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The Korteweg-de Vries-Burgers' equation: a reconstruction of exact solutions
Wave Motion 14 (1991) 267-271 Elsevier
267
The Korteweg-de Vries-Burgers' equation: a reconstruction of exact solutions M. Vlieg-Hulstman and W.D. Halford Department of Mathematics and Statistics, Massey University, Palmerston North, New Zealand
Received 9 January 1991 It is shown that recentlypublished exact solutions to the KdVB equation are essentiallythe same. A Painlev6analysis leads directly to one form of the KdVB solution. There is a linear relationship betweenthe KdVB solution and particular solutions of the KdV and Burgers' equation.
1. Introduction The Korteweg-de Vries (KdV) equation u, + 2auux + cuxxx= 0,
(1.1)
where a and c are constants, has been studied extensively by many authors (see for example [ 1]). It first appeared in the literature in 1895 [2]. One can set c = 1 without loss o f generality, but for our purposes we shall not specify c. The Burgers' equation u, + 2aUUx + buxx = 0,
(1.2)
where a and b are constants, is also important in applied mathematics (see for example [3]). Its name derives from its use by Burgers [4] for studying turbulence about 50 years ago. Again, we could set b = 1 but prefer not to do so here. On the other hand, the Korteweg-de VriesBurgers' (KdVB) equation ut + 2aUUx + buxx + CUxxx= 0,
(1.3)
where a, b and c are constants, is not so well known. Wijngaarden [5] considered it. Johnson [6] used it to model nonlinear waves in an elastic tube with dispersion and dissipation. Travelling wave solutions to the KdVB equation have been studied 0923-5965/91/$03.50 © 1991 - Elsevier SciencePublishers B.V.
by Jeffrey and Kakutani [7], Bona and Schonbek [8], Xiong [9], Jeffrey and Xu [10], and Mclntosh [ 11 ]. However, it appears that exact solutions were unknown until quite recently [9-11]. Furthermore, it seems that the exact solution published by Xiong [9] was unknown to Jeffrey and Xu [10] and Mclntosh [11], and that the latter author was unaware o f the work done by Jeffrey and Xu. We aim to show that the solutions o f Xiong and Mclntosh are algebraically equivalent, and that the solution of Jeffrey and Xu can be related to the others by a Galilean transformation. This paper demonstrates that essentially only one exact solution to the KdVB equation is known. A feature of this solution is that it is a linear combination of particular solutions o f the KdV equation and the Burgers' equation. We also show how a Painlev6 analysis of the KdVB equation leads directly to the transformation used by Jeffrey and Xu [10] and to the corresponding exact solution. This transformation is also a linear combination of that which expresses the KdV in bilinear form and the C o l e - H o p f transformation which linearises Burgers' equation. We show that the Jeffrey and Xu solution is obtained by the vanishing of the second term in the Painlev6 series expansion for the dependent variable.
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
268
2. E x a c t
solutions to the KdVB
An equivalent form of (2.5a) is
equation
Attention is first directed to the exact solution obtained by Xiong [9] in 1987. Travelling wave solutions to the KdVB equation (1.3) are obtained by the substitution u=f(O = f ( x - & ) where 8 is the speed of the wave, which reduces eq. (1.3) to the ordinary differential equation (ODE):
(2.1)
- S f ' + 2aff' + bf" + e l " = 0
C
C
C
(2.2)
C
where a is a constant of integration. translation
a
(1 + e "¢)
_2+1 2a
(8+ 6e2c)
(2.5b)
where 8 = + J36E4c 2+ 4aa. However, we note there is a sign option in the expression for fl in (2.4) which should appear in the last term of (2.5b). Also from (2.4) and (2.6) we have: 8 3e2c /3=--+ 2a a
The first integral of the above ODE (2.1) is
f" +b-f' +a-f2-8- f +a-=O
6dc
u= - - -
The
Mclntosh [11] employed the Ince transformation [12]
f = zZr + fl,
z = eu¢,
r = r(z)
to reduce (2.2) to
f = g + fl
p2cr" + ar2= 0
transforms (2.2) into
provided
g" +bg' +ag2 + 1- (2afl-8)g c
c
b=-51gc,
c
1
+- ( a f t - 8fl+ a) =0.
(2.4)
When a = 0 then aft-Sfl+ a = 0 gives either fl= 0 or fl=8/a. Then g=y(l+e~¢) -z is a solution of (2.3) if and only if (2.6) holds with 8= 6E2e when fl=8/a, or 6=-6~'e when fl=0.
