Symmetry analysis of Generalized Burger's equation

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Symmetry Analysis of GeneralizedBurgers Equation

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Chapter I

Introduction

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1.1 Introduction

The Partial differential equations (PDEs) arising in many physical fields like the

condense matter physics, fluid Mechanics, physics and optics, etc. which exhibit a

rich variety of non linear phenomena. When the inhomogenetics of media and non-

uniformity of boundaries are taken into account in various physical situations, the

variable coefficient PDEs often can provide more powerful and realistic molds than

their constant- coefficient counterparts in describing a large variety of real phenomena.

It is known that to find exact solutions of the PDEs is always one of the central themes

of Mathematics and Physics. In the last few decades, remarkable progress has been

made in understanding the integrability and non-integrability of nonlinear partial

differential equations.

The motivation for the present thesis comes from the work on the GBE with

variable viscosity

ut + uux =(t)

2uxx, (1.1)

by Doyle and Englefield (1990). They arrived at (1.1) by considering an application

of the Burgers equation

ut + uux =

2uxx, (1.2)

in the formation and decay of nonplane shock waves. In (1.2) x is a coordinate moving

with the wave at the speed of sound and u is the velocity fluctuations. The coefficient

of uxx, which allows for viscosity, is approximated by a constant in (1.2), but is

in reality a function of the time t (Lighthill (1956)), say, (t). Doyle and Englefield

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(1990) by using the method for defining an optimal system of group-invariant solutions

(Olver (1986)) showed that (t) satisfies the first order equation

=

c1 + c2c3t2 + (c1 c2)t + c4 , (1.3)

and obtained the following five distinct expressions for (t) by setting certain of the

constants ci equal to zero:

1(t) = et, (1.4)

2(t) = e1/t, (1.5)

3(t) = (t + )r, (1.6)

4(t) =

t +

t +

r, = 0 = , (1.7)

5(t) = exp2tan1(t + )

. (1.8)

Sachdev, Nair and Tikekar (1988) through u = (1+ t)(1n)/2

(1+n)/2(1n)

f(), =

1/2(1 + t)(1n)/2x reduced another GBE

ut + u(1n)/(1+n)ux =

2(1 + t)nuxx, (1.9)

to

f 2f(1n)/(1+n)f + (n + 1)(f + f) = 0, (1.10)

for which an exact solution in terms of an error function is obtained in a manner

described below: Equation (1.10) is integrated once under the conditions f and f

vanish as to give

f (1 + n)f2/(1+n) + (1 + n)f = 0, (1.11)

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which is once again integrated to yield the solution f = e(1+n)2/2[h()](1+n)/(1n),

where h() = [f(0)](1n)/(1+n) 2(1 n)erf[1n2

]. Doyle and Engelfield (1990)

observed that equation (1.11) being the Bernoullis equation can be converted to a

linear equation and therefore solved in terms of an error function. The forms (1.4)-

(1.8) included those of Scott (1981a) and Sachdev, Nair and Tikekar (1988) as special

cases.

Mayil Vaganan (1994) employed the direct method to derive the similarity reduc-

tions of the Burgers equation as well as its generalizations

ut + uux +ju

2t= uxx, (1.12)

ut + u2ux +

ju

2t= uxx, (1.13)

ut + u2ux = uxx, (1.14)

ut + uux + f(x, t) = g(t)uxx, (1.15)

ut + uux + f(t)u

= g(t)uxx. (1.16)

The Burgers equation (1.2) is introduced by Batman (1915). Equation (1.2) is

in fact a special case of mathematical models of turbulance (Burgers (1939, 1940)).

A similarity solution in the form u(x, t) = t1/2S(z), z(x, t) = x/

2t, where S(z)

satisfies a Riccati equation is obtained by Burgers (1950). Hopf (1950) and Cole

(1951) linearized the Burgers equation (1.2) through the Cole-Hopf transformation

u = x/ to the heat equation t = (/2)xx.

Now we describe some important works on the GBEs: Lardner and Arya (1980)

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obtained matched asymptotic solutions of the Burgers equation with linear damping

ut + uux + u =

2

uxx, (1.17)

under the constraint that the shock is thin. They have also considered an extended

form of (1.17) in which the coefficient ofux is p0u + q0u2 + r0c(t), where p0, q0 and r0

are constants. The very presence ofu in (1.17) is due to the fact that the equation of

motion includes a small viscous damping term proportional to the velocity (Crighton

(1979)). The N-wave solutions for (1.17) are obtained by Sachdev and Joseph (1994).

Asymptotic solutions to the modified Burgers equation

ut + u2ux = uxx, (1.18)

with two initial disturbances, namely, N-wave and sinusoid, for small values of the

dissipation coefficient are derived by Lee-Bapty and Crighton (1987).

Lighthill (1956) and Leibovich and Seebass (1974) derived the nonplanar Burgers

equation

ut + uux +ju

2t=

2uxx, (1.19)

where 0 < < 1 are constants and j = 0, 1, 2 to describe the propagation of weakly

nonlinear longitudinal waves in gases or liquids from a non planar source. Sachdev,

Joseph and Nair (1994) have given an exact N-wave solutions for (1.19).

The GBE describing the progressive waves with cylindrical symmetry j = 1 or

spherical symmetry j = 2 is

vx + vv j2x

v = v. (1.20)

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The existence of an exact solution of the form x1/2v(x, ) = (x/2) to (1.20) is first

noted by Chong and Sirovich (1973) and subsequently Rudenko and Soluyan (1977)

provided the closed form expression for . Sinai (1976) found the the profile of the

axisymmetric body which gives rise to a similarity solution in the steady supersonic

flow problem governed by an equation of the form (1.20).

Grundy, Sachdev and Dawson (1994) obtained large time solutions of an initial

value problem (IVP) for a GBE:

ut + (u+1)x +

ju

2t= uxx, u(x, 1) = u1(x). (1.21)

Sachdev and his collobarators (1986, 1987, 1988) by inserting the self-similar form

u = tpf(), = x/

t into the GBEs

ut + uux + u

=

2uxx, (1.22)

ut + uux +

ju

2t=

2uxx, (1.23)

ut + uux =

2g(t)uxx, (1.24)

obtained equations for f() in the form

f a1fqf + a2f + a3fr = 0, (1.25)

which is transformed through H() = fq to an equation of the form

HH + aH2

+ f()HH + g()H2 + bH + c = 0, (1.26)

which they called the Euler-Painleve transcendent (EPT). Here f() and g() are

arbitrary functions and a,b,c are real constants. Equation (1.26) extends the class

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of nonlinear ordinary differential equations (ODEs) studied by Euler and Painleve

(Kamke (1943)) for which b = c = 0. Equation (1.26) with b = c = 0 is exactly

linearizable to

v + f v + (a + 1)gv = 0, (1.27)

through H = v1

a+1 . It has been established that for the Burgers equation and its

generalizations b = 0 and therefore (1.26) is, in general, not linearizable.

Recently Rao, Sachdev and Mythily Ramaswamy (2001, 2002, 2003) obtained the

self-similar reductions of the GBEs (1.22), (1.23) and investigated them in detail

for positive single-hump, monotonic (bounded or unbounded) solutions and also for

solutions with a finite zero by assuming certain asymptotic conditions at .

The following GBE

ut + unux +

j2t +

u +

+ x

un+1 = 2 uxx, (1.28)

has been studied for N-wave solutions by Sachdev, Joseph and Mayil Vaganan (1996)

and for similarity solutions by Mayil Vaganan and Asokan (2003).

We now list below the GBEs which are studied in this project:

ut + f(t)uux + l(t)u + g(t)uxx = 0, (1.29)

ut + unux =

(t)

2uxx, (1.30)

ut + unux(

j

2t+ )u + (+

xun+1) =

(t)

2uxt, (1.31)

where f(t), g(t) and l(t) are variable coefficients, n is a positive integer.

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The structure of the thesis is arranged as follows. In chapter 2, the Painleve

test is extended to equation (1.29) in order to obtain the constraints on the variable

coefficients for it to possess the Painleve property. In chapter 3, auto-Backlund trans-

formation is presented via the truncated Painleve expansion, also analytic solutions

are obtained. In chapter 4 the generalized burgers equation is linearized to the heat

equation under the generalized Cole-Hopf transformation.

References

1. Bianchi, L. Lezioni sulla teoria dei gruppi continui finiti di trasformazioni. Pisa:

Spoerri, 1918.

2. Bluman, G. W. and Cole, J. D. The general similarity solution of the heat

equation, J. Math. Mech. 18: 1025 - 1042 (1969).

3. Bluman, G. W. and Cole, J. D. Similarity methods for Differential Equations.

Applied Mathematical Sciences No. 13, Springer-Verlag, New York, 1974.

4. Bluman, G. W. and Kumei, S. Symmetries and Differential Equations. Springer-

Verlag, New York, 1989.

5. Bluman, G. W. and Anco, S. C. Symmetries and Integration methods for Dif-

ferential Equations, Applied Mathematical Sciences, No. 154, Spriner-Verlag,

New York, 2002.

6. Burgers, J. M. Mathematical examples illustrating relations occuring in the

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theory of turbulent for motion, Trans. Roy. Neth. Acad. Sci. 17: 1 - 53

(1939).

7. Burgers, J. M. Application of a model system to illustrate some points of the

statistical theory of free turbulence, Proc. Roy. Neth. Acad. Sci. 43: 2 - 12

(1940).

8. Burgers, J. M. The formation of vortex sheets in a simplified type of turbulent

motion, Proc. Roy. Neth. Acad. Sci. 53: 122 - 133 (1950).

9. Chong, T. H. and Sirovich, L. Nonlinear effects in steady supersonic dissipative

gasdynamics - II. Three dimensional axisymmetric flow, J. Fluid Mech. 58: 53

- 63 (1973).

10. Chowdhury, A. R. and Naskar, M. J. Phys. A: Math. Gen. 19: 1775 - 1781

(1986).

11. Cole, J. D. On a quasi-linear parabolic equation occurring in aerodynamics,

Quart. Appl. Math. 9: 225 - 236 (1951).

