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Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations

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Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations mathematics Article l-Symmetry and m-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations Yu-Shan Bai 1,2,*, Jian-Ting Pei 1 and Wen-Xiu Ma 2,3,4,5,* 1 Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China; [email protected] 2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA 3 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China 4 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 5 School of Mathematics, South China University of Technology, Guangzhou 510640, China * Correspondence: [email protected] (Y.-S.B.); [email protected] (W.-X.M.) Received: 9 June 2020; Accepted: 9 July 2020; Published: 12 July 2020 Abstract: On one hand, we construct l-symmetries and their corresponding integrating factors and invariant solutions for two kinds of ordinary differential equations. On the other hand, we present m-symmetries for a (2+1)-dimensional diffusion equation and derive group-reductions of a first-order partial differential equation. A few specific group invariant solutions of those two partial differential equations are constructed. Keywords: l-symmetries; m-symmetries; integrating factors; invariant solutions 1. Introduction Lie symmetry method is a powerful technique which can be used to solve nonlinear differential equations algorithmically, and there are many such examples in mathematics, physics and engineering [1,2]. If an nth-order ordinary differential equation (ODE) that admits an n-dimensional solvable Lie algebra of symmetries, then a solution of the ODE, involving n arbitrary constants, can be constructed successfully by quadrature. If a partial differential equation (PDE) admits a Lie point symmetry, then its dimension can be reduced by one, and further its group invariant solution can be systematically constructed. However, there exist some kinds of differential equations which have trivial Lie point symmetries or have no symmetry, and Lie symmetry method cannot be applied directly. It is also known that the existence of nontrivial Lie point symmetries is not necessary for guaranteeing the integrability by quadrature for differential equations [3,4]. In 2001, a new kind of symmetries, called l-symmetries, was introduced by Muriel and Romero [3]. Indeed, ODEs which have trivial Lie point symmetries or no symmetry but possess l-symmetries can be integrated by means of the l-symmetry approach. l-symmetries can also be used to construct first integrals and integrating factors of such equations [5,6]. Gaeta and Morando considered the case of PDEs, and extended l-symmetries to m-symmetries [7,8]. It was proved that m-symmetries are as useful as standard symmetries in respect to symmetry reductions, and the determination of invariant solutions by using m-symmetries is completely similar to the standard one in the Lie symmetry method (see, for example, [9–16] for many other interesting applications and theoretical developments about l- and m-symmetries). Both l-symmetries and m-symmetries are generalizations of Lie point symmetries, which could be viewed as Lie point symmetries of integrable couplings [17], and provide new insights into the development of the Lie symmetry theory. The determination of both symmetries depends on the Mathematics 2020, 8, 1138; doi:10.3390/math8071138 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1138 2 of 13 prolongation formula that generalizes the standard Lie symmetry prolongation of vector fields. The most outstanding factor is that the determining equations are nonlinear, and so calculations are much more complicated. In this paper, we use the package of the differential characteristic set method [18,19] and symbolic computing systems to determine the existence of generalized symmetries and to simplify the corresponding determining equations. The differential characteristic set method, developed by Wentsun Wu [20] in the 1970s, is a fundamental algorithmic method, together with the Gro¨bner base algorithm. The method is very effective in calculating both classical and non-classical symmetries (for further applications, please refer to [21]). This paper is structured as follows. In Section2, we calculate l-symmetries of two kinds of second-order ODEs and construct their integrating factors and invariant solutions by using the obtained l-symmetries. In Section3, we generate m-symmetries of two different PDEs and construct some invariant solutions of the equations through applying the obtained m-symmetries. In Section4, we are devoted to providing some concluding remarks. 