Xiong assumed the solution of (2.3) is of the form g = y(1 +e~C)-2
If
and showed that u is given by 3b2
u = 25ac
[ ,/ 1+
14625ac2
~ 9b
(
2 l + e b~/(5~)
' (2.5a)
( = x - 8t + 7/and where 0 is a constant, provided
b=56c,
6e2c
7=---,
a
8/r85
q ~
4a a
afl2-flS+a=0
which are the same conditions as Xiong used if we identify/t= - e. From (2.7) Mclntosh obtained Xiong's solution as given in (2.5b) if one replaces p by - e and puts 1/= 0 in the expression for (.
Suppose a f t - 8fl+ a= 0 but aGO then 4a a"
f l = 6 + 3uzc, 2a a
(2.3)
c
fl_ 8 + 1 ~ -~a 2Va 5
(2.7)
then
(2.6)
a
u
6e2c 6fie - - (1+e~¢)-2+ - , a
ff
(= x - 6e2ct.
(2.8)
If fl=O,
6e2c a
8 fl=-,
6e2c then
u= - - -
(l+e~¢) -2,
(=x+6~ct.
a
(2.9)
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
Writing U=Vx eq. (1.3) assumes its "potential" form. Jeffrey and Xu [10] turned this potential form into bilinear form via the transformation 6c Fx +6b ln v=-F
a F
(2.10)
5a
Travelling wave solutions were then obtained by writing
3.
269
Linear relationships Burgers' equation in the form ut + 2aUUx + 6~b 5 Uxx = 0
(3.1)
can be written in potential form (put u = o,,) and then reduced to the linear heat equation
F = l + e o, O= k x + rot + r1 F, +6~b5 Fx~=O
where 1/is a constant. This requires 6b3 and t o = - 125c----~
k= + b . 5c
(2.11)
6b TB: o = - - In F. 5a
Jeffrey and Xu's solutions are then U--~
3b2 50ac
[so~h 2
1 ½0+2tanh ~0+2],
by the Cole-Hopf transformation
k = - -b 5c (2.12a)
When F = 1 + ek'~- o,,, the solution to Burgers' equation (3.1) is 3bkl F u~=-~a l tanhTk, /~x--6b~ klt~+
and u=
3b2
[sech z 10-
1 2 tanh ~0 - 2],
50ac
(3.2)
l ]J
(3.3a)
b
k= - -5c
(2.12b) which are not the same as the Xiong-McIntosh solution (2.5a). On the other hand, (2.12a) and (2.12b) are identical with (2.8) and (2.9) respectively. The KdVB equation (1.3) is invariant under the Galilean transformation
The Korteweg-de Vries equation u, + 2auux + 6CU~x~= 0
(3.4)
in potential form (u= v~) is turned into bilinear form FF~t - Fx Ft + 18cF2x + 6cFF. . . . -24cFxFxxx = 0
(3.5)
by the transformation x*=x+mt,
t*=t,
u*=u+n
(2.13)
if and only if m = 2an. By choosing n = - fl we obtain (2.9) from (2.5b) and (2.13), upon dropping the asterisk. Similarly, by choosing n = - fl+ 662c/a we obtain (2.8) from (2.5b) and (2.13), upon dropping the asterisk. Thus by using a Galilean transformation on Xiong's solution (2.5b) we have recovered the solutions (2.8) and (2.9). Hence there is essentially only one exact solution known at this stage.
TKdV:
36c Fx a F
V= - -
(3.6)
When we apply Hirota's method to (3.5) the solution Ur,ov =
is obtained.
18/~2c sech 2 ~ ( x - 6 c ~ t ) 2a 2
(3.7a)
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
270
Further, when we set kl =k2 = b/(5c), solutions (3.3a) and (3.7a) become 3b2 uB = 2-~ac [tanh ( + 1], 18b2 UKd V --
50ac
(3.3b)
For the KdVB equation (1.3) we find a= - 2 and a recursion relation for determining the uy of the form
cuj(j+ 1 ) ( j - 4 ) ( j - 6)¢p~ .....