12. Doyle, J. and Englefield, J. Similarity solutions of a generalized Burgers equa-

tion, IMA J. Appl. Math. 44: 145 - 153 (1990).

13. Grundy, R. E., Sachdev, P. L. and Dawson, C. N. Large time solution of an

initial value problem for a generalised Burgers equation, Nonlinear Diffusion

Phenomenon.(Eds: P. L. Sachdev and R. E. Grundy) pp.68-83. Narosa Pub-

lishing House, New Delhi, 1994.

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14. Gungor, F. Symmetries and invariant solutions of the two-dimensional variable

coefficient Burgers equation, J. Phys. A: Math. Gen. 34: 4313 - 4321 (2001).

15. Hopf, E. The partial differential equation ut + uux = uxx, Commun. Pure Appl.

Math. 3: 201 - 230 (1950).

16. Ibragimov, N. H. CRC Hand Book of Analysis of Differential Equations, Exact

Solutions and Conservation Laws, CRC Press, New York, 1994.

17. Kalyani, R. Some solutions of Burgers equation, J. Maths. Phys. Sci. 5: 109 -

120 (1971).

18. Kamke, E. Differential gleichungen: Losungsmethoden and Losungen, Akademis-

che Verlagagesellschaft, Leipzig, 1943.

19. Kuznetsov, V. P. Soviet Phys. Acoust. 16: 467 - 470 (1971).

20. Lee-Bapty, I. P. and Crighton, D. G. Nonlinear wave motion governed by the

modified Burgers equation, Phil. Trans. R. Soc. Lond. A 323: 173 - 209

(1987).

21. Lighthill, M. J. Viscosity effects in sound waves of finite amplitude. Surveys

in Mechanics (Eds. G. K. Batchelor and R. M. Davies) Cambridge University

Press, pp. 250 - 351 (1956). .

22. Mayil Vaganan, B. Exact Analytic Solutions for some classes of partial differ-

ential equations, Ph. D. Thesis, Indian Institute of Science, Banglore, India,

1994.

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23. Mayil Vaganan, B. and Asokan, R. Direct Similarity Analysis of Generalized

Burgers Equations and Perturbation Solutions of Euler-Painleve Transcendents,

Stud. Appl. Math. 111(4): 435 - 451 (2003).

24. Olver, P. J. Symmetry groups and group invariant solutions of partial differential

equations, J. Diff. Geom. 14: 497 - 542 (1979).

25. Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate

Text in Mathematics No. 107, Springer-Verlag, New York, 1986.

26. Rudenko, O. V. and Soluyan, S. I. Theoretical foundations of nonlinear acoustics

(English translation by R. T. Beyer), New York, Consultants Bureau (Plenum),

1977.

27. Schwarz, F. J. Phys. A: Math. Gen. 20: 1613 - 1614 (1987).

28. Sinai, Y. L. Similarity solution of the axisymmetric Burgers equation, Phys.

Fluids, 19: 1059 - 1060 (1976).

29. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Analysis of self

similar solutions of a generalized Burgers equation with nonlinear damping,

Math. Problems Eng. 7: 253 - 282 (2001).

30. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Analysis of the self

similar solutions of the non-planar Burgers equation, Nonlinear Analysis 51:

1447 - 1472 (2002).

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31. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Self-similar solutions

of a generalized Burgers equation with nonlinear damping, Nonlinear Analysis:

Real World Applications 4: 723 - 741 (2003).

32. Steven Nerney, Edward J. Schmahl and Musielak, Z. E. Analytic solutions of

the vector Burgers equation, Quart. Appl. Math. 1: 63 - 71 (1996).

33. Tajiri, M., Kawamoto, S. and Thushima, K. Reduction of Burgerrs equation to

Riccati equation, Math. Japon. 28: 125 - 133 (1983).

34. Taylor, G. I. The conditions necessary for discontinuous motion in gases, Proc.

R. Soc. Lond. A 84: 371 - 377 (1910).

35. Webb, G. M. and Zank, G. P. Painleve analysis of the two dimensional Burgers

equation, J. Phys. A: Math. Gen. 23: 5465 - 5477 (1990).

36. Zabolotskaya, E. A. and Kholohlov, R. V. Sov. Phys. Acoust. 15: 35 - 40

(1969).

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Chapter II

Painleve Analysis of Generalized Burg-ers Equation

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2.1 Introduction

Nowadays, a wide class of nonlinear evolution equations (NLEEs) has been derived

to describe a variety of nonlinear wave phenomena in different physical contests,

including nonlinear optics, hydrodynamics, condensed matter, plasma physics and

quantum field theory. The study of integrable models continues to attract much

attention from many mathematicians and physicists.

The painleve property for ordinary differential equations is defined as follows.

The solutions of a sytem of ordinary differential equations are regarded as (analytic)

functions of a complex (time) variable. The movable singularities of the solution

(as a function of complex t) whose location depends on the initial conditions and

are, hence movable. Fixed singularities occur at points where the coefficients of the

equation are singular. The system is said to possess the Painleve property when all

the movable singularities are single valued.

Experience has shown that, when a system possess the Painleve property, the

system will be Integrable. Albowitz have proven that when a partial differential

equation is solvable by the inverse scattering transform and a system of ordinary

differential equation is obtained from this pde by an exact similarity reduction then

the solution associated with the Gelfand-Levitan-Marchenko equation will possess

the Painleve property. Furthermore, they conjecture that, when all the odes obtained

by exact similarity transforms from a given pde have the Painleve property, perhaps

after a change of variables,then the pde will be integrable. In another context

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Chudnovsky has observed that a certain pde which arises a compatabality condition

in a generalization of the theory of Isomonodromy. Deformation is meromorphic [in

(x, t)].It is somewhat remarkable that a connection between integrability and the

Painleve property has been noted since the work of Kowalevskaya, and, as of yet, the

precise equivalence of these concepts remains to be determined.

In this chapter we define a Painleve property for partial differential equations

that does not refer to that for ordinary differential equations.Indeed, we believe that

by extending the definition of the Painleve property it is possible to treat the phe-

nomenon of integrable behaviour in a unified manner. For the past few years, several

inhomogeneous NPDEs (INPDEs) have been studied from the soliton point of view.

This chapter is devoted to study the generalized Burgers equation

ut + f(t)uux + g(t)uxxx + l(t)u = 0, (2.1)

where f(t), g(t) and l(t) are time dependent coefficients.

Equation (2.1) arises in various areas of Mathematical Physics and Nonlinear

Dynamics.

2.2 Painleve Property

One major difference between analytic functions of one complex variable and analytic

functions of several complex variables cannot be isolated. If f = f(z1,...,zn) is a

meromorphic function of N complex variable (2N real variables), the singularities of

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f occur along analytic manifolds of dimension 2N-2. These manifolds are determined

by conditions of the form

(z1,...,zn) = 0 (2.2)

where is an analytic function of in a neighbourhood of the manifold. Therefore, we

say that a pde has the Painleve property when the solutions of the pde are single-

valued about the movable, singularity manifold. To be precise, if the singulariy

manifold is determined by (2.1) and u = u(z1,...,zn) is a solution of the pde, then we

assume that

u = u(z1,...,zn) =

j=0

ujj (2.3)

where

= (z1,...,zn)

and

uj = uj(z1,...,zn)

are analytic functions of (z1,...,zn) in a neighbourhood of the manifold (2.1) , and

is an integer.Substituting (2.2) into the pde determines the possible values of and

defines the recursion relations for uj , j = 0, 1, 2,.... The process is analogous to that

for ordinary differential equations.

2.3 Painleve Analysis for (2.1)

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According to the method of Kurskal (1989), the solutions for (2.1) can be expanded

interms of the Laurent series as follows.

u(x, t) = (x, t)j=0

j(x, t)uj , (2.4)

where (x, t) = x + (t), uj(x, t) = uj(t) are analytic functions in a neighbourhood

of (x, t) = 0, the non characteristic movable singularity manifold defined by u0 = 0

and is a positive integer.

Through the leading order analysis, we found that = 1 and

u0 = 2g(t)

f(t)x. (2.5)

The recursion relations are found to be

uj2,t + (j 2)uj1t + fj

m=0

ujm [um1,x + (m 1)xum] + g(t) [uj2,xx

+2(j 2)uj1,xx + (j 2)xxuj1 + (j 1)(j 2)uj2

x

+ l(t)uj2 = 0(2.6)

Collecting terms involving uj, it is found that

2x(j 2)(j + 1)uj = F(uj1,...,u0, t, x, xx,...)forj = 0, 1, 2,... (2.7)

We note that the recursion relations (2.7) are not defined when j = 1, 2. These

values of j are called the resonances of the recursion relation and correspond to

points where arbitrary functions of (x, t) are introduced into the expansion. j = 1

corresponds to the arbitrary singularity manifold ( = 0).On the other hand the

resonance at j=2 introduces an arbitrary function u2 and a compatability condition

on the functions (, u0, u1) that requires the right hand side of (2.7) vanish identically.

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For Burgers equation,we find from (2.6)

j = 0 u0 = 2g(t)

f(t)

x (2.8)

j = 1 u1 = 1f x

(t + gxx) (2.9)

j = 2 u0,t + f(t)(uu0)x + g(t)u0,xx + l(t)u0 = 0 (2.10)

Taking account of (2.8), (2.9) and (2.10)(which represents a compatibility condition

since it involves already determined quantities) reduces to

gft + l

g

f = 0 (2.11)

from which it follows that equation (2.1) possesses the Painleve property if and only

if

g(t) = c1f(t)e

l(t)dt. (2.12)

2.4 Conclusions

The variable-coefficient nonlinear evolution equations, although their coefficient func-

tions often make the studies hard, are of current interests since they are able to

describe the real situations in many fields of physical and engineering sciences. In

this chapter we have considered a generalized Burgers equation containing arbitrary

functions f(t), g(t) and l(t). The Painleve analysis leads to the explicit constraint

on the variable coefficient on the variable coefficients for such a equation to pass the

Painleve test.

