Self-similar solutions of the Burgers hierarchy

Assuming n = 1 in Eq. (1) we have the Burgers equation

Eq. ( 2) was firstly introduced in [5]. It is well known that this equation can be linearized by means of the Cole-Hopf transformation [6][7][8]. Exact solutions of Eq.(2) were considered in many papers (see, for example, [9][10][11][12]).

Assuming n = 2 in Eq. ( 1) we obtain the Sharma -Tasso -Olver equation

The Sharma -Tasso -Olver equation was derived in [1,13]. Some exact solutions of this equation were presented in [14][15][16][17][18][19][20][21].

At n = 3 and n = 4 we obtain the following fourth and fifth order partial differential equations

Assuming

we have that Eq.( 1) is invariant under the dilation group in the case

Assuming C 0 = e -a in (7), we obtain the delation group for the Burgers hierarchy (1) in the form

From transformations (8) we have two invariants for Eq.( 1)

Therefore we look for the solutions of the Burgers hierarchy taking into account the variables

Substituting ( 10) into (1) we obtain the equation for f (z) at

in the form

where β is the constant of integration. Solving Eq.( 12) we obtain solutions of the Burgers hierarchy in the form

Let us study the solutions of nonlinear ordinary differential equation (12).

2 Exact solutions of equation( 12)

First of all let us prove the following lemma. Lemma 1. Equation ( 12) can be transformed to the linear equation of (n + 1) -th order by means of transformation

Proof. The proof of this lemma can be given by means of the mathematical induction method.

Using the transformation ( 14) we have

Assuming that there is equality

Differentiating Eq.( 16) with respect to in z we have

From Eq.( 17) we obtain the equality

Therefore we obtain the formula

Taking this formula into account we have the equality

As result of this lemma we obtain that solutions of Eq. ( 12) can be found by the formula (14), where ψ(z) is the solution of the linear equation

Let us consider the partial cases. Assuming β = 0 in Eq.( 21) we have

Denoting ψ z = y we obtain

In the case n = 1 we get solution of Eq.( 23) in the form

The general solution of Eq.( 23) can be written as

where C 2 and C 3 are arbitrary constants. In the case n = 2 we obtain the general solution of Eq.( 23) in the form

where J 1 3 and Y 1 3 are the Bessel functions. In the case n > 2 solution of Eq.( 23) has n solutions

where E n,m,l is a Mittag -Leffler type special function defined by [22];

Γ (n(ms + l) + 1) Γ(n(ms + l + 1) + 1) (28)

In the case β = 0 solutions of Eq.( 23) can be referred to the type of the Laplace equations [23]. There are partial solutions ψ(z) = -z m of Eq.( 21) at β = m, where 0 < m ≤ n is integer. In the general case solutions of equations ( 21) can be found using the Laplace transformation or taking the expansions in the power series into account.

For a example let us solve the Cauchy problem for linear ordinary differential equation ( 21) at β = -1. We have the following problem

Assuming n = 4 we obtain

z 4k+2 k j=0 (4 j + 3) (4k + 3)! + 6 a 3 ∞ k=0 z 4k+3 k j=0 (4 j + 4) (4k + 4)! .

(34)

One can show that these power series are conversed for any values z. Therefore self-similar solutions of equations for the Burgers hierarchy are found after substitution (34) into formula (14).

Author is grateful to Andrey Polyanin for useful discussion of nonlinear differential equation Eq.( 21).