# Symmetry Classification and Solutions for the Third-Order of Kudryashov–Sinelshchikov Equation

Authors: Jina Li; Hong Li; Tianhao Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper studies nonlinear wave propagation in a bubble-liquid mixture based on the third-order Kudryashov–Sinelshchikov (KS) equation. Symmetry is classified and reduced through the symmetry group method. Through the generalized conditional symmetry method, this equation is classified, and the generalized conditional symmetry considered are second, third, fourth, and fifth-order, respectively. Meanwhile, the classified equations are transformed into a system of ordinary differential equations and solved.


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## Body

## 1. Introduction

The nonlinear evolution equation (NLEE) is essential to describe wave propagation in a bubble-liquid mixture, a typical nonlinear medium. The NLEE includes the Korteweg–de Vries (KdV) [1], the Burgers–Korteweg–de Vries equations. It is of significance to investigate the wave propagation in a bubble-liquid mixture.Obtaining the solutions of the NLEE is vital due to the physical information and insights into the problems that the NLEE can provide [2–9]. In order to solve the NLEE, several methods were proposed, including Inverse Scattering transformation [10], symmetry approach [11–13], and Darboux transformation (DT) [14]. Among them, the symmetry group algorithm has been the most widely used in obtaining the exact solutions of the NLEE. The nonlinear partial differential equation was introduced in [1], which is a vanish dissipation defined as follows:(1)ut+auux+uxxx−uuxxx−βuxuxx=0,where u represents a density that models viscosity and heat transfer. α and β are parameters. Equation (1) is a generalized equation of the KdV and BKdV equations, and was called the Ksudryashov–Sinelshchikov (KS) equation. This equaution was studied for various methods [15–29]. In [19], the characteristics of a bubble-liquid mixture, such as peaked solitons, were studied. Also, the Lie symmetry, optimal system, and solutions of the following KS equation were discussed,(2)ut+auux+buxxx+cuuxxx+duxuxx=0.This paper aims to study the following type of KS equation,(3)ut=Fuux+uxxx+Guuxxx+uxuxx,where Fu and Gu indicate arbitrary functions. By taking some appropriate forms for Fu and Gu, equation (3) is simplified to the KdV equation [24].(4)ut+6uux+uxxx=0,it was widely used to model various mechanical engineering and physical phenomena.The rest of this paper is structured as follows. Section2 presents the symmetry classification for the system (1), admitting the presence of GCS. Section 3 conducts the GCS method to reduce the symmetry and find exact solutions to the classified equations. Lastly, Section 4 is the conclusion.


## 2. Symmetry Classification

The symmetry group methods, especially the Lie point symmetry method, are effectively used to obtain the exact solution of partial differential equations. However, there are many problems that allow us to use symmetry groups. Based on the Lie point symmetry, many other methods were proposed to obtain the exact solutions, such as the conditional symmetry method and generalized conditional symmetry method, which is one of the most effective symmetry method first presented by Zhdanov, Fokas and Liu [30–32]. This paper adopts the generalized conditional symmetry method to analyze KS equation (3) with a physical background and obtain several solutions accordingly. We get the following results.Theorem 1.
Equation (3) admits the second-order GCSs in the form(5)Q=η∂∂u≡u2−∑i=01aiui∂∂uandu0=u.if and only if the equation is equivalent to one of the followings:(i)
(6)ut=Guua0−a0Gu−a0u+f1ux+uxxx+Guuxxx+uxuxxandη=uxx−a0u,(ii)
(7a)ut=uf1+f2ux+uxxx+G1u2+G2u+G3uxxx+uxuxx(7b)η=uxx,(iii)
(8a)ut=uxuf1+f2+uxxx+12uxx2G3a12−ua12−uf1a12x+uxuxx(8b)η=uxx−a1ux,(iv)
(9)ut=−3G1u2a0−G2ua0−a0u+f2ux+uxxx+G1u2+G2u+G3uxxx+uxuxxandη=uxx−a0u,(v)
(10)ut=427ua12+f2ux+uxxx+G3−23uuxxx+uxuxxandη=uxx−a1ux+29a12u.Proof.
According to the definition of the GCS method and the computation procedure, we obtain the following equation:(11)F4ux4+F3ux3+F2ux2+F1u+F0=0,where(12)F4=a1Guuu,F3=a0uGuuu+5a12Guu+3a0Guu+Fuu,F2=8a0a1uGuu+4a13Gu+7a0a1Gu+2a13+2a1Fu+3a0a1,F1=3a02u2Guu+7a0a12Gu+6a02Gu+4a0a12+3a0Fu+3a02u,andF0=3a02a1Gu+3a02a1.
We solve the system of differential equationsF0=0, F1=0, F2=0, F3=0, andF4=0. When F4=0, we obtain two cases, a1=0 or Guuu=0.Case 1.
Whena1=0, we can show Fu=Guua0−a0Gu−au+f1.Case 2.
WhenGuuu=0, we can obtain Gu=G1u2+G2u+G3,Fu=−5G1u2a12−3G1u2a0+uf1+f2. By substituting Fu,Gu into Fii=0,1,2,3,4, so the classification results are shown above.Theorem 2.
The equation admits the third-order GCSs in the form(13)Q=η∂∂u≡u3−∑i=02aiui∂∂u,u0=u.if and only if the equation is equivalent to one of the followings(i)
(14)ut=f2ux+uxxx+G1u+G2uxxx+uxuxxandη=uxxx,(ii)
(15)ut=f2−a1+2a1G1uux+uxxx+G1u+G2uxxx+uxuxxandη=uxxx−a1ux,(iii)
(16a)ut=f2+a22uux+uxxx+G2−uuxxx+uxuxx(16b)η=uxxx−a2uxx,(iv)
(17)ut=f2+427a22uux+uxxx+G2−23uuxxx+uxuxxandη=uxxx−a2uxx+29a22ux.Theorem 3.
The equation admits the fourth- and fifth-order GCSs, if and only if the equation satisfies one of the followings(i)
(18)ut=f2+f1uux+uxxx+G2+G1uuxxx+uxuxxandη=uxxxx,(ii)
(19)ut=f2+a2uux+uxxx+G2−uuxxx+uxuxxandη=uxxxx−a2uxx,(iii)
(20)ut=f2+415a2uux+uxxx+G2−23uuxxx+uxuxxandη=uxxxx−a2uxx+425a22u,(iv)
(21)ut=f2+f1uux+uxxx+G2−23uuxxx+uxuxxandη=uxxxxx−154f1uxxx+94f12ux.