We have applied directly the PDE Painleve test to equation (2.1) and provided

constrain (2.12) which is the necessary and sufficient condition for equation (2.1)

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to have the Painleve property. Thus, equation (2.1) is predicted to be completely

integrable if and only if the variable coefficients of equation (2.1) satisfy constraint

(2.12).

References

1. J. Weiss, M. Tabor, G. Carnevale, The Painleve property for partial differential

equations J. Math. Phys. 24 (1983) 522-526.

2. J. Weiss, The Painleve property for partial differential equations, J. Math. Phys.

24 (1983) 1405-1413.

3. P. A. Clarkson, Painleve analysis and the complete integrability of a generalized

variable-coefficient Kadomtsev-Petviashvili equation, IMA J. Appl. Math. 44

(1990) 27-53

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Chapter III

Auto-Backlund transformation andExact Solutions of generalized Burg-ers equation

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3.1 Introduction

It has been shown that for the generalized Burgers equation of the form

ut + uux + g(t)uxx. (3.1)

which is a well established nonlinear equation for the study of acoustic and shock

waves. Backlund transformations exist only if g(t) is constant when (3.1) reduces to

the ordinary Burgers equation. The equation has not been solved for non-constant

a(t). However, Doyle and Englefield(1990) have found that similarity solutions of

(3.1) for specific g(t), (i.e). et or e1/t for which is infinitesimal invariant transfoma-

tions exist. Later, Kington and Sophocleous(1991) have shown that in addition to the

finite point transformation, a reciprocal point transformation as well as transforma-

tions relating equations with different functions g(t) exist. In this work, we further

generalize the equation (3.1)is of the form

ut + f(t)uux + g(t)uxx + l(t)u = 0. (3.2)

The generalized Burgers equation with the nonlinear f(t), damping term l(t) and

dispersion term a(t) can model propagation of a long shock-wave in a two layer

shallow liquid. Malomed and Shira(1991) have qualitatively demonstrated that for

the special case of the GBE with g(t) = -1 a shock wave solution reverses its velocity

and disintegrate after the passage of the critical point where f(t) changes its sign.

However, any analytic solitary wave solutions for the GBE have not been found. In

the following, we make use of both the truncated Painleve expansion and symbolic

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computation method to obtain an auto-Backlund transformation and certain soliton-

typed explicit solutions with some constraints between f(t) and g(t).

A non-linear pde is said to possess the Painleve property when the solutions of

the NPDE are single valued about the movable singularity manifold which is non-

characteristic. To be more precise, if the singularity manifold is determined by

(z1, z2,...,zn) = 0. (3.3)

and u = u(z1, z2,...,zn) is a solution of the NPD, then it is required that

u = j=0

ujj (3.4)

where u0 = 0, = (z1, z2,...,zn), uj = uj(z1, z2,...,zn) are analytic function of (zj)

in a neighbourhood of the manifold and is a negative rational number. Substitution

of equation(3.4) into the NPDE determines the allowed values of , and defines the

recursion relation for uj, j=0,1,2,...When the equation (3.4) is correct, the NPDE is

said to possess the Painleve property and is conjectured to be integrable.

3.2 Auto-Backlund Transformation

In order to find the solitonic solutions for equation (3.2), we truncate the Painleve

expansion (3.4) at the constant-level term in the senses of Tian and Gao(1995) and

Hong and Jung(1999)

u(x, t) = j(x, t)jl=0

ut(x, t)t(x, t). (3.5)

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On balancing the highest-order contributions from the linear term (uxx) with the

highest-order contributions from the nonlinear terms (uux), we get J = 1, so that

u(x, t) =u0(x, t) + u1(x, t)(x, t)

(x, t)(3.6)

We will stay with the general assumption then x = 0 but will not initially impose

any constraint on the model parameter g(t) and f(t). When substituting the above

expression into (3.2), we let the coefficients of like powers of to vanish so as to get

the set of Painleve- Backlund equations,

3 : f(t)u20x + 2g(t)u02x = 0 (3.7)

2 : f(t)u0u0,x f(t)u1u0x 2g(t)xu0, x g(t)u0xx u0t = 0. (3.8)

1 : u0,t + b(t)u0u1,x + f(t)u1u0,x + g(t)u0,xx + lu0 = 0. (3.9)

0 : u1,t + f(t)u1u1,x + g(t)u1,xx + iu1 = 0 (3.10)

The set of equations (3.6)-(3.10)contitutes an auto-backlund transformation, if

the set is solvable with respect to (x, t), u0(x, t) and u1(x, t). Equation (3.7) brings

out two solutions:

u0(x, t) =

2g(t)(x)

f(t) or u0(x, t) = 0 (3.11)

After substituting the non trivial solution into (3.8), we obtain

u1(x, t) = g(t)xx + txf(t)

. (3.12)

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Subsequently, we find the constraint equation for variable coefficients g(t) and f(t)

from equation (3.9)

g(t) = c1f(t)e

l(t)dt (3.13)

Where c1 is an arbitrary constant.Thus, we are able to find a family of exact analytic

solutions to Eq.(3.2) as follows.

u(x, t) = g(t)xx + txf(t)

+

2g(t)x

f(t)

(x, t)1 (3.14)

with the constraint equation for (x, t) and g(t)

22xg(t)txx + 2xg(t)tt 4txxg(t)2xx 2txxg(t)t

ddt

g(t)2xt 4g(t)3xxxxxx + 3g(t)33xx

+4g(t)22xxt 2txg(t)2xxx

+g(t)xx2

t + g(t)3

2

xxxx + g(t)l2

xt = 0 (3.15)

we note that once a Backlund transformation discovered, and a set of seed solutions

is given, one will be able to find an infinite number of solutions by the repeated

applications of the transformation, (i.e.) to generate a heirarchy of solutions with

increasing complexity. In the rest, we will find a family of exact analytic solutions to

(3.2).

Sample solution:

(x, t) = 1 + e[A(t)x+B(t)] (3.16)

Is subtituted into the constraint Eq.(3.15). Equating to zero the coefficients of

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like powers of x yield

x1 :ddt

A(t) ddt

a(t)

a(t) d2

dt2A(t) + 2

( ddt

A(t))2

A(t) lA(t) = 0 (3.17)

x0 :ddt

B(t) ddt

a(t)

a(t) d

2

dt2B(t) + 2

ddt

B(t) ddt

A(t)

A(t) lB (t) = 0 (3.18)

By setting A(t) = e1t a a trial function to Eq.(3.17),we find an ordinary differ-

ential equation for g(t) as

d

dta(t)1 + g(t)

21 = 0

g(t) = c2e

1tel(t)dt (3.19)Finally, B(t) is obtained from Eq.(3.18) as

B(t) = c4 +

c5e1te

l(t)dtdt (3.20)

where ci are all non-zero arbitrary constants. Combining all terms, we find a family

of the analytical solutions of Eq.(3.2) as

u(x, t) = e1t

c2c1

c2e

l(t)dt + 1xe

2l(t)dt + c5e

l(t)tl(t)dt

+2

c1e

l(t)dt e[1t+e

1tx+c5e1te

l(t)dt

dt+c4] 1(x, t) (3.21)

References

1. J. Doyle, M.J. Englefield, IMAJ Appl. Math. 44 (1990). 145.

2. J.G. Kingston, C. Sophocleous, Phys. Lett. A 155 (1991). 15.

3. N. Joshi, Phys. Lett. A 125 (1987). 456.

4. W.P. Hong, Y.D. Jung, Phys. Lett. A 257 (1999). 149.

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5. B.A. Malomed, V.I. Shrira, Physica D 53 (1991). 1.

6. B. Tian, Y.T. Gao, Phys. Lett. A 209 (1995). 297.

7. W.P. Hong, Nuovo Cimeto B 114 (1999). 845.

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Chapter IV

Exact Linearization of GeneralizedBurgers Equation via the General-ized Cole-Hopf Transformations

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4.1 Introduction

The search of exact solutions of nonlinear partial differential equations is of great

importance, because these equations appear in complex physics phenomena, mechan-

ics, chemistry, biology and engineering.

In this chapter we obtain new solutions for the generalized Burgers equation

ut + f(t)uux + g(t)uxx + l(t)u = 0, (3.22)

Equation (3.22) is a generalization of the well-known Burgers equation(Burgers (1974))

ut + uux = uxx. (3.23)

Owing to the assumptions of the constant coefficients and unforced turbulence, the

physical situations in which the classical Burgers equation arises tend to be highly

idealized. In practice, the generalized Burgers equation (3.22) may provide us with

more realistic models in many different physical contextslike the long-wave propaga-

tion in an inhomogeneous two-layer shallow liquid, directed polymers in a random

medium, pinning of vortex lines in superconductors etc.

Harry Bateman(1915) considered a nonlinear equation whose steady solutions

were thought to describe certain viscous flows. This equation, modeling a diffusive

nonlinear wave, is now widely known as the Burgers equation, and is given by

ut + uux = uxx. (3.24)

where is a constant measuring the viscosity of the fluid. It is nonlinear parabolic

equation, simply describing a temporal evolution where nonlinear convection and

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linear diffusions are combined, and it can be derived as a weakly nonlinear approxi-

mation to the equations of gas dynamics. Although nonlinear, Eq.(3) is very simple,

and interest in it was revived in the 1940s, when Dutch physicist Jan Burgers pro-

posed it to describe a mathematical model of turbulence in gas(Burgers(1940)). As

a model for gas dynamics, it was then studied extensively by (Burgers(1948)), Eber-

hard Hopf(1950), Julian Cole(1951), and others, in particular; after the discovery of a

coodinate transformation that maps it to the heat equation. While as a model for gas

turbulence the equation was soon rivaled by more complicated models, the linearizing

transformation just mentioned added importance to the equation as a mathematical

model, which has since been extensively studied. The limit 0 is an hyperbolic

equation, called the inviscid Burgers equation.

ut + uux = 0. (3.25)

This limiting equation is important because it provides a simple example of a con-

servation law, capturing the crucial phenomenon of shock formation. Indeed, it was

oriinally introduced as a model to describe the formation of shock waves in gas dy-

namics. A first-order partial differential equation for u(x, t) is called a conservation

law if it can be written in the form ut+(f(u))x = 0. For equation (3.24), f(u) = u2/2.

