution of (2.5) provided co + 6 b k 2 = O,
and
(2.9a, b)
b2k 2 - c2k 4 = O,
with/3 arbitary. From (2.9a) and (2.9b) we have 6b 3 co~--
¢2,
b k = + -C.
(2.10a, b)
Thus, corresponding to k = b / c and 0 = ( b / c ) x - ( 6b3/ c2)t +/3, we have u = Vx =6b{ln(1 + e a
o
6c )Ix +-b--{In(1 +e°)}xx
- }3b2 (sech2 2 ( 0a) + 2 tanh c ( 2 ) +2 ,
(2.11a)
similarly, and corresponding to k = - b~ c, 0 = - ( b / c ) x - (6b2/c 2) t +/3, we have u=vx=~a c
seth 2
-2tanh
-2
.
(2.11b)
Equations (2.11) represents travelling wave solutions to (1.1), and their forms are shown as the full lines in Fig. 1 for the ease a = l , b = l , c = 3 and/3---0.
562
A. Jeffrey, S. Xu / Solutions to the KdVB equation 2.5-
KdVB-UI 2.0-
1,5-
J
Burgers
5"/\,,
'1" \,// ................... .~ -,2
-,0
-~
-,
-,
-2
/ ~v'l ,u
", 2
KdV
4
,
8
,0
~2
KDVB-U2 -2.5
Fig. I. Travelling wave solutions for the case a = ], b = I, e =3,/3 =0. KdVB-UI is given by (2.]Ia), KdVB-U2 is given by (2.11b).
Notice that the travelling wave solution u given by (2.11a) has the limit 0 as O ~ - o o and the limit - 6 b 2 / a c as O~oo, and the travelling wave solution given by (2.11b) has the limit 0 as O o - o o and the limit - 6 b 2 / a c as 0 ~ oo. All orders o f derivatives o f u with respect to 0 tend to zero as 101 ~ oo. This type o f travelling wave solution to the KdVB equation is t h e one studied by Bona and S c h o n b e k [1] w h o s h o w e d it is unique, but did not find its functional form. We remark that, as it stands, the travelling wave solution (2.11) cannot be r e d u c e d to either the sech 2 type o f travelling wave solution to the K d V equation in the limit b ~ 0 or to the tanh type travelling wave solution to the Burgers' equation in the limit c ~ 0. If such a reduction is to be c a r d e d out, it is necessary to return to (2.5) and to take the limit b ~ 0 or c ~ 0 , which will yield the desired solutions. The amplitudes, wave numbers, and frequencies o f the travelling wave solution given by (2.11) d e p e n d on the coefficients a, b, and c. Since the amplitudes are inversely proportional to a and c, the shock strengthens if either a or c becomes small. I f the dissipative effect is weak, (2.11) gives rise to a weak shock wave provided that a and c are o(1) quantities. N o other solutions to (1.1) can be obtained from (2.5) by e x p a n d i n g F in the form N
F=
~ e °',
(2.12)
i=1
with Oi = to~t + k~x + ~i,
and N > 0
(2.13)
an integer, a n d took k~,/3~ constants. We n o w show this is indeed the case. We first consider
A. Jeffrey, S. Xu / Solutions to the KdVB equation
563
the case N = 3. Setting N = 3 and substituting (2.12), (2.13) into (2.5) we obtain L(01) e 2°~ ÷ L(02) e2°z+ L(03) e2°3÷ P(O1, 02) e °1+°2÷ P(O1, 03) e 01+03÷ P(O2, 03) e °2+°3 = 0, (2.14) where L(0,)=6b(to,+6bk2),
(2.15)
i=1,2,3,
P(0,, 0fl = 6(5 b2k 2 + 2b2k~kj + 5 b2k 2 + 6bck 3 - 6bckEkj - 6bck~k 2 + 6bck ] + btoj + bto, + c E k ~ - 4c2k~ kj + 6cEk2kf - 4c2k~k ] + c2k4+ ctoiki - cto,kj + ctojkj - cwjk~),
i,j = 1, 2, 3.
(2.16)
Notice that P ( O,, Oj) = P ( Oj, Oi). Thus for (2.14) to hold we require that L(0s) = 0,
(2.17)
i = 1,2,3
P ( 0 1 , 02)=0,
P(OI, 03) = 0 ,
(2.18)
P(02, 03) = 0 .