2. l-Symmetries of Ordinary Differential Equations 2.1. The Basic Concept of l-symmetries Consider an n-th order ordinary differential equation (ODE) D(x, u(n)) = 0, (1) (k) where (x, u ) = (x, u, u1, ··· , uk) and for i = 1, ..., k, ui denotes the derivative of order i of the dependent variable u with respect to the independent variable x. The canonical form of this equation reads as follows u(n) = Y(x, u(n−1)). (2) ¶ ¶ Recall [3] that if v = x(x, u) ¶x + h(x, u) ¶u is a vector field on M, where M is an open subset of the independent and dependent variables, and l is an arbitrary smooth function defined on the jet space C¥(M(k)), then the l-prolongation of order n of v, denoted by v[l,(n)], is the vector field defined on M(n) by ¶ n ¶ v[l,(n)] = x(x, u) + ∑ h[l,(i)](x, u(i)) , (3) ¶x i=0 ¶ui where h[l,(0)] = h(x, u) and [l,(i)] (i) [l,(i−1)] (i−1) h (x, u ) = Dx(h (x, u )) − Dx(x(x, u))ui (4) [l,(i−1)] (i−1) + l(h (x, u ) − x(x, u)ui), for 1 ≤ i ≤ n, where total derivative Dx = ¶x + ux¶u + uxx¶ux + ··· . If there exists a function l 2 C¥(M(k)) such that [l,(n)] (n) v (4(x, u ))j4(x,u(n))=0 = 0, (5) we will say that a vector field v, defined on M, is a l-symmetry of the Equation (1). Obviously, if l = 0, the l-prolongation of order n of v is exactly the classical nth prolongation of v [1]. Mathematics 2020, 8, 1138 3 of 13 2.2. Applications of l-Symmetries 2.2.1. l-Symmetries Reductions and Integrating Factors without Using Lie Symmetries Consider the following ordinary differential equation u2 x 1 u = x + u ( + ) + axu, a 2 R. (6) xx u x u3 x We can use the differential characteristic set method [18] to determine that this equation has no Lie point symmetries easily. Assume that l-symmetry generator of Equation (6) is ¶ ¶ v = x(x, u) + h(x, u) , ¶x ¶u and the second prolongation formula is of the form ¶ ¶ ¶ ¶ v[l,(2)] = x + h + h[l,(1)] + h[l,(2)] . ¶x ¶u ¶ux ¶uxx From Equations (5), we know that v satisfies the following l-symmetry condition: 2 [l,(2)] ux x 1 v [u − − u ( + ) − axu] j 2 = 0. (7) xx x 3 ux x 1 u u x uxx− −ux( + )−axu=0 u u3 x The determining equation of (6) is 2 [l,(2)] [l,(1)] 2ux 1 x ux 3xux 1 1 h + h (− − − ) + h( + − ax) + x(−ux( − ) − ax) = 0, (8) u x u3 u2 u4 u3 x2 where [l,(1)] 2 h = hx + (hu − xx − lx)ux − xu(ux) + lh, [l,(2)] 2 h = hxx + lxh + 2lhx + l h + (2hxu − xxx − 2lxx − lxx + luh 2 + 2lhu − l x)ux + (hu − 2xx − 2lx + lux h)uxx + (huu − 2xxu 2 3 − lux − 2lxu)(ux) − (3xu + lux x)uxuxx − xuu(ux) . Substituting the above h[l,(1)], h[l,(2)] into the Equation (8), one can get a set of over-determined homogeneous differential equations for x, h 8 xu xlux − − xuu − = 0, > u u > h hu 2xu 2xxu hlux xlux xxlux > 2 − − − 3 − 2lxu + huu − 2xxu + − − 3 − xlu = 0, > u u x u u x u <> 3xh − 2lh − x + x − lx − xlx − 2 + − xu − 2hx − xx − xxx u4 u u3 x2 x u3 l x 2lhu 3a xu u x u3 hlux xhlux > −2lxx + 2hxu − xxx + + − axuxlu + hlu − xlx = 0, > x u3 x > − x − lh − xlh + 2 − u − xu + xu − hx − xhx + > a h x u3 l h a x 2a lx a hu x u3 2lhx :> −2axuxx + hxx + axuhlux + lxh = 0. It can be checked that these equations, whose unknowns are x, h and l, admit the solution = = = x = x = ¶ x 0, h u, l u3 . Hence, if l u3 , the vector field v u ¶u is a l-symmetry of Equation (6). Now, we use the prolongation formula (4) to construct invariant solutions. We can determine [l,(2)] = x v with l u3 and obtain ¶ x ¶ x2 1 xu ¶ [l,(2)] = + ( + ) + ( + + x + ) v u ux 2 5 2 3 uxx . ¶u u ¶ux u u u ¶uxx Mathematics 2020, 8, 1138 4 of 13 It can be checked that u x y = x, w = x + , u 3u3 are two functionally independent invariants for v[l,(1)]. Upon calculating an additional invariant by derivation [1] D w u − 3xu − 3u2u2 + 3u3u = x = x x xx wy 4 , Dxy 3u Equation (6) can be reduced to the equation of y, w, wy, w w − − ay = 0. (9) y y Solving (9), one can get 2 w = ay + c1y, c1 2 R. We recover the invariant solution of Equation (6) by solving the auxiliary first-order differential equation 2 2 3 3 3u ux + x − 3ax u − 3c1xu = 0. (10) Let u˜ = u3. The equation of (10) turns into 2 u˜x + x − 3ax u˜ − 3c1xu˜ = 0, (11) and by integrating this equation, we get the invariant solution of the Equation (6): Z x 3 2 3 3 2 3 1 u = [exp( c1x + ax )(c2 − exp(− c1 p − ap )pdp)] 3 , 2 1 2 where c1, c2 are arbitrary constants.
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