Such conservation laws may exhibit the formation of shocks, which are discontinuities

appearing in the solution after a finite time and then propagating in a regular manner.

When this phenomenon arises, an initially smooth wave becomes steeper and steeper

as time progresses, until it forms a jumb discontinuity-the shock.

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Nowadays, the Burgers equation is used as a simplified model of a kind of hydro-

dynamic turbulence(Case & Chiu(1969)), called Burgers turbulence. Burers himself

wrote a treatise on the equation now known by his name (Burgers (1974)), where

several variants are proposed to describe this particular kind of turbulence. Equa-

tion (3.23) was originally derived to describe the propagation of nonlinear waves in

dissipative media, where (> 0) is the kinematic viscosity, and u(x, t) represents the

fluid velocity field. It plays an active role in explaining two fundamental effects char-

acteristic of any turbulence: the nonlinear redistribution of energy over the spectrum

and the action of viscosity in small scales. Over the decades, the Burgers equation

has been widely used to model a large calss of physical systems in which the non-

linearity is fairly weak(quadratic) and the dispersion is negligible compared to the

linear damping. Hopf (1950) and Cole(1951) independently discovered a transforma-

tion that reduces the Burgers equation (3.23) to a linear diffusion equation. First, we

write (3.23) in a form similar to a conservation law

ut +

x

1

2u2 ux

= 0. (3.26)

This may be regarded as the compatibility condition for a function ti exist, such

that

u = x, (3.27)

ux 12

u2 = t. (3.28)

We substitue the value of u from (3.27) in (3.28) to obtain

xx 12

x2 = t. (3.29)

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Next, we introduce = 2log so that

u = x = 2x

. (3.30)

This is called the Cole-Hopf transformation which, by differentiating, gives

xx = 2

x

2 2

xxandt = 2t

. (3.31)

Consequently, (3.29) reduces to the linear heat equation

t = xx. (3.32)

The relation between the Burgers and the heat equation was already mentioned in

an earlier book(Forsyth(1906), but the former had not been recognized as physically

releant; hence, the importance of this connection was seemingly not noticed at the

time. Using the transformaion of Equation (3.23), known as the Cole-Hopf trans-

formation, it is easy to solve the initial value problem for this equation. Recently,

a generalization of the Cole-Hopf transformation has been successfully used to lin-

earize the boundary value problem for the Burgers equation posed on the semiline

x > 0Calogero(1989).

Many solutions of equation (3.32) are well-known in the literature. For a more

complete exposition, we exhibit some of them in next sections.

To solve equation (3.23), we simply substitute the given solution for in (3.30).

Motivated by this idea, we intend to extend the Cole-Hopf transformaion to lin-

earize equation (3.22). This is our aim in the Second section.

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This chapter is organized as follows: In section 4.2 we successfully apply the

generalized Cole-Hopf transformation to generalized Burgers equation and we reduce

the problem of solving this equation to solve the linear heat equation. Section 4.3

is dedicated to show some exact solutions to linear heat equation by transformation

groups. At the end, we give some conclusions.

4.2 Exact Linearization of Generalized Burgers Equation

In this section we construct the generalized Cole-Hopf transformation from equa-

tion (3.23) to the standard heat equation. In order to look for soltions to generalized

Burgers equation (3.22), we apply the generalized Cole-Hopf transformation given by

u(x, t) = A(x, t)G(z, ) + B(x, t), z = (t)x + (t), = (t), (3.33)

where A(x, t) = 0, B(x, t), (t), (t) and (t) are some functions to be determinedlater. We use the above ansatz into (1) to obtain

G[AT] + GGz[f A

2] + Gzz [gA2] + G[At + f BAx + f ABx + lA]

+Gz[A(x + ) + fBA + 2gAx] + [Bt + f BBx + gAxx + gBxx + lB] = 0, (3.34)

Now we assume the following conditions

f A2

A= 1 (3.35)

gA2

A= 1 (3.36)

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At + f BAx + f ABx + lA = 0 (3.37)

A(x + ) + fBA + 2gAx = 0, (3.38)

Bt + f BBx + gAxx + gBxx + lB = 0, (3.39)

f AAx = 0 (3.40)

In view of (14)-(18), equation (13) reduces to the Burgers equation

G + GGz + Gzz = 0 (3.41)

Application of x to (17)gives

Bx =f

(3.42)

Equations (18) and (19) lead to

(

f) + f(

f)2 l

f = 0 (3.43)

(

f) + f(

(f)2) l

f = 0 (3.44)

from which we find that

(t) =c1

c0 +

f eldtdt

(3.45)

(t) =

c2f 2e

ldtdt (3.46)

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Inserting (3.46) in equations (3.33) and (3.38) we have

z = (c1

c0 +

f eldtdt

)x + c2c2

1 f e

ldtdt

[c0 +

f eldtdt]2

(3.47)

B =e

ldt[x c2c1]

c0 +

f eldtdt

(3.48)

Using (3.42) in (3.37) and solving for A(t), we get

A(t) = c3e

l(t)dt (3.49)

Now (3.35) gives = A, and its solution is (after inserting for A, )

= c3c21

e

ldtdt

[c0

f eldtdt]2

+ c4 (3.50)

Substituting all the known terms into (3.33) we find that

u = c3c1

c0 + f eldtdte

l(t)dtG(, z) +

eldt[x c2c1]

c0 + f eldtdt (3.51)

If we replace G in (3.51) by

G =z(, z)

(, z)(3.52)

We find that (3.41) and (3.51) become

= zz (3.53)

which is the linear heat equation and

u =e

ldt

[c0 +

f eldtdt]

[c1c3z(, z)

(, z)+ (x c2c1)] (3.54)

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4.3 Exact Solutions to linear heat equation

Suppose that = (x, t) is a solution to equation

t = xx (3.55)

Then = (x,t) is a solution to heat equation

t = xx (3.56)

Conversely, if = (x, t) is a solution to (3.56), then = (x, t/) is a solution to

(3.55).Thus, equation (3.55) and (3.56) are the same. We shall call Equation (3.55)

the normalized heat equation. We already obtained some solutions to equation (3.56)

in a form of travelling wave.In thi section we give solutions to this equation by uing

Lie group theory[12].

Definition : Let be a system of differential equations. A symmetry group of

the system is a local group of transformations G acting on an open subset M of

the space of independent and dependent variables for the system with the property

that whenever u = f(x) is a solution of, and whenever g.f is defined for g G,

then u = g.f(x) is also a solution of the system. (By Solution we mean any smooth

solution u = f(x) defined on any subdomain X).

Following are Symmetry groups of the heat equation (1):

G1 (x + ,t,u),

G2 (x, t + , u),

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G3 (x,t, exp(e)u),

G4 (exp()x, exp(2)t, u),

G5 (x + 2t), t , u exp(x 2t),

G6

x

1 4t ,t

1 4t , u

1 4t exp x2

1 4t

,

G (x,t,u + (x, t))

This means that if = (x, t)is a solution of the heat equation (1), so are the functions

(1) = (x , t),

(2) = (x, t ),

(3) = exp()(x, t),

(4) = (exp()x, exp(2)t) ,

(5) = exp(x + 2t)(x 2t,t),

(6) =1

1 + 4texp[

x21 + 4t

](x

1 + 4t,

t

1 + 4t),

(7) = (x, t) + x(x, t)

Where is any real number and (x, t)is any other solution to the heat equation. The

symmetry groups G3 and G thus reflect the linearity of the heat equation;we can add

solutions and multiply them by constants. The groups G1

and G2

demonstrate the

time- and space-in variance of the equation, reflecting the fact that the heat equation

has constant coefficients.The well-known scaling symmetry turns up in G4, while G5

represents a kind of Galilean boost to a moving coordinate frame. The last group G6

is a genuinely local group of transformations. Its appearance is far from obvious from

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basic physical principles, but it has the following nice consequence If we let = c be

just a constant solution, then we immediately conclude that the function

=c

1 + 4texp(

x21 + 4t

), (3.57)

is also a solution. In particular, if we set c =

/ we obtain the fundamental solution

to the heat equation at the point (x0, y0) = (0, (1/(4)))To obtain the fundamental

solution

=14t

exp(x2

4t), (3.58)

We need to translate this solution in t using the group G2 (with replaced by (1/4))

It can be shown that the most general solution obtainable from a given solution

= (x, t) by group transformations is of the form

(x, t) =1

1 + 46texp[3 5x + 6x

2 5t1 + 46t

]

(exp(4)(x 25t)

1 + 46 1, exp(24)t

1 + 46t 2) + (x, t) (3.59)

Where 1,...,6are real constants and = (x, t) an arbitrary olution to the heat

equation.

Previous considerations allow to obtain particular solutions to the heat equations

t = xx.Some of them are

w(t) = k = constant, (3.60)

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w(x) = A(x) + B, (3.61)

w(x, t) = A(x2 + 2t) + B, (3.62)

w(x, t) = A(x3 + 6tx) + B, (3.63)

w(x, t) = A(x4 + 12tx2 + 122t2) + B, (3.64)

w(x, t) = x2n +n

k=1

(2n)(2n 1)...(2n 2k + 1)(t)kx2n2kk!

(3.65)

w(x, t) = x2n+1 +n

k=1

(2n + 1)(2n)...(2n 2k + 2)(t)kx2n2k+1k!

(3.66)

w(x, t) = Aexp(2

t x) + B, (3.67)w(x, t) = A

1t

exp( x2

4t) + B, (3.68)

w(x, t) = A exp(2tcos(x + B), (3.69)

w(x, t) = A exp(2tcos(x + B) + C, (3.70)

w(X, t) = A exp(x)cos(x 22t + B) + K, (3.71)

w(x, t) = A erf(x

2t ) + B, (3.72)

Where A,B,C and are arbitrary constants, n is a positive integer,

erf(z) 2

z0

exp(2)d, (3.73)

is the error function ( probability integral).