sech 2 (
(3.7b)
respectively where
By choosing Burgers' equation in the form (3.1) and the KdV equation in the form (3.4) one can observe that the transformation (2.10) is a linear combination of (3.2) and (3.6) since
U K d V B ~- U B "[- ~1U K d v •
TKdvB=TB--ITKdv and
1
UKdVB ~ UB -- ~ UKdV
(4.2)
j=0,
6c Uo= - - - ~
(4.3)
j=l,
6b ~x + 6c ~bx~ ul=~a a
(4.4)
j=2,
u2=2aL
a
1 ~_ ~ , _ 4 c ~bxxX+3c*;:x
Ox
6b
(3.8)
A similar expression for (3.8) has been derived by Xiong. The same argument applied to solution (2.12b) yields
uj_ 1)
where h is a nonlinear function. Resonances occur at j = - 1 , 4, 6 as in the KdV case. The us at the resonances are arbitrary. From (4.2) with the specific function for the KdVB equation, the first three terms are
TKdVB ---- TB + / T K d v •
Also, the solution (2.12a) is a linear combination of (3.3b) and (3.7b) since
Uo, u, . . . . .
~ 25cb2] -~ .
(4.5)
Assume uj=0 for allj~>2. Then (4.1) is simply
U=Uor~- Z+ ul ~-l
(4.6)
and substituting (4.3) and (4.4) into this gives u = -6c - (dpxI +6b (In q~)~.
a \q~/x 5a
(4.7)
Writing u = vx we can integrate (4.7) to obtain 4. Painlev6 analysis
Weiss et al. [13] have shown how the integrability of a partial differential equation (pde) is intimately related to the "Painlev6 property" of the equation, viz. the dependent variable u can be expressed as
u=r~~ ~ uj~
(4.1)
j=0
where a is a (negative) integer determined by comparing the lowest powers of ~ corresponding to the nonlinear and linear terms in the differential equation.
6c ~bx+6b in ~ a ~b 5a
(4.8)
upon making the arbitrary integration constant zero. We observe that (4.8) is precisely the Jeffrey and Xu transformation (2.10), having replaced by F. The Jeffrey and Xu solution (2.12b) is obtained directly from (4.3) and (4.4) by putting ul =0. For then (4.4) yields ~b= A - l [ e ° ' J r
r(t)]
(4.9)
where 0= 2x+f(t), ~.=-b/5c, while f(t) and r(t) are arbitrary functions.
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
F r o m (4.3) we n o w have 6c 20 u0= - - - e
(4.10)
a
Substituting (4.9) into (4.5) we obtain u2 = (2aA,) - 1[6c,~3 - f ' ( t ) ] .
(4.11)
C h o o s i n g f ( t ) = tot + r/, where 1/is an arbitrary constant and o9= 6c~,- 3=
_6b3/125c2,
we find u2 = 0. We can set u4 = u6 = 0 at the resonances. A s s u m i n g uj=O for all j>~3, the series expansion (4.1) is just the single term u=u0~b -2
(4.12)
which is solution (2.12b) if we choose r(t)= 1. The solution (2.12a) is obtained f r o m (2.12b) by a Galilean transformation.
5. References [1] R.M. Miura, "The Korteweg--de Vries equation: a survey of results", SlAM Review 18, 412-459 (1976). [2] D.J. Korteweg and G. de Vries, "On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves", Phil. Mag. 39, 422443 (1895).
271
[3] E.R. Benton and G.W. Platzman, "A table of solutions of the one-dimensional Burgers equation", Quart. Appl. Math 30, 195-212 (1972). [4] J.M. Burgers, "Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion", Trans. Roy. Neth. Acad. Sci. Amsterdam 17, 1-53 (1939). [5] L. van Wijngaarden, "One-dimensional flow of liquids containing small gas bubbles", Ann. Rev. Fluid Mech. 4, 369-395 (1972). [6] R.S. Johnson, "A non-linear equation incorporating damping and dispersion", J. Fluid Mech. 42, 49-60 (1970). [7] A. Jeffrey and T. Kakutani, "Weak nonlinear dispersive waves: a discussion centred around the Korteweg-de Vries equation", SlAM Review 14, 582~43 0972). [8] 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). [9] S.L. Xiong, "An analytic solution of Burgers-KdV equation", Chinese Sci. Bull. 34, 1158-1162 (1989). [10] A. Jeffrey and S. Xu, "Exact solutions to the Kortewegde Vries-Burgers equation", Wave Motion 11, 559-564 (1989). [11] I. Mclntosh, "Single phase averaging and travelling wave solutions of the modified Burgers--Korteweg--de Vries equation", Phys. Lett. A 143, 57-61 (1990). [12] E.L. Ince, Ordinary Differential Equations, Dover, New York, 1956. [13] J. Weiss, M. Tabor and G. Carnevale, "The Painlev6 property for partial differential equations", J. Math. Phys. 24, 522-526 (1983).