The three equations given by (2.17) can be satisfied if toi = - 6 b k 2,
(2.19)
i = 1, 2, 3.
On substitution of cos into (2.18), we obtain a system of nonlinear algebraic equations P(Os, Oj) = P(Oj, 0i) = 6{-b2(ks - kj)2+ c2(ks - ~)4},
i,j = 1, 2, 3.
(2.20)
Equation (2.20) involves three unknowns in terms of the differences k l - k2, k l - k3 and k 2 - k 3 . For solutions which are distinct from the travelling wave solutions given by (2.11), we require that kl ~ k2 ~ k3. Clearly, equation (2.20) does not have such a set of solutions ks, i = 1, 2, 3. Thus there is no solution of the form (2.12) when N = 3. When N = 4 , 5, it follows that for an F of the form (2.12) to be a solution to (2.5), equations (2.17) and (2.18) must be satisfied by distinct ks, i = 1, 2 , . . . , N with L(Os), P(Oi, Oj) given by (2.15) and (2.16), respectively. Proceeding in a similar fashion to that of the case when N = 3, we can show that there are no distinct solutions ks, and hence that there is no solution for F of the form (2.12) with N = 4, 5 to (2.5). If N > 5, it is easy to verify that equation (2.17) and (2.18) must hold for F to be a solution of (2.5). L(Oi) and P(Oi, Oj) are again given by (2.15) and (2.16). In this case, there cannot be distinct solutions k~, i = 1, 2 , . . . , N to equations (2.17) and (2.18), because (2.18) only comprise a set of algebraic equations, each of which is of degree 4. The possibility of solving (2.5) for further solutions and the question of the stability of travelling wave solutions (2.11) is currently under examination.
References [ 1] J.L. Bona and M.E. Schonbek, "Travelling wave solutions to the Korteweg-de Vries-Burgers equation", Proc. Roy. Soc. Edinburgh, 101A, 207-226 (1985). [2l J. Canosa and J. Gazdag, "The Korteweg-de Vries-Burgers equation", J. Comput. Phys. 23, 393-403 (1977). [3] H. Grad and P.W. Hu, "Unified shock profile in a plasma", Phys. Fluids I0, 2596-2602 (1967). [4] J.D. Gibbon, et al., "The Painleve property and Hirota's method", Stud. Appl. Math. 72, 39-63, (1985).
564
A. Jeffrey, S. Xu / Solutions to the KdVB equation
[5] A. Jeffrey and T. Kakutani, "Weak nonlinear dispersive waves: a discussion centred around the Korteweg-de Vries equation", SIAM Rev. 14, 582-643 (1972). [6] R.S. Johnson, "Shallow water waves on a viscous fluid--the undular bore", Phys. Fluids, 15, 1693-1699, (1972). [7] R.S. Johnson, "A nonlinear equation incorporating damping and dispersion", J. Fluid Mech. 42, 49-60 (1970). [8] A. Jeffrey, "Some aspects of the mathematical modelling of long nonlinear waves", Arch. of Mechanics, 31, 559-574 (1979). [9] A.C. Newell, Solitons in Mathematical Physics, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA (1985).
EXACT
SOLUTIONS
559
TO THE
KORTEWEG-DE
VRIES-BURGERS
EQUATION
Alan J E F F R E Y and Siqing XU Department of Engineering Mathematics, University of Newcastle Upon Tyne, England, NE1 7R U Received 7 October 1988
We introduce a transormation which reduces the Korteweg-de Vries-Burgers (KdVB) equation, u, + 2auux + 5bux,. + cu~x~ = 0, to a quadratic form involving a new dependent variable and its partial derivatives. Exact solutions o f the KdVB equation can be obtained by solving this equation. The exact form of the travelling wave solution to the KdVB equation is obtained in this paper, and its nature depends on the direction o f propagation of the wave.