These solutions are useful in solving generalized Burgers equation (3.22).

4.4 Conclusions

We successfully applied the generalized Cole-Hopf transformation to generalized Burg-

ers equation. As a particular case, we obtained some solutions to linear heat equation.

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We think that the application of the generalized Cole-Hopf transformation is a useful

tool in the search of solutions to other generalized burgers equation.

References

1. M. Scott, Encyclopedia of Nonlinear Science, Taylor and Francis, 2005.

2. H. Bateman, Some recent research on the motion of fluids, Monthly Weather

Review 43 (1915) 163-170.

3. J. Burgers, Application of a model system to illustrate some points of the sta-

tistical theory of free turbulence, Proceedings of the Nederlandse Akademie van

Wetenschappen 43 (1940) 2-12.

4. J. Burgers, A mathematical model illustrating the theory of turbulence, Ad-

vances in Applied Mechanics 1 (1948) 171-199.

5. E. Hopf, The partial differential equation ut uux luxx, Communications in

Pure and Applied Mathematics 3 (1950) 201-230.

6. J. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quar-

terly Journal of Applied Mathematics 9 (1951) 225-236.

7. K.M. Case, S.C. Chiu, Burgers turbulence models, Physics of Fluids 12 (1969)

1799-1808.

8. J. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Sta-

tistical Problems, Reidel, Dordrecht and Boston, 1974.

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9. A.R. Forsyth, Theory of Differential Equations, Cambridge University Press,

Cambridge, 1906.

10. F. Calogero, S. De Lillo, The Burgers equation on the semiline, Inverse Problems

5 (1989) L37.

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Chapter V

SIMILARITY SOLUTIONS OF THE GENERALIZED

BURGERS EQUATION ut + unux +

+ j

2t

u +

+

x

un+1 =

(t)2 uxx

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5.1 Introduction

The generalized Burgers equation

ut + unux +

+

j

2t

u +

+

x

un+1 =

(t)

2uxx, (5.1)

where , , are non negative constants, n is a positive integer, j = 0, 1, 2 and

(t) is the variable viscosity, has been studied recently by several authors. When

n = 1, = = = j = 0 and (t) = , a constant, (5.1) becomes the well-known

Burgers equation

ut + uux =

2uxx. (5.2)

The Hopf-Cole transformation (Hopf (1950), Cole (1951)) changes (5.2) to the linear

heat equation. When n = 1, = = = 0 and (t) = , a constant, (5.1) becomes

the nonplanar Burgers equation

ut + uux +j

2tu =

2uxx. (5.3)

When n = 2, = = = j = 0 and (t) = , a constant, (5.1) becomes the

modified Burgers equation

ut + u2ux =

2uxx. (5.4)

When n = 1, = = j = 0 and (t) = , a constant, (5.1) becomes the Burgers

equation with linear damping term:

ut + uux + u =

2uxx. (5.5)

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In fact equation (5.1), with (t) = , a constant, includes as special cases the equa-

tions

Ur + 12a20

U U +jU

2r=

2a30

U, (5.6)

vr + 12c20

vv +v

r=

b

2c200v, (5.7)

ut + uux +ju

2t=

2uxx and ut + uux + u =

2uxx, (5.8)

ut + uux + u =

2uxx, (5.9)

vx +

jv

2x vv = v, (5.10)ut + uux + H(x,t,u,ux) = 0, (5.11)

ut + uux +

j

2tu =

2uxx, (5.12)

studied respectively by Scott (1981), Enflo (1985), Sachdev, Enflo, Srinivasa Rao,

Mayil Vaganan and Poonam Goyal (2003), Lardner and Arya (1980), Crighton and

Scott [4], Nimmo and Crighton (1979), and Sachdev and Nair (1987). The Burgers

equation with linear damping (5.9) is recently investigated by Mayil Vaganan and

Senthil Kumaran (2004) for its symmetries by the Lies classical method (Lie (1967),

Bluman and Kumei (1989)).

When (t) = , a constant, (5.1) becomes

ut + unux + ( +

j

2t)u + (+

x)un+1 =

2uxx. (5.13)

Sachdev, Joseph and Mayil Vaganan (1996) studied (5.13) for exact N-wave solutions.

Mayil Vaganan and Asokan (2003) obtained similarity solutions of (5.13) by the direct

method of Clarkson and Kruskal (1989).

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But the viscosity is actually a function of t (Lighthill (1956)). In this case the

Burgers equation takes the form

ut + uux =(t)

2uxx. (5.14)

Doyle and Englefield (1990) determined similarity reductions of the generalized Burg-

ers equation (GBE) (5.14) using the method for defining an optimal system of group-

invariant solutions (Olver (1995)). Mayil Vaganan and Senthil Kumaran (2003) ob-

tained invariant solutions of (5.14) by the direct method. Scott (1981) obtained the

long time asymptotics of solutions of (5.14) when (t) = t, > 0. Parker (1980)

also derived large-time asymptotics of the generalized Burgers equation

V

t+ (t)V (t)V V

x= (t)

2V

x2, (5.15)

subject to the initial condition

V(x, 0) = r(x) = A1 sin(x/l), 0 < x < l. (5.16)

Later Sachdev, Nair and Tikekar (1988) introduced the Euler-Painleve transcendents

in the form

HH + aH2

+ A(z)HH + B(z)H2 + bH + c = 0, (5.17)

as the self-similar reductions of the GBE

ut + uux =

2g(t)uxx, (5.18)

when g(t) is given by either (1 + t)n or emt.

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Mayil Vaganan and Senthil Kumaran (2004, 2005, 2005) in a series of papers

obtained similarity solutions in terms of exponential, error and Kummer functions of

the GBEs

ut + unux =

2uxx, (5.19)

ut + unux + u =

2uxx, (5.20)

ut + uux + u =

2uxx. (5.21)

This chapter gives detailed account of group-invariant solutions of the generalized

Burgers equation (5.1), recover previously obtained solutions and determine new so-

lutions and this is carried out in section 2. The results and discussions of the present

chapter are contained in section 3.

2. Similarity solutions of (5.1)

If (5.1) is invariant under a one parameter group of infinitesimal transformations

x = x + X(x,t,u) + (2),

t = t + T(x,t,u) + (2),

u = u + U(x,t,u) +

(2), (5.22)

then

u

jT

2t2 T

j

2t+

(Uu 2Xx)

j

2t+

+ut

T

Tt +

2Txx + 2Xx

+ unux

T

+ Xx

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un+1T

+

x

(Uu 2Xx)

+

x

X

x2

+un1ux [nU] + U j2t

+ + Ut

2Uxx

+un

U(n + 1)

+

x

+ Ux

+ ux

Xt

2(2Uxu Xxx)

+uxut

2Txu + 2Xu

+ unut[Tx] + unu2x[2Xu] + u2x

2(Uuu 2Xxu)

+u3x

2Xuu

+ u2xut

2Tuuuux

3Xu

j

2t+

+ un+1ux

3Xu

+

x

+uxt[Tx] + uut

Tu

j

2t+

+ un+1ut

Tu

+

x

+ uxuxt[Tu] = 0.(5.23)

We consider two cases: 1) The case Xu = Tu = Uu = 0 and 2) The general case.

1: The case Xu = Tu = Uu = 0.

Here (5.23) simplifies to

u T j

2t2

T

j

2t

+ (Uu

2Xx)

j

2t

+ +ut

T

Tt +

2Txx + 2Xx

+ unux

T

+ Xx

+un+1T

+

x

(Uu 2Xx)

+

x

X

x2

+nun1uxU +

U

j

2t+

+ Ut

2Uxx

+un

U(n + 1)

+

x

+ Ux

+ ux

Xt

2(2Uxu Xxx)

unutTx + uxtTx = 0. (5.24)

Equating the coefficients of uxt, un1ux to zero in (5.24), we get Tx = U = 0. Setting

the coefficients of u, ut, unux, u

n+1, ux in (5.24) equal to zero, we have

jT2t2

+

j

2t+

2Xx T

= 0, (5.25)

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T

Tt + 2Xx = 0, (5.26)

T

+ Xx = 0, (5.27)2Xx T

+

x

X

x2= 0, (5.28)

Xt + 2

Xxx = 0. (5.29)

We obtain from (5.26) and (5.27) that

Tt = Xx. (5.30)

Using (5.27) and (5.30) in equation (5.25) we get

jTt jt

T = 0. (5.31)

On inserting (5.27), equation (5.28) becomes

+

x

Xx

x2X = 0. (5.32)

Now we consider three cases: In the foregoing analysis c0, r1, x0, t0, f0, c1, r2, c2, A0, B0, r3, r0,

C0, D0, r4, r5, r6, r7, r8 are arbitrary constants and the relation between f and H is

f = H1/n.

Case 1.1: = = 0

Equation (5.1) with = = 0 is

ut + unux +

j

2tu +

xun+1 =

(t)

2uxx. (5.33)

In this case the solutions of (5.30)- (5.32) are

T = c0t, X = c0x. (5.34)

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The characteristic equations are

dx

c0x

=dt

c0t

=du

0

. (5.35)

Solving (5.35) we obtain

z =x

tand u = f(z). (5.36)

Substituting (5.34) in (5.27) and solving for , we get

(t) = r1t. (5.37)

Using (5.36) and (5.37) in (5.33), we thus arrive at

f +2

r1zf 2

r1fnf j

r1f 2

r1zfn+1 = 0. (5.38)

Through the transformation f = H1/n equation (5.38) is changed to

HH

n + 1

n

H

2+

2

r1zH H 2

r1H +

jn

r1H2 +

2n

r1zH = 0. (5.39)

We call (5.39) the generalized Euler-Painleve transcendent because it contains a term

2nr1z

H which is not present in the general form of the Euler-Painleve transcendent

(5.17).

