## Abstract

The fifth order non-linear partial differential equation in generalized form is analyzed for Lie symmetries. The classical Lie group method is performed to derive similarity variables of this equation and the ordinary differential equations (ODEs) are deduced. These ordinary differential equations are further studied and some exact solutions are obtained.

### Keywords

- generalized fifth order non-linear partial differential equation
- lie symmetries
- exact solutions

## 1. Introduction

The theories of modern physics mainly include a mathematical structure, defined by a certain set of differential equations and extended by a set of rules for translating the mathematical results into meaningful statements about the physical work. Theories of non-linear science have been widely developed over the past century. In particular, non-linear systems have fascinated much interest among mathematicians and physicists. A lot of study has been conducted in the area of non-linear partial differential equations (NLPDEs) that arise in various areas of applied mathematics, mathematical physics, and many other areas. Apart from their theoretical importance, they have sensational applications to various physical systems such as hydrodynamics, non-linear optics, continuum mechanics, plasma physics and so on. A large variety of physical, chemical, and biological phenomena is governed by nonlinear partial differential equations (NLPDEs). A number of methods has been introduced for finding solutions of these equations such as Homotopy method [1],

Lie group method is one of the most effective methods for finding exact solutions of NLPDEs [17, 18]. This method was basically initiated by Norwegian mathematician Sophus Lie [19]. He developed the theory of “Continuous Groups” known as Lie groups. This method is orderly used in various fields of non-linear science. Shopus Lie was the first who arranged differential equations in terms of their symmetry groups, thereby analyzing the set of equations, which could be integrated or reduced to lower order equations by group theoretic algorithms. The Lie group analysis is a mathematical theory that synthesizes symmetry of differential equations. In this method, the differential structure is studied for their invariance by acting one or several parameter continuous group of transformations on the space of dependent and independent variables. We observe a plenty books and research article about Lie group method [17, 18, 20, 21, 22].

Wazwaz [23] introduced a fifth order non-linear evolution equation as follows:

In this chapter, he obtained multiple soliton solutions of this equation.

We will consider the generalized fifth order non-linear evolution equation of the form:

where

In this chapter, we will study the Eq. (2) by the Lie classical method. Firstly, Lie classical method will be used to obtain symmetries of generalized fifth order non-linear evolution Eq. (2). Symmetries will be used to reduce the Eq. (2) to ordinary differential equations (ODEs) and corresponding exact solutions of the generalized fifth order non-linear evolution Eq. (2) will be obtained.

## 2. Symmetry analysis

Lie classical method of infinitesimal transformation groups reduces the number of independent variables in partial differential equations (PDEs) and reduces the order of ODEs. Lie’s method has been widely used in equations of mathematical physics and many other fields [11, 24]. In this chapter, we will perform Lie symmetry analysis [17, 18, 19, 24] for the generalized fifth order non-linear evolution Eq. (2).

Let the group of infinitesimal transformations be defined as:

which leaves the Eq. (2) invariant. The infinitesimal transformations (3) are such that if

Herein, on invoking the invariance criterion as mentioned in [18], the following relation is deduced:

where

where

Now, after computing (5) we get:

The Lie classical method for determining the similarity variables of (2) is mainly consists of finding the infinitesimals

where

Corresponding vector fields can be written as:

## 3. Symmetry reductions and invariant solutions

To obtain the symmetry reductions of Eq. (2), we have to solve the characteristic equation:

where

To solve Eq. (9), following cases will be considered: (*i*) *ii*)

Case (i)

On solving Eqs. (9) we have,

where

where primes

Let assume the solution of ODE (11) in following form:

where

Corresponding solution of ODE (11) can be written as:

where

Corresponding solution of main Eq. (2) is given by:

with

Some more solutions of ODE (11) are given by:

where

Corresponding solutions of main Eq. (2) are given by:

where

Case (ii)

On solving Eq. (9) for vector field

where

where primes

Let assume the solution of ODE (21) in following form:

where

Substituting (22) into ODE (21) and equating coefficients of the different powers of

Corresponding solution of ODE (21) can be written as:

where

Corresponding solution of main Eq. (2) can be written as:

where

## 4. Conclusion

In this chapter, we derived the symmetry variables and symmetry transformations of the generalized fifth order non-linear partial differential equation. We applied Lie symmetry analysis for investigating considered nonlinear partial differential equation and using similarity variables, given equation is reduced into ordinary differential equations. We derived explicit exact solutions of considered partial differential equation corresponding to each ordinary differential equation obtained by reduction.