267
The Korteweg-de Vries-Burgers' equation: a reconstruction of exact solutions M. Vlieg-Hulstman and W.D. Halford Department of Mathematics and Statistics, Massey University, Palmerston North, New Zealand
Received 9 January 1991 It is shown that recentlypublished exact solutions to the KdVB equation are essentiallythe same. A Painlev6analysis leads directly to one form of the KdVB solution. There is a linear relationship betweenthe KdVB solution and particular solutions of the KdV and Burgers' equation.
1. Introduction The Korteweg-de Vries (KdV) equation u, + 2auux + cuxxx= 0,
(1.1)
where a and c are constants, has been studied extensively by many authors (see for example [ 1]). It first appeared in the literature in 1895 [2]. One can set c = 1 without loss o f generality, but for our purposes we shall not specify c. The Burgers' equation u, + 2aUUx + buxx = 0,
(1.2)
where a and b are constants, is also important in applied mathematics (see for example [3]). Its name derives from its use by Burgers [4] for studying turbulence about 50 years ago. Again, we could set b = 1 but prefer not to do so here. On the other hand, the Korteweg-de VriesBurgers' (KdVB) equation ut + 2aUUx + buxx + CUxxx= 0,
(1.3)
where a, b and c are constants, is not so well known. Wijngaarden [5] considered it. Johnson [6] used it to model nonlinear waves in an elastic tube with dispersion and dissipation. Travelling wave solutions to the KdVB equation have been studied 0923-5965/91/$03.50 © 1991 - Elsevier SciencePublishers B.V.
by Jeffrey and Kakutani [7], Bona and Schonbek [8], Xiong [9], Jeffrey and Xu [10], and Mclntosh [ 11 ]. However, it appears that exact solutions were unknown until quite recently [9-11]. Furthermore, it seems that the exact solution published by Xiong [9] was unknown to Jeffrey and Xu [10] and Mclntosh [11], and that the latter author was unaware o f the work done by Jeffrey and Xu. We aim to show that the solutions o f Xiong and Mclntosh are algebraically equivalent, and that the solution of Jeffrey and Xu can be related to the others by a Galilean transformation. This paper demonstrates that essentially only one exact solution to the KdVB equation is known. A feature of this solution is that it is a linear combination of particular solutions o f the KdV equation and the Burgers' equation. We also show how a Painlev6 analysis of the KdVB equation leads directly to the transformation used by Jeffrey and Xu [10] and to the corresponding exact solution. This transformation is also a linear combination of that which expresses the KdV in bilinear form and the C o l e - H o p f transformation which linearises Burgers' equation. We show that the Jeffrey and Xu solution is obtained by the vanishing of the second term in the Painlev6 series expansion for the dependent variable.
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
268
2. E x a c t
solutions to the KdVB
An equivalent form of (2.5a) is
equation
Attention is first directed to the exact solution obtained by Xiong [9] in 1987. Travelling wave solutions to the KdVB equation (1.3) are obtained by the substitution u=f(O = f ( x - & ) where 8 is the speed of the wave, which reduces eq. (1.3) to the ordinary differential equation (ODE):
(2.1)
- S f ' + 2aff' + bf" + e l " = 0
C
C
C
(2.2)
C
where a is a constant of integration. translation
a
(1 + e "¢)
_2+1 2a
(8+ 6e2c)
(2.5b)
where 8 = + J36E4c 2+ 4aa. However, we note there is a sign option in the expression for fl in (2.4) which should appear in the last term of (2.5b). Also from (2.4) and (2.6) we have: 8 3e2c /3=--+ 2a a
The first integral of the above ODE (2.1) is
f" +b-f' +a-f2-8- f +a-=O
6dc
u= - - -
The
Mclntosh [11] employed the Ince transformation [12]
f = zZr + fl,
z = eu¢,
r = r(z)
to reduce (2.2) to
f = g + fl
p2cr" + ar2= 0
transforms (2.2) into
provided
g" +bg' +ag2 + 1- (2afl-8)g c
c
b=-51gc,
c
1
+- ( a f t - 8fl+ a) =0.
(2.4)
When a = 0 then aft-Sfl+ a = 0 gives either fl= 0 or fl=8/a. Then g=y(l+e~¢) -z is a solution of (2.3) if and only if (2.6) holds with 8= 6E2e when fl=8/a, or 6=-6~'e when fl=0.
