Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system

In this paper we perform Lie group classification of a generalized coupled (2+1)-dimensional hyperbolic system, viz., \begin{document}$ u_{tt}-u_{xx}-u_{yy}+f(v) = 0,\,v_{tt}-v_{xx}-v_{yy}+g(u) = 0, $\end{document} which models many physical phenomena in nonlinear sciences. We show that the Lie group classification of the system provides us with an eleven-dimensional equivalence Lie algebra, whereas the principal Lie algebra is six-dimensional and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary functions \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} , the forms of which include, amongst others, the power and exponential functions. Finally, for three cases we carry out symmetry reductions for the coupled system.

1. Introduction. The blow up problem for positive solutions of parabolic and hyperbolic problems with reaction terms of local and nonlocal type involving a variable exponent was studied in [9]. Parabolic problems appear in many branches of applied mathematics and can be used to model, for example, chemical reactions, heat transfer and population dynamics (see [9] and references therein). Escobedo and Herrero [1] extended the work of [9] and studied the system of equations where p, q are arbitrary constants and investigated the boundedness and blow-up of its solutions. The uniqueness and global existence of solutions of the system (1) was studied in [2]. Recently, the authors of [4] considered nonlinear parabolic and hyperbolic systems with variable exponents and obtained results concerning the existence and blow-up property of solutions. Inspired by the works done in [1,2,4], more recently the authors of [7] studied the coupled (2+1)-dimensional hyperbolic system where q, p, α and β are non-zero constants. A complete Noether symmetry classification was carried out in [7] and it was shown that four main cases arose in the Noether classification with respect to the standard Lagrangian. The conservation laws were also constructed for the cases which admitted Noether point symmetries.
In this work we consider the generalization of the system (2), namely where f (v) and g(u) are nonzero arbitrary functions of their respective arguments. The aim of this work is to perform Lie group classification of the system (3).
2. Equivalence transformations. An equivalence transformation (see for example [5]) of (3) is an invertible transformation involving the independent variables t, x, y and the dependant variables u and v that map (3) into itself. The vector field is the generator of the equivalence group for (3) provided it is admitted by the extended system [6,8] The prolonged operator of (4) for the extended system (5)-(6) is given by where Y [2] is the second-prolongation of (4) given by Here the variables ζ's and ω's are defined by and are the usual total differentiation operators and are the new total differentiation operators for the extended system. The application of the operator (7) and the invariance conditions of system (5)-(6), after some lengthy calculations, leads to the following equivalence generators: which yields the eleven-parameter equivalence group given by The composition of the above transformations gives t = a 1 + a 4 x + a 5 y + te a9 , x = a 2 + a 4 t − a 6 y + xe a9 , y = a 3 + a 5 t + a 6 x + ye a9 , which are the equivalence transformations.
3. Principal Lie algebra. According to Lie's theory the system of partial differential equations (PDEs) (3) is invariant under the group with generator if and only if where Γ [2] denotes the second prolongation of the generator (10) and the symbol | (3) means that it is evaluated on system (3). As the ξ's and η's do not depend on any derivatives of u and v, the determining equations (11) split with respect to the derivatives of u and v, yielding the following overdetermined system of thirty-one linear PDEs: Solving the above system for arbitrary f and g, we find that the system (3) admits the six-dimensional Lie algebra spanned by which is the principal Lie algebra of the system (3).
These classifying relations lead to twelve cases for the functions f and g and for each case we also provide the associated extended Lie point symmetries.
Case 1. f (v) and g(u) arbitrary but not of the form in Cases 2-12 given below.
In this case, we obtain the principal Lie algebra Case 2. f (v) = nv + σ and g(u) = mu + θ, where n, σ, m and θ are constants This case extends the principal Lie algebra by four Lie point symmetries, namely where H(t, x, y) is any solution of the PDE and C 1 , C 4 , C 8 are arbitrary constants.
Case 3. f (v) = αv n and g(u) = θu m , where α, n, θ and m are constants We have four subcases. The principal Lie algebra is extended by one Lie point symmetry Case 3.2. n = m = −1 This subcase extends the principal Lie algebra by two Lie point symmetries, viz., Case 3.3. mn = 1, where m and n are non-zero constants Here the principal Lie algebra extends by one Lie point symmetry Case 3.4. n = 5 and m = 5 In this subcase the principal Lie algebra extends by the following four Lie point symmetries: Case 4. n = −1 and g(u) is arbitrary This case extends the principal Lie algebra by one Lie point symmetry Case 5. f (v) is arbitrary and m = −1 Here the principal Lie algebra extends by one Lie point symmetry Case 6. f (v) = αe nv and g(u) = θe mu , where α, n, θ and m are constants This case extends the principal Lie algebra by one Lie point symmetry Case 7. f (v) = αv n and g(u) = θe mu , where α, n, θ and m are constants This case extends the principal Lie algebra by one Lie point symmetry Case 8. f (v) = αe nv and g(u) = θu m , where α, n, θ and m are constants This case extends the principal Lie algebra by one Lie point symmetry Case 9. f (v) = nv + σ and g(u) = θu m , where n, σ, θ and m are constants with m = n = 1 In this case the principal Lie algebra extends by one Lie point symmetry Case 10. f (v) = αv n and g(u) = mu + θ, where α, n, m and θ are constants with m = n = 1 This case extends the principal Lie algebra by one Lie point symmetry Case 11. f (v) = nv + σ and g(u) = θe mu , where n, σ, θ and m are constants This case extends the principal Lie algebra by one Lie point symmetry Case 12. f (v) = αe nv and g(u) = mu + θ, where α, n, m and θ are constants This case extends the principal Lie algebra by one Lie point symmetry 5. Symmetry reductions of system (3). In this section, we present a few symmetry reductions of system (3) using some of the symmetries derived in Section 3 [3]. We first consider Case 1 and use the rotational symmetry Γ 6 to perform symmetry reduction of system (3). The associated Lagrange system of Γ 6 is yields the invariants z = t, r = x 2 + y 2 and the group-invariant solution of system (3) is u(t, x, y) = F (z, r), v(t, x, y) = G(z, r), where F (z, r) and G(z, r) satisfy As a second example of symmetry reduction of system (3) we consider Case 3.1 with f (v) = αv n and g(u) = θu m and use the rotational symmetry Γ 5 . In this case the associated Lagrange system of Γ 5 is dt which gives the four invariants z = t, r = t 2 − y 2 , u = F (z, r), v = G(z, r). Following the same procedure as above, system (3) reduces to derive the invariants z = t, r = t 2 − x 2 , u = G(z, r), v = H(z, r). Invoking these