Case 1.2 = = j = 0

For this case equation (5.1) is

ut + unux + u

n+1 =(t)

2uxx. (5.40)

The infinitesimals are

X = x0, T = t0 and U = 0. (5.41)

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Substituting (5.41) in (5.27) and solving for , we get

= , a constant. (5.42)

Thus the similarity form of equation (5.40) is

u = f(z), z = x x0t0

t. (5.43)

Inserting (5.43) in (5.40) and using (5.42) we find that f(z) satisfies the equation

f 2x0t0

f 2

fnf 2

fn+1 = 0. (5.44)

A solution of equation (5.44) is

f = f0e2x0z/t0, (5.45)

provided = 2x0t0

.

Under the transform f = H1/n equation (5.44) changes again to a generalized

Euler-Painleve transcendent(GEPT)

HH

n + 1

n

H

2 2x0t0

HH 2

H +2n

H = 0. (5.46)

It follows from f = H1/n and the solution (5.45) of (5.44) that the corresponding

solution of the GEPT (5.46) is

H =1

fn0e2nx0z/t0. (5.47)

The reduction of (5.33) with (t) given by (5.37) to the GEPT (5.46) facilitates the

determination of an exact closed form solution in terms of an exponential function.

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The extra term 2n

H in (5.46) (recall the EPT (5.17)) together with 2

H helps

to find an exact solution. As for (5.44) the terms f,2x0t0

f and 2

fnf,2

fn+1

separately lead to this exponential solution.

Case 1.3 = = = 0

In this case equation (5.1) takes the form

ut + unux +j

2tu =(t)

2 uxx. (5.48)


T = c0t, X = c0x + c1, U = 0. (5.49)

Inserting (5.49) in (5.27) and solving for , we get

(t) = r2t. (5.50)

The similarity transformation of (5.48) is

u = f(z), z =c0x + c1

t. (5.51)

Putting (5.50) and (5.51) in (5.48), the latter reduces to

f +2

r2c20zf 2

r2c0fnf j

r2c20f = 0. (5.52)

Using the transformation f = H1/n, equation (5.52) becomes an EPT

HH

n + 1

n

H

2+

2

r2c20zH H 2

r2c0H +

jn

r2c20H2 = 0. (5.53)

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2. The general case

Now setting the coefficients of uxuxt, uxt in (5.23) equal to zero, we have

Tu = Tx = 0. (5.54)

Equating the coefficients of u2x, uxut in (5.23) to zero and using (5.54), we obtain

Xu = Uuu = 0. (5.55)

Equation (5.55) lead to

U = A(x, t)u + B(x, t). (5.56)

Again equating the coefficients ofux, ut and the remaining terms in (5.23) to zero, we

have

unT

+ Xx

+ nun1UXt

2(2Uxu Xxx) = 0,(5.57)

T

Tt + 2Xx = 0,(5.58)u

jT

2t2 T

j

2t+

(Uu 2Xx)

j

2t+

+ U

j

2t+

+Ut 2

Uxx + un+1

T

+

x

(Uu 2Xx)

+

x

X

x2

+un

U(n + 1)

+

x

+ Ux

= 0.(5.59)

Using (5.56) and (5.58) in (5.57) and setting the coefficients of un, un1 and u0 to

zero, we get

Tt Xx + nA = 0, (5.60)

B = 0, (5.61)

Xt Ax + 2

Xxx = 0. (5.62)

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Substituting (5.56) and (5.58) in (5.59) and equating the coefficient of un+1 and u to

zero, we have

T

+

x

(A 2Xx)

+

x

X

x2+ A(n + 1)

+

x

+ Ax = 0,(5.63)

jT2t2

T

j

2t+

+ 2Xx

j

2t+

+ At

2Axx = 0.(5.64)

Inserting (5.58) in (5.63) and (5.64), we get

Xxx + n

+

x

Xx n

x2x = 0, (5.65)

j2t +

T j2t2 T + At Axx = 0. (5.66)

Integration of equation (5.60) with respect to x gives

X = (T + nA) x + C1(t), (5.67)

where C1(t) is the function of integration. Substituting (5.67) in (5.62) and equating

the coefficients of x and x0 to zero, we get

T + nAt = 0, (5.68)

C1 + Ax = 0. (5.69)

Solving (5.68) for A, we get

A = T

n+ c2. (5.70)

Using (5.70) in (5.69), we find that

C1(t) = c1. (5.71)

In view of (5.70) and (5.71), (5.67) becomes

X = nc2x + c1. (5.72)

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On inserting (5.70), equation (5.66) becomes

T

nj

2t +

T

+

jn

2t2 T = 0. (5.73)

Equations (5.56) and (5.61) lead to

U = Au. (5.74)

We reduce (5.1) to ODEs under the cases (1) j = = 0, (2) = = = 0, (3)

= = 0, (4) j = = 0, (5) = = j = 0 and (6) = j = 0. For brevity we

suppress the details of the determination of X , T , U .

Case 2.1 j = = 0

For this case equation (5.1) is

ut + unux + u + u

n+1 = (t)2

uxx. (5.75)

Substituting j = 0 in (5.73) and solving for T, we get

T = A0ent + B0. (5.76)

On using (5.76), equation (5.70) gives

A = A0ent + c2. (5.77)

Putting = 0 in (5.65), we get

Xxx + nXx = 0. (5.78)

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Inserting (5.72) in (5.78), we get c2 = 0. Therefore (5.77) and (5.72) can be written

as

A = A0ent, X = c1. (5.79)

Using (5.79) in (5.76), we get

U = A0entu. (5.80)

Substituting (5.76) and (5.79) in (5.58), we have

= A0nent

A0ent + B0. (5.81)

The exact solution of equation (5.81) is

(t) = r3

A0ent + B0

1

(5.82)

The characteristic differential equation dxX

= dtT

= duU

takes the form

dx

c1=

dt

A0ent + B0=

du

A0entu, A0 = 0. (5.83)

The case A0 = 0 is studied in case 2.1(b).

The similarity form of (5.75) with (t) given by (5.82) namely

ut

+ unux

+ u + un+1 =r3

2 (A0ent + B0)uxx

, (5.84)

is

u =

A0ent + B0

1/n

f(z), (5.85)

z = x c1nB0

log

ent

A0ent + B0

, B0 = 0. (5.86)

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The case B0 = 0 is subsequently considered in Case 2.1(a).

Substituting (5.86) in (5.84), we get the ordinary differential equation

f +2c1r3

f 2r3

fnf 2r3

fn+1 2B0r3

f = 0. (5.87)


f(z) = expc1

r3 1

r3

c21 + 2B0r3

z

. (5.88)

Through f = H1/n equation (5.87) is transformed again to an GEPT

HH

n + 1

n

H

2+

2c1r3

HH 2r3

H +2n

r3H+

2nB0r3

H2 = 0. (5.89)

The solution of (5.89) corresponding to (5.88) is

H(z) = exp

nc1r3 n

r3

c21 + 2B0r3

z

. (5.90)

Case 2.1(a) B0 = 0

In this case the infinitesimals are

T = A0ent, X = c1, U = A0entu. (5.91)

The viscosity is

(t) = r0ent, (5.92)

The similarity form of equation (5.75) when (t) is given by (5.92) namely

ut + unux + u + u

n+1 =r02

entuxx, (5.93)

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is

u = etf(z), z(x, t) = x +c1

A0nent. (5.94)

Here f is governed by

f 2c1r0A0

f +2

r0fnf 2

r0fn+1 = 0, (5.95)

whose solution is

f(z) = e2c1r0A0

z, (5.96)

provided that = 2c1r0A0 .

Through the transform f = H1/n equation (5.95) becomes a GEPT

HH

n + 1

n

H

2 2c1A0r0

HH +2

r0H +

2n

r0H = 0, (5.97)

with the solution

H(z) = e2nc1r0A0

z. (5.98)

Case 2.1(b) A0 = 0

The infinitesimals and (t) are

T = B0, X = c1, U = 0, (t) = r1. (5.99)

The similarity transformation is

u = f(z), z = x c1B0

t, (5.100)

where f satisfies

f 2r1

fnf +2c1

B0r1f 2

r1f

r1fn+1 = 0. (5.101)

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f(z) = exp c1

B0r1 c21

B0r12+

2

r1 z , (5.102)

provided that =

c1

B0r1

c21B0r12

+ 2r1

.

Through the transform f = H1/n equation (5.101) becomes a GEPT

HH

n + 1

n

H

2+

2c1B0r1

H +2

r1H2 +

2

r1H = 0. (5.103)

Therefore the solution of (5.103) is

H(z) = exp

n

c1

B0r1

c21B0r12

+2

r1

z

. (5.104)

Case 2.2 = = = 0

Equation (5.1) reduces to

ut + unux +

j

2tu =

(t)

2uxx. (5.105)


T = C0 + D0tjn2 , X = nc2x + c1, U =

c2 c0

n

D0j

2tjn21

u. (5.106)

The viscosity equation is

=

2nc2

C0 +jnD02

tjn21

C0t + D0tjn/2. (5.107)

The general solution of equation (5.107) is

(t) = r4t2nc2C0

1C0 + D0t

jn2 1

2nc2C0[ jn2 1]

1

. (5.108)

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The similarity form of the equation (5.105) with (t) is given by (5.108), viz.,

ut + unux +

j

2tu =

r4

2t2nc2C0

1 C0 + D0t jn2 1

2nc2

C0[ jn2 1]1

uxx, (5.109)

is

u = tc2C0

1n

C0 + D0t

jn21 c2

C0( jn2 1)+ 1n

f(z), (5.110)

z = (c2x + c1)C0 + D0t

jn21 nc2C0[ jn2 1] . (5.111)

For this case the reduced ODE is

f 2C0r4(n + 1)

jn

2 1

zf +

2C0r4(n + 1)

jn

2 1

fnf 2C0

r4(n + 1)

jn

2 1

f = 0.

