Exact solutions of equations for the Burgers hierarchy

At n = 1 Eq. ( 1) is the Burgers equation

Eq. ( 2) was firstly introduced in [1]. It’s well known that the Burgers equation can be linearized by the Cole-Hopf transformation [2,3]. Exact solutions of Eq.( 2) were discussed in many papers( see for example [4][5][6][7] ).

In the case n = 2 from Eq. ( 1) we have the Sharma -Tasso -Olver (STO) equation

The STO equation was derived in [8,9]. Some exact solutions of this equation was obtained in [10][11][12][13][14][15][16].

At n = 3 and n = 4 we have the following fourth and fifth order partial differential equations u t + α u xxxx + 10 α u x u xx + 4 α uu xxx + 12 α uu 2

x + +6 α u 2 u xx + 4α u 3 u x = 0, (4)

In this paper we present the generalized Cole-Hopf transformation which we use for finding different types of exact solutions: the solitary wave solutions, the periodic solutions and the rational solutions. The advantage of our approach is that we can find the exact solutions for whole Burgers hierarchy. We can construct them without using the traveling wave. This fact allows us to obtain solutions of different types.

Eq. ( 1) can be linearized by the Cole-Hopf transformation [9,17,18]

Taking this transformation into account, we have [18]

where Ψ n,x -n-th derivative of Ψ with respect to x. Exact solutions of the Burgers equation can be obtained by using a generalization of the Cole-Hopf transformation [17][18][19][20][21][22][23]. This transformation can be written as

where F (x, t) satisfies the Burgers equation. Let us show that transformation (8) is valid for all hierarchy (1). First of all, we prove the following lemma. Lemma 1 The following identity takes place

where Ψ n,x is n-th derivative of Ψ with respect to x.

Proof. Let us apply the method of mathematical induction. When n = 1 we get

At n = 2 we have

By the induction, assuming n = k -1, we obtain

Finally, when n = k we have

This equality completes the proof. Theorem 1 Let F (x, t) be a solution of Eq. (1). Then

is the solution of the Burgers hierarchy (1).

Proof. Using the Cole-Hopf transformation (6), we obtain

Substituting transformation (14) into hierarchy (1) and taking the Lemma 1 and Eq. ( 15) into account we have following set of equalities

Thus, we have that if F (x, t) satisfies equation (1) then u(x, t) by formula (14), is solution of (1) as well.

Let us show that the Burgers hierarchy has the solution in the form

where k j and x (j) 0 are arbitrary constants. This result follows from the theorem. Theorem. Let

be a solution of the Burgers hierarchy. Then

is a solution of the Burgers hierarchy as well.

Proof. This theorem follows from the generalized transformation ( 14) for the solution of the hierarchy (1). Let

be the solution of the hierarchy (1) equation, then

is also the solution of the Burgers hierarchy by the generalized transformation for the solution of the hierarchy (1). Formulae ( 22) and ( 23) can be written in the form

Assume that

is the solution of the hierarchy (1) and substituting U m into the generalized transformation (14) we obtain that

is a solution of hierarchy (1). This equality completes the proof. This theorem allows us to have the solutions of the Burgers hierarchy in the form (19).

It is obvious, that function

is the solution of the

By the Cole -Hopf transformation (6) we have the solution of the Burgers hierarchy in the form

Taking the theorem into account we have the solution of the hierarchy (1) in the form (19).

Let us present some examples. When N = 2, n = l = 1 we have the solution of the Burgers equation

In the case N = 2, n = l = 2 we have the solution of the Sharma-Tasso-Olver equation in the form

When N = 3, n = 2 and l = 5 we obtain the following solution of the Sharma-Tasso-Olver equation

In the case N = 2, n = 3 and l = 3 we have solitary wave solution for the Eq. ( 4)

We can see that Eq. ( 28) is linear and has polynomial solutions. Thus, we can present its solution in the form

(n = 1, 2, . . .), N ∈ N,

From the transformation (6) and formula (34) we have the exact solution of the hierarchy (1) in the form (

For the Burgers equation (n=1) from ( 38) we obtain the solution in the form

We demonstrate solution (37) when

For the Eq. ( 4) from (38) we have the following solution

By analogy with solution (19) we can look for the periodic solutions of the equation for the Burgers hierarchy taking the trigonometric functions into consideration. Equation (28) has trigonometric solutions at n + 1 = 2l + 1

(j = 0, 1, 2, . . . , N), (l = 1, 2, . . .), N ∈ N,

From the transformation (6) we have the exact solution of the hierarchy in the form ( 1)

For example, we can write following solution for the Sharma -Tasso -Olver equation (l=1)

Assuming From formula (39) we obtain following solution for the Eq. ( 5) (l=2)

Other solutions can be written using the formula (21).

Using Eq. (28) and transf