Xiong assumed the solution of (2.3) is of the form g = y(1 +e~C)-2
If
and showed that u is given by 3b2
u = 25ac
[ ,/ 1+
14625ac2
~ 9b
(
2 l + e b~/(5~)
' (2.5a)
( = x - 8t + 7/and where 0 is a constant, provided
b=56c,
6e2c
7=---,
a
8/r85
q ~
4a a
afl2-flS+a=0
which are the same conditions as Xiong used if we identify/t= - e. From (2.7) Mclntosh obtained Xiong's solution as given in (2.5b) if one replaces p by - e and puts 1/= 0 in the expression for (.
Suppose a f t - 8fl+ a= 0 but aGO then 4a a"
f l = 6 + 3uzc, 2a a
(2.3)
c
fl_ 8 + 1 ~ -~a 2Va 5
(2.7)
then
(2.6)
a
u
6e2c 6fie - - (1+e~¢)-2+ - , a
ff
(= x - 6e2ct.
(2.8)
If fl=O,
6e2c a
8 fl=-,
6e2c then
u= - - -
(l+e~¢) -2,
(=x+6~ct.
a
(2.9)
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
Writing U=Vx eq. (1.3) assumes its "potential" form. Jeffrey and Xu [10] turned this potential form into bilinear form via the transformation 6c Fx +6b ln v=-F
a F
(2.10)
5a
Travelling wave solutions were then obtained by writing
3.
269
Linear relationships Burgers' equation in the form ut + 2aUUx + 6~b 5 Uxx = 0
(3.1)
can be written in potential form (put u = o,,) and then reduced to the linear heat equation
F = l + e o, O= k x + rot + r1 F, +6~b5 Fx~=O
where 1/is a constant. This requires 6b3 and t o = - 125c----~
k= + b . 5c
(2.11)
6b TB: o = - - In F. 5a
Jeffrey and Xu's solutions are then U--~
3b2 50ac
[so~h 2
1 ½0+2tanh ~0+2],
by the Cole-Hopf transformation
k = - -b 5c (2.12a)
When F = 1 + ek'~- o,,, the solution to Burgers' equation (3.1) is 3bkl F u~=-~a l tanhTk, /~x--6b~ klt~+
and u=
3b2
[sech z 10-
1 2 tanh ~0 - 2],
50ac
(3.2)
l ]J
(3.3a)
b
k= - -5c
(2.12b) which are not the same as the Xiong-McIntosh solution (2.5a). On the other hand, (2.12a) and (2.12b) are identical with (2.8) and (2.9) respectively. The KdVB equation (1.3) is invariant under the Galilean transformation
The Korteweg-de Vries equation u, + 2auux + 6CU~x~= 0
(3.4)
in potential form (u= v~) is turned into bilinear form FF~t - Fx Ft + 18cF2x + 6cFF. . . . -24cFxFxxx = 0
(3.5)
by the transformation x*=x+mt,
t*=t,
u*=u+n
(2.13)
if and only if m = 2an. By choosing n = - fl we obtain (2.9) from (2.5b) and (2.13), upon dropping the asterisk. Similarly, by choosing n = - fl+ 662c/a we obtain (2.8) from (2.5b) and (2.13), upon dropping the asterisk. Thus by using a Galilean transformation on Xiong's solution (2.5b) we have recovered the solutions (2.8) and (2.9). Hence there is essentially only one exact solution known at this stage.
TKdV:
36c Fx a F
V= - -
(3.6)
When we apply Hirota's method to (3.5) the solution Ur,ov =
is obtained.
18/~2c sech 2 ~ ( x - 6 c ~ t ) 2a 2
(3.7a)
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
270
Further, when we set kl =k2 = b/(5c), solutions (3.3a) and (3.7a) become 3b2 uB = 2-~ac [tanh ( + 1], 18b2 UKd V --
50ac
(3.3b)
For the KdVB equation (1.3) we find a= - 2 and a recursion relation for determining the uy of the form
cuj(j+ 1 ) ( j - 4 ) ( j - 6)¢p~ .....
sech 2 (
(3.7b)
respectively where
By choosing Burgers' equation in the form (3.1) and the KdV equation in the form (3.4) one can observe that the transformation (2.10) is a linear combination of (3.2) and (3.6) since
U K d V B ~- U B "[- ~1U K d v •
TKdvB=TB--ITKdv and
1
UKdVB ~ UB -- ~ UKdV
(4.2)
j=0,
6c Uo= - - - ~
(4.3)
j=l,
6b ~x + 6c ~bx~ ul=~a a
(4.4)
j=2,
u2=2aL
a
1 ~_ ~ , _ 4 c ~bxxX+3c*;:x
Ox
6b
(3.8)
A similar expression for (3.8) has been derived by Xiong. The same argument applied to solution (2.12b) yields
uj_ 1)
where h is a nonlinear function. Resonances occur at j = - 1 , 4, 6 as in the KdV case. The us at the resonances are arbitrary. From (4.2) with the specific function for the KdVB equation, the first three terms are
TKdVB ---- TB + / T K d v •
Also, the solution (2.12a) is a linear combination of (3.3b) and (3.7b) since
Uo, u, . . . . .