## 3. Symmetry Reduction and Solution

Example 1.
Symmetry reduction of equation (7a), which admits symmetry operator (7b).
We obtain the solutions form by integratinguxx=0,(22)ut=ϕ1tx+ϕ2t.
In the following, solution (22) is inserted into equation (7a), yielding the system of ordinary differential equations as follows:(23)dϕ1dt=f1ϕ12anddϕ2dt=ϕ1f1ϕ2+f2,and the solutions are(24)ϕ1=1−f1t+c1andϕ2=f2t+c2−f1t+c1.
The solution of equation (7a) is u=x/−f1t+c1+f2t+c2/−f1t+c1.
Whenf2=0,G1=0,andG2=0, equation (7a) can turn into(25)ut=f1uux+uxxx+G2uuxxx+uxuxx,the solution is u=x+c2/−f1t+c1.Example 2.
Symmetry reduction of equation (8a), which admits operator (8b).
We obtain the solutions form by integratinguxx−a1ux=0,(26)u=ϕ1t+ϕ2teax.
Equation (8a) can be reduced to the following system of ordinary differential equations:(27)dϕ1dt=0anddϕ2dt=−12a1a12−f1ϕ1ϕ2+a1G3a12+a12+f2ϕ2,the solutions of (27) are(28)ϕ1=c1andϕ2=c2e−12a1c1a12−2G3a12−c1f1−2a12−2f2t.
So we can obtain the solution of (8a) u=c1+c2e−1/2a1c1a12−2G3a12−c1f1−2a12−2f2t+a1x. When f2=0andG3=0, the special solution is u=c1+c2e−a1c1a12−c1f1−2a12t/2+a1x.Example 3.
Symmetry reduction of equation (16a), which admits operator (16b).
By integratinguxxx−a2uxx=0, the solutions can be obtained as follows:(29)u=ϕ1t+ϕ2tx+ϕ3tea2x.
Accordingly, we can reduce the equation (16a) to the system of ODEs,(30)dϕ1dt=ϕ1ϕ2a22+ϕ2f2,dϕ2dt=ϕ22a22, anddϕ3dt=ϕ3ϕ2a22+ϕ3a2G2a22+a22+f2,the solutions of above equations are(31)ϕ1t=f2t+C3−a22t+C1,ϕ2t=1−a22t+C1,andϕ3t=C2ea2fG2a22+a22+f2−a22t+C1.
By insertingϕ1t,ϕ2t into (29), we get the solution of (16a).


## 4. Conclusion

In this paper, using the symmetry group theory and maple software, we investigated the symmetry classification, symmetry reduction, and solutions of the Kudryashov–Sinelshchikov equation for the third-order. We also obtain the polynomial and exponential solutions of the classical KS equation and enrich the results of KS equations. We will study the other type of KS equations and extend the application scope of the symmetry group method in the future.



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