New soliton solutions of the Burgers equation with

additional time-dependent variable coeﬃcient

1BAZAR BABAJANOV, 2FAKHRIDDIN ABDIKARIMOV

1DSc., Department of Applied Mathematics and Mathematical Physics,

Urgench State University, Urgench, 220100,

UZBEKISTAN

2Phd student, Khorezm Mamun Academy, Khiva, 220900,

UZBEKISTAN

Abstract: In this article, we use the functional variational method to solve the Burgers equation with

an additional time-dependent variable coeﬃcient. The main advantage of the proposed method over

other methods is that it allows to obtain more new solutions of the equation. Among the solutions

obtained, new soliton solutions should be noted, which are of great importance for revealing the internal

mechanism of physical phenomena. Three-dimensional graphs of solutions are constructed using the

mathematical program Matlab. For a better understanding of the physical properties of some of the

resulting solutions, their graphical representations are shown. This method is eﬀective for ﬁnding exact

solutions to many other similar wave equations.

Key-Words: Burgers equation, nonlinear evolution equations, variable coeﬃcient, functional variable

method, soliton solutions, ordinary diﬀerential equation.

Received: February 9, 2023. Revised: November 22, 2023. Accepted: December 19, 2023. Published: February 14, 2024.

1 Introduction

The Burger equation is a fundamental partial dif-

ferential equation in ﬂuid mechanics. It is also

a very important model used in several areas

of applied mathematics such as heat conduction,

acoustic waves, gas dynamics and traﬃc ﬂow[1].

In[2], the Burgers equation was ﬁrst introduced in

1915. This equation is expressed in the following

basic form

ut+uux−uxx = 0.

It was later proposed by Burger as one of

the classes of equations describing mathematical

models of turbulence[3]. In 1951, Cole studied the

Burgers equation and gave a theoretical solution

to this equation[4].

In[5], the Burgers equation with variable coef-

ﬁcient

ut+α(t)uux−β(t)uxx = 0

is investigated, where α(t) and β(t) are given con-

tinuous diﬀerentiable functions. These variable

coeﬃcients of Burgers equation with the nonlin-

ear α(t) and dispersion β(t) can model propaga-

tion of a long shock-wave in a two-layer shallow

liquid. In addition, this equation appears in ion

acoustic waves in plasma[6], traﬃc ﬂow[7, 8], dy-

namics of soil water[9], shock formation in elastic

gas[10], turbulence in ﬂuid dynamics[11, 12].

The shock wave, multi-shock solitons and

voltera integral type wave solutions[13, 14] for

the Burgers equation with variable coeﬃcient are

obtained by using Cole-Hopf transformation[15].

Zhang obtained not trivial and time dependent

conservation laws with the presence of admissible

transformation and Lie symmetry for the Burgers

equation with variable coeﬃcient[16]. Christov

obtained the kink or shock type traveling waves

solution of the Burgers equation with variable co-

eﬃcient by using the Crank-Nicolson numerical

scheme.

To ﬁnd exact solutions to nonlinear evolu-

tion equations, many direct methods are used.

For example: tanh-function method[17], func-

tional variable method[18], Hirota method[19],

Backlund transform method[20], G/G′expansion

method[21] and extended tanh-method[22].

In this paper, we consider the Burgers equa-

tion with additional time-dependent variable co-

eﬃcient

ut+h1(t)uux−h2(t)uxx +ω(t)ux= 0,(1)

where u(x, t) is an unknown function, x∈R,t≥

0, h1(t)= 0, h2(t)= 0 and ω(t)= 0 are given

continuous diﬀerentiable functions.

The main aim of this paper is to ﬁnd the exact

soliton solutions of the Eq.1 via functional vari-

able method. The main advantage of the pro-

posed method over other methods is that it al-

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DOI: 10.37394/232013.2024.19.6

Bazar Babajanov, Fakhriddin Abdikarimov

E-ISSN: 2224-347X

Volume 19, 2024

lows to obtain more new solutions of the equa-

tion. Among the solutions obtained, new soliton

solutions should be noted, which are of great im-

portance for revealing the internal mechanism of

physical phenomena. Three-dimensional graphs

of solutions are constructed using the mathemat-

ical program Matlab. For a better understanding

of the physical properties of some of the result-

ing solutions, their graphical representations are

shown. This method is eﬀective for ﬁnding exact

solutions to many other similar wave equations.