~ 25cb2] -~ .
(4.5)
Assume uj=0 for allj~>2. Then (4.1) is simply
U=Uor~- Z+ ul ~-l
(4.6)
and substituting (4.3) and (4.4) into this gives u = -6c - (dpxI +6b (In q~)~.
a \q~/x 5a
(4.7)
Writing u = vx we can integrate (4.7) to obtain 4. Painlev6 analysis
Weiss et al. [13] have shown how the integrability of a partial differential equation (pde) is intimately related to the "Painlev6 property" of the equation, viz. the dependent variable u can be expressed as
u=r~~ ~ uj~
(4.1)
j=0
where a is a (negative) integer determined by comparing the lowest powers of ~ corresponding to the nonlinear and linear terms in the differential equation.
6c ~bx+6b in ~ a ~b 5a
(4.8)
upon making the arbitrary integration constant zero. We observe that (4.8) is precisely the Jeffrey and Xu transformation (2.10), having replaced by F. The Jeffrey and Xu solution (2.12b) is obtained directly from (4.3) and (4.4) by putting ul =0. For then (4.4) yields ~b= A - l [ e ° ' J r
r(t)]
(4.9)
where 0= 2x+f(t), ~.=-b/5c, while f(t) and r(t) are arbitrary functions.
M. Vlieg-Hulstman, W.D. Halford / Exact solutions of K-de V-B equation
F r o m (4.3) we n o w have 6c 20 u0= - - - e
(4.10)
a
Substituting (4.9) into (4.5) we obtain u2 = (2aA,) - 1[6c,~3 - f ' ( t ) ] .
(4.11)
C h o o s i n g f ( t ) = tot + r/, where 1/is an arbitrary constant and o9= 6c~,- 3=
_6b3/125c2,
we find u2 = 0. We can set u4 = u6 = 0 at the resonances. A s s u m i n g uj=O for all j>~3, the series expansion (4.1) is just the single term u=u0~b -2
(4.12)
which is solution (2.12b) if we choose r(t)= 1. The solution (2.12a) is obtained f r o m (2.12b) by a Galilean transformation.
5. References [1] R.M. Miura, "The Korteweg--de Vries equation: a survey of results", SlAM Review 18, 412-459 (1976). [2] D.J. Korteweg and G. de Vries, "On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves", Phil. Mag. 39, 422443 (1895).
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[3] E.R. Benton and G.W. Platzman, "A table of solutions of the one-dimensional Burgers equation", Quart. Appl. Math 30, 195-212 (1972). [4] J.M. Burgers, "Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion", Trans. Roy. Neth. Acad. Sci. Amsterdam 17, 1-53 (1939). [5] L. van Wijngaarden, "One-dimensional flow of liquids containing small gas bubbles", Ann. Rev. Fluid Mech. 4, 369-395 (1972). [6] R.S. Johnson, "A non-linear equation incorporating damping and dispersion", J. Fluid Mech. 42, 49-60 (1970). [7] A. Jeffrey and T. Kakutani, "Weak nonlinear dispersive waves: a discussion centred around the Korteweg-de Vries equation", SlAM Review 14, 582~43 0972). [8] 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). [9] S.L. Xiong, "An analytic solution of Burgers-KdV equation", Chinese Sci. Bull. 34, 1158-1162 (1989). [10] A. Jeffrey and S. Xu, "Exact solutions to the Kortewegde Vries-Burgers equation", Wave Motion 11, 559-564 (1989). [11] I. Mclntosh, "Single phase averaging and travelling wave solutions of the modified Burgers--Korteweg--de Vries equation", Phys. Lett. A 143, 57-61 (1990). [12] E.L. Ince, Ordinary Differential Equations, Dover, New York, 1956. [13] J. Weiss, M. Tabor and G. Carnevale, "The Painlev6 property for partial differential equations", J. Math. Phys. 24, 522-526 (1983).