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

is paper studies nonlinear wave propagation in a bubble-liquid mixture based on the third-order Kudryashov–Sinelshchikov (KS) equation. Symmetry is classied and reduced through the symmetry group method. rough the generalized conditional symmetry method, this equation is classied, and the generalized conditional symmetry considered are second, third, fourth, and fth-order, respectively. Meanwhile, the classied equations are transformed into a system of ordinary dierential equations and solved.


Introduction
e nonlinear evolution equation (NLEE) is essential to describe wave propagation in a bubble-liquid mixture, a typical nonlinear medium. e NLEE includes the Korteweg-de Vries (KdV) [1], the Burgers-Korteweg-de Vries equations. It is of signi cance 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][3][4][5][6][7][8][9]. In order to solve the NLEE, several methods were proposed, including Inverse Scattering transformation [10], symmetry approach [11][12][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. e nonlinear partial di erential equation was introduced in [1], which is a vanish dissipation de ned as follows: 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. is equaution was studied for various methods [15][16][17][18][19][20][21][22][23][24][25][26][27][28][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, is paper aims to study the following type of KS equation, where F(u) and G(u) indicate arbitrary functions. By taking some appropriate forms for F(u) and G(u), equation (3) is simpli ed to the KdV equation [24].
it was widely used to model various mechanical engineering and physical phenomena. e rest of this paper is structured as follows. Section 2 presents the symmetry classi cation for the system (1), admitting the presence of GCS. Section 3 conducts the GCS method to reduce the symmetry and nd exact solutions to the classi ed equations. Lastly, Section 4 is the conclusion.

Symmetry Classification
e symmetry group methods, especially the Lie point symmetry method, are e ectively used to obtain the exact solution of partial di erential 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][31][32]. is 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
if and only if the equation is equivalent to one of the followings: (ii) (v) Proof. According to the definition of the GCS method and the computation procedure, we obtain the following equation: where We solve the system of differential equations F 0 � 0, 0, 1, 2, 3, 4), so the classification results are shown above.

Theorem 2. e equation admits the third-order GCSs in the form
if and only if the equation is equivalent to one of the followings (ii) (iii)

Mathematical Problems in Engineering
(iv)

Theorem 3. e equation admits the fourth-and fifth-order GCSs, if and only if the equation satisfies one of the followings
(iv)

Symmetry Reduction and Solution
Example 1. Symmetry reduction of equation (7a), which admits symmetry operator (7b). We obtain the solutions form by integrating u xx � 0, In the following, solution (22) is inserted into equation (7a), yielding the system of ordinary differential equations as follows: and the solutions are 1 and When f 2 � 0, G 1 � 0, and G 2 � 0, equation (7a) can turn into the solution is u � (x + c 2 )/(−f 1 t + c 1 ). (8a), which admits operator (8b). We obtain the solutions form by integrating

Example 2. Symmetry reduction of equation
Equation (8a) can be reduced to the following system of ordinary differential equations: the solutions of (27) are So we can obtain the solution of (8a) u � c 1 + c 2 e (− 1/2)a 1 (c 1 a 2 Example 3. Symmetry reduction of equation (16a), which admits operator (16b).
By integrating u xxx − a 2 u xx � 0, the solutions can be obtained as follows: Accordingly, we can reduce the equation (16a) to the system of ODEs, Mathematical Problems in Engineering the solutions of above equations are and By inserting ϕ 1 (t, ϕ 2 (t) into (29), we get the solution of (16a).

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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.