2 Description of the method

The basic idea of the functional variable method

proposed in[23]. Let us consider the nonlinear

diﬀerential equation with independent variables

x,y,z,tand a dependent variable u

P(u, ut, ux, uy, uz, uxy, uyz, uxz , ...)=0,(2)

where Pis a polynomial in u(t, x, y, z, ...) and

its partial derivatives.

The following transformation

ξ=

i=0

αiχi+δ, (3)

is used for the new wave variable.

Now, we can introduce the following transfor-

mation for the travelling wave solution of Eq.2

u(χ0, χ1, ...) = u(ξ),(4)

and the chain rule

∂u

∂χi

=αi

dξ ,∂2u

∂χi∂χj

=αiαj

d2u

dξ2, .... (5)

After this transformation, the Eq.2 is trans-

formed into an ordinary diﬀerential equa-

tion(ODE) of the form

Q(u, u′, u′′, u′′′ , ...)=0,(6)

where Qis a polynomial in u(ξ) and its total

derivatives, while u′=du

dξ .

Let us make a transformation in which the un-

known function uis considered as a functional

variable of the form

u′=F(u),(7)

then, the solution can be found by the relation

Zdu

F(u)=ξ+C, (8)

Some successive diﬀerentiations of uin terms

of Fare given as

u′′ =dF (u)

dξ =dF (u)

du F(u) = 1

d(F2(u))

du ,

u′′′ =1

d2(F2(u))

du2pF2(u),

u(IV )=1

2hd3(F2(u))

du3F2(u) + d2(F2(u))

du2

d(F2(u))

du i,

........................................................................

(9)

The Eq.6 can be reduced in terms of u,Fand

its derivatives upon using the expressions of Eq.7

and Eq.9 into Eq.2 gives

R(u, dF (u)

du ,d2F(u)

du2,d3F(u)

du3, ...)=0.(10)

After integration, Eq.10 provides the expres-

sion of F(u) and this, together with Eq.7, give ap-

propriate solutions to the being considered prob-

lem.

3 Algorithm for ﬁnding

solutions

Let us consider the Burgers equation with addi-

tional time-dependent variable coeﬃcient

ut+h1(t)uux−h2(t)uxx +ω(t)ux= 0.(11)

This equation appears in many physical prob-

lems, including the behavior of waves in nonlin-

ear optics, plasma or liquids, water waves, ion-

acoustic waves in collisions, and less commonly

in plasma. The ﬁrst term utrepresents the evo-

lution term and the second represents the disper-

sion term.

The wave variable

ξ=a(t) + b(t)x(12)

will convert Eq.11 to the following form

u′

t+ (at(t) + bt(t)) u′

ξ+h1(t)b(t)uu′

ξ−

−h2(t)b2(t)u′′

ξ+ω(t)b(t)u′

ξ= 0,(13)

where a(t) and b(t) are an unknown time-

dependent functions, to be determined later.

Let a(t), b(t), h1(t), h2(t) and ω(t) are con-

stant functions in Eq.11, that is, a(t) = a,b(t) =

b,h1(t) = h1,h2(t) = h2and ω(t) = ω. In this

case, the following transformation

ξ=a+bx. (14)

is used in Eq.11 to get an ordinary diﬀerential

equation of the form

h1buu′

ξ−h2b2u′′

ξ+ωbu′

ξ= 0.(15)

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Bazar Babajanov, Fakhriddin Abdikarimov

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Volume 19, 2024

Integrating Eq.15, we have

h1b

2u2−h2b2u′

ξ+ωbu = 0.(16)

From Eq.16 follows the expression for the func-

tion u′

ξ:

u′

ξ=su +nu2,(17)

where s=ω

h2b,n=h1

2h2b.

Now we move on to the question of determin-

ing unknown functions a(t) and b(t). To do this,

we search the solution of Eq.11 in the form

u(t, ξ) =

k=0

qk(t)Φk(ξ),(18)

where Φ satisﬁes Eq.19

Φ′=λΦ + µΦ2,(19)

where λand µare free parameters and mis an

undetermined integer and qk(t) are coeﬃcients to

be determined later.