1. Introduction
The KdVB equation for u ( x , t) has the form ut + 2 auux + 5 buxx + cuxxx = 0,
(1.1)
with a, b and c constant coefficients. It arises in many different physical contexts as a model equation incorporating the effects of dispersion, dissipation and nonlinearity. Johnson [7] derived (1.1) as the governing equation for waves propagating in a liquid-filled elastic tube in which the weak effects of dispersion, dissipation and nonlinearity are present. Grad and Hu [3] used a steady state version of (1.1) to describe a weak shock profile in plasmas. In the limits b ~ 0 or c-* 0, that is when the weak effect of dissipation or dispersion is comparatively insignificant and so can be neglected, the KdVB equation can be approximated by the KdV equation u, + 2auu,, + cUxxx = 0,
(1.2)
or the Burgers' equation ut + 2auux + 5buxx = 0,
(1.3)
respectively. The KdV equation (1.2) and the Burgers equation (1.3) are both exactly soluble and they each have a wide range o f applications in physical problems. A number of theoretical issues relating to the KdVB equation have received considerable attention. In particular, the travelling wave solution to the KdVB equation has been studied extensively. Johnson [7] examined the travelling wave solution to the KdVB equation in the phase plane by means of a perturbation method in the regimes where b ,~ c and c ,t: b, and developed formal asymptotic expansions for the solution. Grad and Hu [3] studied the same problem using a method similar to that used by Johnson [7] and a related problem was considered by Jeffrey [8]. A numerical investigation of the problem was carried out by Canosa and Gazdag [2]. More recently, Bona and Schonbeck [1] studied the existence and uniqueness of bounded travelling wave solution to (1.1) which tend to constant states at plus and minus infinity. They also considered the limiting behaviour of the travelling wave solution of (1.1) as b tends to zero while c 0165-8641/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
A. Jeffrey, S. Xu / Solutions to the KdVB equation
560
is O(1), and as c tends to zero while b is o(1). The case where both b and c tend to zero with the ratio b/c held fixed was also examined. Qualitative results concerning the travelling wave solutions to the KdVB equation were also obtained by the above mentioned authors and others, but they did not find the exact functional form of the travelling wave solution, or any other exact solutions. The KdVB equation was examined by G i b b o n et al. [4] within the framework of soliton theory. A comprehensive account of the travelling wave solution to the KdVB equation can also be found in the review p a p e r by Jeffrey and Kakutani [5]. For other theoretical issues concerning the KdVB equation, we refer the reader to Bona and Schonbek [1] and to the reference cited there. In the present study we solve (1.1) foi" exact solutions by means of a direct method. In Section 2, we will introduce a transformation of the dependent variable u which reduces (1.1) to a quadratic form in a new dependent variable and its partial derivatives. Exact solutions are then obtained by solving this equation in terms of a series of exponentials. Two travelling wave solutions to (1.1) are obtained in this fashion. The situation is similar to that of the KdV equation in bilinear form in terms of Hirota's bilinear operator. The KdV equation (1.3) can be written in its bilinear form by making a suitable change of the dependent variable, and an infinite sequence of exact solutions, the N-soliton solutions, can be then obtained by solving the bilinear equation in terms of a series of exponentials. For the KdVB equation, however, we will show that the quadratic form yields only the two travelling wave solutions, and not the infinite n u m b e r found for the KdV equation. A similar situation may occur with equations which can be put into a bilinear form in terms of Hirota's bilinear operator. In such cases, at least two solutions can be obtained by solving the bilinear equations in terms of a series of exponentials; see, for example, Newell [9].
2. The transformation and exact solutions
To introduce our transformation of the dependent variable u, it proves convenient to use a variable representing an integral over the pulse profile that satisfies equation (1.1). Let
= I x u(X, t) dX,
(2.1)
then integrating (1.1) with respect to x once gives, in terms of ~3,
~t+a~E + 5b~xx+c~xxx+~(t)=O,
(2.2)
where ~ ( t ) is an arbitrary function of t arising from the integration. The function if(t) can be eliminated by introducing a new variable v(x, t) defined as
v(x,t)=~-f~
~b(X) dX,
(2.3)
which reduces (2.2) to the form
v , + a v 2 + 5bvxx+cVxxx=O. Equation (2.4) is equivalent to (1.1), and hereafter we will work with (2.4). The main result of the present study is stated in the following theorem.