(5.112)

The first integral of equation (5.112) with c2 = C0n(n+1)jn2 1

is

f 2C0r4(n + 1)

jn

2 1

zf +

2C0r4(n + 1)2

jn

2 1

fn+1 = 0. (5.113)

The transformation f = H1/n linearizes (5.113) to

H 2nC0r4(n + 1)

1 jn

2

zH +

2nC0r4(n + 1)2

1 jn

2

= 0. (5.114)


H(z) =

H(0)

nC0(2jn)

2r4(n + 1)2erf

nC0(2jn)2r4(n + 1)

z

exp

nC0(2jn)

2r4(n + 1)z2

.

(5.115)

The corresponding solution of (5.112) with c2 = C0n(n+1)jn2 1

is

f(z) =

H(0)

nC0(2jn)

2r4(n + 1)2erf

nC0(2jn)

2r4(n + 1)z

1/n

exp

C0(2jn)2r4(n + 1)

z2

.

(5.116)

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Thus an exact closed form solution of (5.109) is given by using (5.111); (5.116) in

(5.110):

u(x, t) = tc2C0

1n

C0 + D0t

jn21 c2C0( jn2 1)

+ 1n

H(0)

nC0(2jn)

2r4(n + 1)2

erf

nC0(2jn)

2r4(n + 1)(c2x + c1)

C0 + D0t

jn21 nc2C0[ jn2 1]

1/n

exp

C0(2jn)

2r4(n + 1)(c2x + c1)

2C0 + D0t

jn2 1

2nc2C0[ jn2 1]

. (5.117)

Case 2.2(a) c2 = 0

When c2 = 0, (5.106), (5.108) reduce to

T = C0t+D0tjn2 , X = c1 U =

C0n

+D0j

2tjn21

u, (t) = r41

C0t + D0tjn2

1.

(5.118)

The corresponding similarity transformation is

u = t1/nC0 + D0t

jn211/n

f(z), (5.119)

z = x c1C0

jn2 1

log tjn21

C0 + D0tjn21

, (5.120)

where f(z) is any solution of

f +2

r4fnf +

2c1

r4f

2C0

r4j

2 1

n f = 0. (5.121)

The equation (5.121) changes to an EPT

HH

n + 1

n

H

2+

2c1r4

HH 2r4

H +2C0r4

H2 = 0, (5.122)

through f = H1/n. The case n = 2, j = 1 is investigated in case 2.2(d).

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Case 2.2(b) c2 = C0 = 0

Setting c2 = C0 = 0 in (5.107) and solving we obtain

(t) = r4tjn2 . (5.123)


T = D0tjn2 , X = c1, U = D0j

2tjn2 1u, (5.124)

give rise to the similarity transformation

u = tj/2f(z), z = x +c1

D0jn2 1

t1 jn2 . (5.125)

The reduced ODE for this case is

f 2r4

fnf +2C1

D0r4

f = 0, (5.126)

whose first integral is the Bernoullis equation

f 2r4(n + 1)

fn+1 +2C1

D0r4f = 0, (5.127)

where we have taken the constant of integration equal to zero. The transformation

f = H1/n linearizes (5.127) to

H 2nc1D0r4

H+2n

r4(n + 1)= 0. (5.128)

The general solution of (5.128) is

H(z) = H(0)e2nc1D0r4

z D0c1(n + 1)

. (5.129)

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The corresponding solution of (5.126) is

f(z) =

H(0)e

2nc1D0r4

z

D0

c1(n + 1)1/n

. (5.130)

In fact (5.129) is a solution of

HH

n + 1

n

H

2+

2c1D0r4

HH 2r4

H = 0, (5.131)

which is derived from (5.126) through f = H1/n.

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Case 2.2(c) D0 = 0

The infinitesimals and viscosity are

T = C0t, X = nc2x + c1, U =

c2 C0n

u, (t) = r4t

2nc2C0

1

. (5.132)


u = tc2C0

1n f(z), z = (nc2x + c1) t

nc2/C0. (5.133)

The similarity function f is governed by

f 2nc2r4

fnf 2nc2C0r4

zf

j

n2c22r4+

2(nc2 C0)n3C0c22r4

f = 0. (5.134)

A first integral of (5.134) with c2 =C0(2jn)2n(n+1)

is the Bernoullis equation

f 4C0r4(2jn) f

n+1 +4(n + 1)

C20r4(2jn)zf = 0, n = 2, j = 1. (5.135)

Equation (5.135) is lenearized via f = H1/n to

H 4(n + 1)nC20r4(2jn)

zH +4n

C0r4(2jn) = 0, (5.136)

whose general solution is

H(z) =

H(0)

2n

r4(n + 1)(2jn)erf

2n(n + 1)

C20r4(2jn)z

exp

2n(n + 1)

C20r4(2jn)z2

.

(5.137)

Therefore the corresponding solution of (5.134) is

f(z) =

H(0)

2n

r4(n + 1)(2jn)erf

2n(n + 1)

C20r4(2jn)z

1/n

exp

2(n + 1)

C20r4(2jn)z2

. (5.138)

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Case 2.2(d) c2 = D0 = 0

The variable viscosity and the infinitesimals are

(t) =r4t

, T = C0t, X = c1, U = C0n

u. (5.139)

The similarity form is

u = t1/nf(z), z = x c1C0

log t, (5.140)

where f is satisfied by

f 2r4

fnf +2c1

C0r4f +

1

r4

2jn

2

f = 0. (5.141)

When n = 2, j = 1 (5.141) is integrated to (after setting the constant of integration

equal to zero)

f 2r4(n + 1) fn+1 + 2c1C0r4

f = 0, (5.142)

a Bernoulli equation and therefore linearizes through f = H1/n to

H 2nc1C0r4

H+2n

r4(n + 1)= 0. (5.143)

The general solution of (5.143) is

H(z) =

H(0)e2nc1C0r4

z + C0c1(n + 1)

, (5.144)

and the corresponding solution of equation (5.141) with n = 2, j = 1 is

f(z) =

H(0)e

2nc1C0r4

z+

C0c1(n + 1)

1/n

. (5.145)

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When n = 2, j = 1, (5.105) becomes the modified cylindrical Burgers equation with

variable viscosity

ut + u2ux + 1

2tu = r

4

2tuxx. (5.146)

Its closed form solution is written by using (5.140), (5.145):

u(x, t) = t1/n

H(0)tc1/C0e(4c1/C0r4)x +C03c1

1/2. (5.147)

Case 2.2(e) C0 = 0


T = D0tjn2 , X = nc2x + c1, U =

c2 D0j

2tjn21

. (5.148)

The viscosity is

(t) = r4tjn2 exp 4nc2

D0(2jn)t1

jn2 , n = 2, j = 1. (5.149)

The similarity form is

u = exp

2c2

D0(2jn)t1 jn2

t

j2 f(z), (5.150)

z = (nc2x + c1)exp

2nc2D0(2jn) t

1 jn2

. (5.151)

The reduced ODE for this case is

f 2nc2r4

fnf +2

nc2D0r4zf 2

n2c2D0r4f = 0. (5.152)

Through f = H1/n equation (5.152) changes to

HH

n + 1

n

H

2+

2

nc2D0r4zH H 2

nc2r4H +

2

n2c2r4D0H2 = 0. (5.153)

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We note that for (5.105) we have found exact closed form solutions in terms of an

exponential and error functions.

Case 2.3 = = 0

In this case equation (5.1) becomes

ut + unux +

j

2tu + un+1 =

(t)

2uxx. (5.154)


T = C0t + D0tjn2 , X = c1, U =

C0

n D0j

2tjn21

u. (5.155)

The function (t) is governed by

= C0 +

jnD02

tjn2 1

C0t + D0tjn2

. (5.156)


(t) =r5

C0t + D0tjn2

. (5.157)

We remark that a special case of (5.154) with = 0, namely (5.105) also admits (t)

in the form same as (5.157) (see equation (5.118) in Case 2.2(a)).

The similarity form of solutions of (5.154) is

u = t1/n

C0 + D0tjn211/n

f(z), (5.158)

z = x c1(jn2 1)C0

logtjn2 1

C0 + D0tjn2 1

, (5.159)

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where f(z) satisfies

f

2

r5 f

n

f

2c1

r5 f

+

2

r5 f

n+1

2C0

r5j

2 1

n

f = 0. (5.160)

A solution of (5.160) is

f(z) = exp

c1

r5c21

r25+

2C0r5

j

2 1

n

z , (5.161)

provided that = c1r5

c21r25

+ 2C0r5

j2 1

n

.

Through f = H1/n equation (5.160) is transformed to a GEPT

HH

n + 1

n

H

2 2c1r5

HH 2r5

H +2n

r5H+

2nC0r5

j

2 1

n

H2 = 0. (5.162)

Therefore the solution of (5.162) is

H(z) = exp

n

c1r5

c21r25

+2C0r5

j

2 1

n

z

. (5.163)

A special but physically interesting situation arises if we take j = n = 2. Under this

assumption (5.154) become the modified spherical Burgers equation with nonlinear

damping term and variable viscosity:

ut + u2ux +

1

tu + u3 =

r5t2

uxx. (5.164)

Case 2.3(a) C0 = 0

In this case the infinitesimals and (t) are

T = D0tjn2 , X = c1, U = D0j

2tjn21u, (t) = r5t

jn2 . (5.165)

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The similarity transformations are

u = t

j/2

f(z), z = x +

c1

D0jn2 1

t

1 jn2

. (5.166)

The f- equation is found to be

f 2r5

fnf +2c1

D0r5f 2

r5fn+1 = 0, (5.167)

for which we obtain the solution

f(z) = e2c1

D0r51 z, (5.168)

under the condition = 2c1D0r5

. We assume that c1/D0 < 0 to ensure that > 0.