One of the most useful ways to obtain the m

parameter is in Eq.18 is the homogeneous balance

method. Substituting Eq.18 into Eq.13 and by

making balance between the linear term u′′

ξand

the nonlinear term uu′

ξ, we will get that 2m+1 =

m+ 2, this in turn gives m= 1, and the solution

Eq.18 takes the form

u(t, ξ) = q0(t) + q1(t)Φ(ξ).(20)

Now, we substitute Eq.20 into Eq.13 and set

each coeﬃcient of Φk(Φ′)p(k= 0,1,2 and p=

0,1 ) to zero to obtain a set of algebraic equations

for q0(t), q1(t), a(t) and b(t):











q0t(t)=0, q1t(t)=0,

h1(t)b(t)q1(t)−2µh2(t)b2(t) = 0,

at(t) + bt(t)x+h1(t)b(t)q0(t)−

−λh2(t)b2(t) + ω(t)b(t) = 0.

(21)

Solving the system of algebraic equations, we ob-

tain

q0(t) = const, q1(t) = const, (22)

b(t) = q1(t)

2µ

h1(t)

h2(t), a(t) = Rt

0(λh2(τ)b2(τ)−

−bτ(τ)x+h1(τ)b(τ)q0(τ)−ω(τ)b(τ))dτ. (23)

Putting determined parameters q0(t), q1(t),

a(t) and b(t) into Eq.20 and taking into account

Eq.12 and Eq.19 we get the soliton solution of

Eq.1:

u(x, t) = Acth a(t) + b(t)x

2,(24)

where b(t) = q1(t)h1(t)

2µh2(t), q0(t) = −A=const,

q1(t) = −2A=const,A > 0,

a(t) = Rt

0(λh2(τ)b2(τ)−bτ(τ)x+h1(τ)b(τ)q0(τ)−

−ω(τ)b(τ))dτ.

4 Example

We illustrate the application of algorithm to solv-

ing the eq.11. Exact soliton solution of the

eq.11 can be deﬁned explicitly for exact values

of h1(t) = t,h2(t) = t,ω(t) = −2t,λ= 1, µ= 1.

In this case, the soliton solution of the eq.11 has

the form

u(x, t) = cth t2+x

2!.(25)

This solution of Eq.11 is veriﬁed and 3D

plots of the solutions obtained using Matlab

mathematical software are shown. Soliton wave

solutions are an important class of solutions

to nonlinear partial diﬀerential equations, since

many nonlinear partial diﬀerential equations have

been found to have diﬀerent soliton wave solu-

tions(Figure 1.).

Figure 1: Soliton wave solution of the eq.11 for

h1(t) = t,h2(t) = t,ω(t) = −2t,λ= 1, µ= 1.

5 Conclusion

In this work, the functional variational method

is successfully used to solve the Burgers equation

with additional variables and time-dependent co-

eﬃcients. Soliton solutions of this equation and

three-dimensional graphs of the resulting solu-

tions are found. The main advantage of the

proposed method over other methods is that it

gives more accurate traveling wave solutions. It

is concluded that exact solutions are of great im-

portance for revealing the internal mechanism of

physical phenomena.

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Declaration of competing

interest

The authors declare that they have no known

competing ﬁnancial interests or personal re-

lationships that could have appeared to in-

ﬂuence the work reported in this paper.

Data availability

The original contributions presented

in the study are included in the arti-

cle/supplementary material, further in-

quiries can be directed to the corresponding

author/s.

Conﬂict of Interest

The author declare that the research was

conducted in the absence of any commercial

or ﬁnancial relationships that could be con-

strued as a potential conﬂict of interest.

Contribution of individual

authors to the creation of a

scientiﬁc article (ghostwriting

policy)

Bazar Babajanov and Fakhriddin Abdikari-

mov conceived of the presented idea. Bazar

Babajanov developed the theory and per-

formed the computations. Fakhriddin Ab-

dikarimov veriﬁed the analytical methods.

Both authors discussed the results and con-

tributed to the ﬁnal manuscript. Both au-

thors contributed to the article and approved

the submitted version.

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