(2.4)
A. Jeffrey, S. X u / Solutions to the K d V B equation
561
Theorem 1. The K d V B equation (2.4) is transformed into the following quadratic f o r m f o r F ( x , t) 6 b F ( Ft + 5bFxx + cF~x) + 6cF( F, + 5 b F ~ + c F ~ ) x - 6 c F x ( F , + 5bFxx + dFxxx)
+ 18c2(FEx - F~Fxxx) + 6 b ( b F 2 - cFxF~x) = 0
(2.5)
by the transformation v(x, t) = 6cFx+ 6b In F. aF a
(2.6)
The proof of Theorem 1 is straightforward. By replacing the t and x derivatives of v in (2.4) by the corresponding t and x derivatives of F using the transformation (2.6), it is readily established that F satisfies (2,5). We omit the detail here. Notice that (2.5) cannot be expressed in terms of Hirota's bilinear operator. Equation (2.5) looks even more complicated than (2.4) and it is not necessarily easier to solve. However, the advantage of dealing with (2.5) rather than (2.4) is that (2.5) admits particular solutions in terms of a series of exponentials. We now show how this can be accomplished. First, we look for a solution of (2.5) of the form F = 1 + e o,
(2.7)
O = ioc + cot + fl,
(2.8)
where
with to, k, B constants to be determined. On substitution of (2.7) into (2.5), we find that a function F of the form (2.7) will be a solution of (2.5) provided co + 6 b k 2 = O,
and
(2.9a, b)
b2k 2 - c2k 4 = O,
with/3 arbitary. From (2.9a) and (2.9b) we have 6b 3 co~--
¢2,
b k = + -C.
(2.10a, b)
Thus, corresponding to k = b / c and 0 = ( b / c ) x - ( 6b3/ c2)t +/3, we have u = Vx =6b{ln(1 + e a
o
6c )Ix +-b--{In(1 +e°)}xx
- }3b2 (sech2 2 ( 0a) + 2 tanh c ( 2 ) +2 ,
(2.11a)
similarly, and corresponding to k = - b~ c, 0 = - ( b / c ) x - (6b2/c 2) t +/3, we have u=vx=~a c
seth 2
-2tanh
-2
.
(2.11b)
Equations (2.11) represents travelling wave solutions to (1.1), and their forms are shown as the full lines in Fig. 1 for the ease a = l , b = l , c = 3 and/3---0.
562
A. Jeffrey, S. Xu / Solutions to the KdVB equation 2.5-
KdVB-UI 2.0-
1,5-
J
Burgers
5"/\,,
'1" \,// ................... .~ -,2
-,0
-~
-,
-,
-2
/ ~v'l ,u
", 2
KdV
4
,
8
,0
~2
KDVB-U2 -2.5
Fig. I. Travelling wave solutions for the case a = ], b = I, e =3,/3 =0. KdVB-UI is given by (2.]Ia), KdVB-U2 is given by (2.11b).
Notice that the travelling wave solution u given by (2.11a) has the limit 0 as O ~ - o o and the limit - 6 b 2 / a c as O~oo, and the travelling wave solution given by (2.11b) has the limit 0 as O o - o o and the limit - 6 b 2 / a c as 0 ~ oo. All orders o f derivatives o f u with respect to 0 tend to zero as 101 ~ oo. This type o f travelling wave solution to the KdVB equation is t h e one studied by Bona and S c h o n b e k [1] w h o s h o w e d it is unique, but did not find its functional form. We remark that, as it stands, the travelling wave solution (2.11) cannot be r e d u c e d to either the sech 2 type o f travelling wave solution to the K d V equation in the limit b ~ 0 or to the tanh type travelling wave solution to the Burgers' equation in the limit c ~ 0. If such a reduction is to be c a r d e d out, it is necessary to return to (2.5) and to take the limit b ~ 0 or c ~ 0 , which will yield the desired solutions. The amplitudes, wave numbers, and frequencies o f the travelling wave solution given by (2.11) d e p e n d on the coefficients a, b, and c. Since the amplitudes are inversely proportional to a and c, the shock strengthens if either a or c becomes small. I f the dissipative effect is weak, (2.11) gives rise to a weak shock wave provided that a and c are o(1) quantities. N o other solutions to (1.1) can be obtained from (2.5) by e x p a n d i n g F in the form N
F=
~ e °',
(2.12)
i=1
with Oi = to~t + k~x + ~i,
and N > 0
(2.13)
an integer, a n d took k~,/3~ constants. We n o w show this is indeed the case. We first consider
A. Jeffrey, S. Xu / Solutions to the KdVB equation
563
the case N = 3. Setting N = 3 and substituting (2.12), (2.13) into (2.5) we obtain L(01) e 2°~ ÷ L(02) e2°z+ L(03) e2°3÷ P(O1, 02) e °1+°2÷ P(O1, 03) e 01+03÷ P(O2, 03) e °2+°3 = 0, (2.14) where L(0,)=6b(to,+6bk2),
(2.15)
i=1,2,3,
P(0,, 0fl = 6(5 b2k 2 + 2b2k~kj + 5 b2k 2 + 6bck 3 - 6bckEkj - 6bck~k 2 + 6bck ] + btoj + bto, + c E k ~ - 4c2k~ kj + 6cEk2kf - 4c2k~k ] + c2k4+ ctoiki - cto,kj + ctojkj - cwjk~),
i,j = 1, 2, 3.