Equation (5.167) is transformed via f = H1/n to GEPT

HH

n + 1

n

H

2+

2c1D0r5

HH 2r5

H +2n

r5H = 0, (5.169)

with the solution

H(z) = e2c1nD0r51

z. (5.170)

Case 2.3(b) D0 = 0

In this case the infinitesimal transformations are

T = C0t, X = c1, U = C0n

u. (5.171)

Inserting D0 = 0 in (5.156) and solving, we obtain

(t) =r5t

. (5.172)

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u = t1/nf(z), z = x

c1

C0log t. (5.173)

Equations satisfied by f and H are

f 2r5

fnf +2c1

C0r5f +

1

r5

2jn

n

f 2

r5fn+1 = 0.(5.174)

HH

n + 1

n

H

2+

2c1C0r5

HH 2r5

H

2jnr5

H2 +

2n

r5H = 0. (5.175)

It is easily verify that

f(z) = exp

c1

C0r5 c21

C20r25

+jn 2

nr5

z

. (5.176)

is a solution of (5.174) provided = c1C0r5

c21C20r

25

+ jn2nr5

and therefore the solution

of (5.175) is

H(z) = exp

n

c1C0r5

c21C20r

25

+jn 2

nr5

z

. (5.177)

All the 3 reductions of (5.154) given in Case 2.3, 2.3(a), 2.3(b) are GEPTs for which

exact closed form solutions in terms of exponential function is found.

Case 2.4 = = 0

Putting = = 0, (5.1) takes the form

ut + unux +

j

2tu +

xun+1 =

(t)

2uxx. (5.178)


T = C0t + D0tjn2 , X = nc2x, U =

c2 C0

n

D0j

2tjn21

u. (5.179)

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=

2nc2 C0 + D0jn2

tjn21

C0t + D0t jn2 . (5.180)

Solving equation (5.180), we get

(t) = r6t2nc2C0

1

C0 + D0tjn21 2nc2

C0(jn2 1)

+1

. (5.181)


u = t c2C0 1n

C0 + D0tjn21 c2

C0( jn2 1)+1/n

f(z), (5.182)

z = xtnc2C0

C0 + D0t

jn21 nc2C0(

jn2

1) . (5.183)

The f and H equations are

f 2r6

fnf +2nc2r6C0

zf 2r6z

fn+1 2r6

c2 C0

n+

jC02

f = 0,(5.184)

HH

n + 1n H2 +2nc2r6C0 zH H

2

r6 H

+2n

r6zH+2n

r6

c2 C0n +

jC02

H

2

= 0.(5.185)

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Case 2.4(a) C0 = 0

The infinitesimal transformations are

T = D0tjn2 , X = nc2x, U =

c2 D0j

2tjn2 1

u. (5.186)

The general solution of (5.180) with C0 = 0 is

(t) = r6exp

4nc2

D0(2jn)t1 jn2

t

jn2 . (5.187)

The form of the similarity solution of (5.178) is

u = exp

2c2

D0(2jn)t1 jn2

t

jn2 f(z),

z = xexp

nc2

D0(2jn)t1

jn2

, (5.188)

where f(z) satisfies

f 2r6

fnf +2nc2D0r6

zf 2r6z

fn+1 2c2D0r6

f = 0. (5.189)

The H equation is

HH

n + 1

n

H

2 2nc2r6D0

zH H 2r6

H 2nr6z

H+2nc2D0r6

H2 = 0. (5.190)

Case 2.5 = = j = 0

Equation (5.1) with = = j = 0 is

ut + unux +

xun+1 =

(t)

2uxx. (5.191)

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The function (t) is any solution of

=

2c2

s1

s1t + s2 . (5.192)

Solving (5.192), we get

(t) = r7 (s1t + s2)2c2s1

s1 . (5.193)


T = s1t + s2, X = c2x, U = c2 s1

n u. (5.194)


u = (s1t + s2)(c2s1)/(ns1) f(z), z = x (s1t + s2)

c2/s1 , (5.195)

where f(z) is satisfied by

f

2

r7 fn

f

+

2c2r7 zf

2

r7zfn+1

2

r7c2 s1s1 f = 0. (5.196)

The H equation is a GEPT :

HH

n + 1

n

H

2+

2c2r7

zH H 2r7

H +2n

r7zH+

2n

r7

c2 s1

s1

H2 = 0. (5.197)

Case 2.5(a) s1 = 0

In this case the infinitesimals, (t) and similarity transformation are

T = s2, X = c2x, U =c2n

u, (t) = r7e2c2s2

t, u = e

c2ns2

tf(z), z = xe

c2s2t.

(5.198)

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The f and H equations are

f

2

r7 f

n

f

+

2c2

s2r7 zf

2

r7zf

n+1

2c2

ns2r7 f = 0,(5.199)

HH

n + 1

n

H

2+

2c2s2r7

zH H 2r7

H +2

r7zH+

2n

r7

c2 s1

s1

H2 = 0.(5.200)

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Case 2.6 = j = 0

Equation (5.1) when = j = 0 is of the form

ut + unux + u +

xun+1 =

(t)

2uxx. (5.201)


T = A0ent + B0, X = nc2x, U =

A0ent + c2

u. (5.202)


=

2nc2 A0entA0ent + B0

, (5.203)

which admits the following general solution

(t) = r8e2nc2B0

tA0e

nt + B0

2c2B0

+1

. (5.204)


u = ec2B0

tA0e

nt + B0

c2

nB0+1/n

, z = xe

nc2B0

tA0e

nt + B0 c2B0 , (5.205)

where f and H = fn satisfy

f 2r8

fnf +2nc2

r8zf 2

r8zfn+1 2

r8(c2 + B0) f = 0.(5.206)

HH

n + 1

n

H

2+

2nc2r8

zH H 2r8

H +2n

r8zH+

2n

r8(c2 + B0) H

2 = 0.(5.207)

Case 2.6(a) B0 = 0


T = A0ent, X = nc2x, U =

A0ent + c2

u. (5.208)

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The function (t) and the similarity transformation are

(t) = r8exp

2nc2

A0 e

nte

nt

, (5.209)

u = etexp c2

nA0ent

f(z), z = x exp

c2

A0ent

. (5.210)

Here f and H are governed by

f 2r8

fnf +2nc2A0r8

zf 2r8z

fn+1 2c2r8A0

f = 0, (5.211)

HH

n + 1

nH

2+

2nc2

A0r8

zH H

2

r8

H +2

r8z

H+2c2

r8A0

H2 = 0. (5.212)

3. Result and Discussions

We applied the Lies classical method to equation (5.1) and considered two case

Tu = Xu = Uu = 0 and the general case separately. We list below the conditions

imposed on the parameters appearing in the GBE

ut + unux + ( +

j

2t)u + (+

x)un+1 =

(t)

2uxx, (5.213)

and all possible (t)s obtained for each such condition:

1. = = j = 0

1 = .

2. j = = 0

2 = r3

A0ent + B0

1

, 3 = r0ent, 4 = r1.

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3. = = = 0

5

= r4t2nc2C0

1 C0

+ D0tjn21

2nc2

C0[ jn2 1]1

, 6

= r4C

0t + D

0tjn2 1 ,

7= r

4t

jn2 ,

8 = r4t2nc2C0

1, 9 =

r4t

, 10 = tjn2 exp

4nc2

D0(2jn)t1 jn2

, 11 = r2t.

4. = = 0

12 = r5

C0t + D0tjn2

1, 13 = r5t

jn2 , 14 =

r5t

.

5. = = 0

15 = r6t2nc2C0

1

C0 + D0tjn2 1

2nc2C0(

jn2 1)

1, 16 = r6t

jn2 exp

4nc2

D0(2jn)t1 jn2

,

18 = r1t.

6. = = j = 0

19 = r7 (s1t + s2)2c2s1

s1 , 20 = r7e2c2s2

t.

7. = j = 0

21 = r8e2nc2B0

tA0e

nt + B0

2c2B0

+1

, 23 = r8exp

2nc2

A0ent

et.

An important contribution of the present work is the generalization of the Euler-

Painleve transcendent (5.17) to a generalized EPT in the form

HH + aH2

+ A(z)HH + B(z)H2 + bH + C(z)H+ c = 0, (5.214)

(see for instance (5.39), (5.46),(5.89) (5.97)). Indeed the similarity reductions of

ut + unux +

j

2tu +

xun+1 =

r1t

2uxx,

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ut + unux +

j

2tu +

xun+1 =

r62

t2nc2C0

1

C0 + D0tjn21 2nc2

C0(jn2 1)

+1

uxx,

ut + un

ux +j

2tu +

xun+1

=r62 exp

4nc2D0(2jn) t

1 jn

2

tjn

2 uxx,

ut + unux +

j

2tu +

xun+1 =

r62

t2nc2C0

1uxx,

ut + unux + u

n+1 =

2uxx,

ut + unux + u + u

n+1 =r32

A0e

nt + B01

uxx,

ut + unux + u + u

n+1 =r02

uxx,

ut + unux + u + un+1 = r02

entuxx,

ut + unux +

j

2tu + un+1 =

r52

tjn/2uxx,

ut + unux +

j

2tu + un+1 =

r52t

uxx,

ut + unux +

xun+1 =

r72

(s1 + s2)2c2s11

uxx,

ut + unux +

xun+1 =

r72

e2c2s2

tuxx,

ut + unux + u +

x

un+1 = r8

2e2nc2B0

tA0e

nt + B0

2c2B0

+1

uxx,

ut + unux + u +

xun+1 =

r82

exp2nc2

A0ent

etuxx,

are transformed via f = H1/n to (5.214). Equation (5.214) is more general than

(5.17) in the sense that the former contains an extra term C(z)H. And when

A(z), B(z), C(z) are constants then an exact solutions of (5.214) in terms of an ex-

ponential function can be determined. It is also evident from the general form of the

f-equation viz.,

f + p(z)f + qf + rfnf + s(z)fn+1 = 0. (5.215)

The condition A,B,C are constants is equivalent to p(z), s(z) are constants. And in

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this case by splitting (5.215) into two equations f+p(z)f+qf = 0, rfnf+sfn+1 = 0

one can easily see that f-equation admits an exponential solution. We also obtain

solutions of f-equations and hence those of H-equations in terms of error functions

by linearizing the f equation when p(z) is linear in z and s(z) is a constant in (5.215).

Here we first obtain an intermediate integral of the f-equation in terms of Bernoullis

equation and is easily linearized. The linear first order equation is shown to have

exact solutions in terms of an error function.(see (5.116), (5.136)).

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Symmetry analysis of Generalized Burger's equation