New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group method

Abstract In this paper, a generalized variable-coefficients KdV equation (gvcKdV) arising in fluid mechanics, plasma physics and ocean dynamics is investigated by using symmetry group analysis. Two basic generators are determined, and for every generator, the admissible forms of the variable coefficients and the corresponding reduced ordinary differential equations are obtained. Finally, by searching for solutions to those reduced ordinary differential equations, many new exact solutions for the gvcKdV equation have been found.


Introduction
This paper is devoted to studying the generalized variable-coefficients KdV equation (gvcKdV), which is given by (Wang, 2006) u t þ g 1 u xxx þ g 2 u 3 þ g 3 u 2 þ g 4 u þ g 5 À Á u x þ g 6 u þ g 7 ¼ 0; (1) where, g i ðtÞ with i ¼ 1; 2; . . . 7 are arbitrary functions of t. When g 2 ¼ 0; Equation (1) is derived by considering the time-dependent basic flow and boundary conditions from the well-known Euler equation with an earth rotation see (Tang, Huang, & Lou, 2006) and analytical solitonic solution is obtained for it by considering g 3 ¼ g 5 ¼ g 7 ¼ 0, which means that Equation (1) not really solved but only the known famous variable coefficients KdV equation. Furthermore, many physical and mechanical situations governed by Equation (1)

Symmetry method
Recently, many methods have been investigated to deal with nonlinear partial differential equations like bilinear representation, B€ acklund transformation methods (El-Shiekh, 2015;L€ u & Peng, 2013;L€ u, Lin, & Qi, 2015a;L€ u & Lin, 2016), tanh function and sinecosine methods (Bibi & Mohyud-Din, 2014;El-Shiekh, 2015;El-Wakil, Abulwafa, El-hanbaly, El-Shewy, & Abd-El-Hamid, 2016;Hu et al., 2016;Moussa & El-Shiekh, 2011), direct reduction method (El-Shiekh, 2012, 2017 and the symmetry group analysis (El-Sayed, Moatimid, Moussa, El-Shiekh, & El-Satar, 2014;El-Sayed et al., 2015;Moatimid, El-Shiekh, & Al-Nowehy, 2013;Moussa & El-Shiekh, 2010, 2012. Symmetry method is one of the new modification of Lie group analysis; it is more easy and simple in calculations than Lie method (Moatimid et al., 2013;Wang, Liu, & Zhang, 2013;Wang, Kara, & Fakhar, 2015;Wang, 2016) and can be briefly described in the following steps: Suppose that the differential operator L can be written in the form where, u ¼ uðt; xÞ and H may depend on t, x, u and any derivative of u as long the derivative of u does not contain more than ðp À 1Þ; t derivatives. We will consider the symmetry operator (called infinitesimal symmetry) in the form (3) and the Fr echet derivative of L(u) is given by With these definitions, we will compute the following: i. FðL; u; vÞ; ii. FðL; u; SðuÞÞ; iii. Substitute H(u) for ( o p u ot p Þ in FðL; u; SðuÞÞ; iv. Set this expression to zero and perform a polynomial expansion; v. Solve the resulting partial differential equations. Once this system of partial differential equations is solved for the coefficients of S(u), Equation (2) can be used to obtain the functional form of the solutions.

Determination of symmetries
In order to find the symmetries of Equation (1), we set the following symmetry operator Calculating the Fr echet derivative F L; u; v ð Þ of L u ð Þ in the direction of v, given by Equation 4 ð Þ, and replacing v by S u ð Þ in F, we get Substituting the values of different derivatives of S(u) in F with the aid of Maple program, we get a polynomial expansion in u x ; u t ; u y ; u x u t , … ,etc. On making use of Equation (1) in the polynomial expression for F, rearranging terms of various powers of derivatives of u and equating them to zero, we obtain On solving system 7 ð Þ; the infinitesimal A, B and C in the above equations are: where, c i ; i ¼ 1; 2; . . .; 5; are arbitrary constants. The functions g i ¼ g i t ð Þ; i ¼ 1; 2; . . .; 7; are governed by the following equations: Ag 7 ð Þ t þ c 4 g 7 þ c 5 g 6 ¼ 0: The symmetry Lie algebra of Equation (1) is generated by the operators and the commutator table of it is given by Now, we are going to search for a one-dimensional optimal system of the Lie algebra generated by the operators (10) as follows: Consider a general element of V ¼ X 5 i¼1 a i V i , and checking whether V can be mapped to a new element V Ã under the general adjoint transformation ; to simplify it as much as possible.
The adjoint table Following Olver (1986), we can deduce the following basic fields which form an optimal system for the gvcKdV, where, k i ; i ¼ 1; . . .; 4 are arbitrary constants. The cases (iii), (iv) and (v) give trivial reductions. Therefore, we will discuss the first and second case by only using the following characteristic equation: By solving Equation (11) for both generators (I) and (II) where, n 1 ; . . .; n 6 and m 1 ; . . .; m 6 are arbitrary constants.

Reductions and exact solutions
In this section, the primary focus is on the reductions associated with the two generators (i) and (ii), and their solutions.

Generator (I)
Corresponding to this generator, the gvcKdV is reduced to the following ordinary differential equation.
3F 000 þ n 1 F 3 F 0 þ n 2 F 2 F 0 þ n 3 FF 0 þ ðn 4 À fÞF 0 þ ðn 5 À k 1 ÞF þ n 6 ¼ 0: To solve Equation (12), we seek a special solution in the form where, A 0 and A 1 are arbitrary constants to be determined. Substituting Equation (13) into Equation (12) and equating the coefficients of different powers of f to zero, we get a system of algebraic equations, solutions of which give rise to the relations on the constants as Finally, we get the following exact solution for the gvcKdV

Generator (II)
The reduced nonlinear ordinary differential equation corresponding to this case is Herein, we apply the modified extended tanh function method (El-Shiekh, 2015) to obtain rational exact solitary wave solutions to Equation (16). Let us assume that Equation (16) has a solution in the form where, / f ð Þ is a solution of the following Riccati equation Substituting Equation (17) into Equation (16) and by balancing the linear term with the greatest nonlinear term, we get Therefore, The similarity variable f Similarity solution u x; t ð Þ integrability conditions The first generator Substituting Equation (21) into Equation (16) and equating the powers of / j ; j ¼ 0; À 11 3 ; À3; À 7 3 ; . . . to zero, we obtain a system of algebraic equations. By solving that system with maple program yields the following solution