(2.16)
Notice that P ( O,, Oj) = P ( Oj, Oi). Thus for (2.14) to hold we require that L(0s) = 0,
(2.17)
i = 1,2,3
P ( 0 1 , 02)=0,
P(OI, 03) = 0 ,
(2.18)
P(02, 03) = 0 .
The three equations given by (2.17) can be satisfied if toi = - 6 b k 2,
(2.19)
i = 1, 2, 3.
On substitution of cos into (2.18), we obtain a system of nonlinear algebraic equations P(Os, Oj) = P(Oj, 0i) = 6{-b2(ks - kj)2+ c2(ks - ~)4},
i,j = 1, 2, 3.
(2.20)
Equation (2.20) involves three unknowns in terms of the differences k l - k2, k l - k3 and k 2 - k 3 . For solutions which are distinct from the travelling wave solutions given by (2.11), we require that kl ~ k2 ~ k3. Clearly, equation (2.20) does not have such a set of solutions ks, i = 1, 2, 3. Thus there is no solution of the form (2.12) when N = 3. When N = 4 , 5, it follows that for an F of the form (2.12) to be a solution to (2.5), equations (2.17) and (2.18) must be satisfied by distinct ks, i = 1, 2 , . . . , N with L(Os), P(Oi, Oj) given by (2.15) and (2.16), respectively. Proceeding in a similar fashion to that of the case when N = 3, we can show that there are no distinct solutions ks, and hence that there is no solution for F of the form (2.12) with N = 4, 5 to (2.5). If N > 5, it is easy to verify that equation (2.17) and (2.18) must hold for F to be a solution of (2.5). L(Oi) and P(Oi, Oj) are again given by (2.15) and (2.16). In this case, there cannot be distinct solutions k~, i = 1, 2 , . . . , N to equations (2.17) and (2.18), because (2.18) only comprise a set of algebraic equations, each of which is of degree 4. The possibility of solving (2.5) for further solutions and the question of the stability of travelling wave solutions (2.11) is currently under examination.
References [ 1] J.L. Bona and M.E. Schonbek, "Travelling wave solutions to the Korteweg-de Vries-Burgers equation", Proc. Roy. Soc. Edinburgh, 101A, 207-226 (1985). [2l J. Canosa and J. Gazdag, "The Korteweg-de Vries-Burgers equation", J. Comput. Phys. 23, 393-403 (1977). [3] H. Grad and P.W. Hu, "Unified shock profile in a plasma", Phys. Fluids I0, 2596-2602 (1967). [4] J.D. Gibbon, et al., "The Painleve property and Hirota's method", Stud. Appl. Math. 72, 39-63, (1985).
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[5] A. Jeffrey and T. Kakutani, "Weak nonlinear dispersive waves: a discussion centred around the Korteweg-de Vries equation", SIAM Rev. 14, 582-643 (1972). [6] R.S. Johnson, "Shallow water waves on a viscous fluid--the undular bore", Phys. Fluids, 15, 1693-1699, (1972). [7] R.S. Johnson, "A nonlinear equation incorporating damping and dispersion", J. Fluid Mech. 42, 49-60 (1970). [8] A. Jeffrey, "Some aspects of the mathematical modelling of long nonlinear waves", Arch. of Mechanics, 31, 559-574 (1979). [9] A.C. Newell, Solitons in Mathematical Physics, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA (1985).