Abstract
We carry out a classification of Lie symmetries for the ()-dimensional nonlinear damped wave equation with variable damping. Similarity reductions of the equation are performed using the admitted Lie symmetries of the equation and some interesting solutions are presented. Employing the multiplier approach, admitted conservation laws of the equation are constructed for some new, interesting cases.
1. Introduction
The wave equation is a mathematical model that is used to describe the propagations of waves as they occur in physics. It arises in many fields like acoustics, electromagnetism, fluid mechanics, general relativity, hydrodynamics, and quantum mechanics [1, 2]. Therefore, investigating the wave equations remains one of the hottest areas of research in applied mathematics.
Historically, the first serious attempt to understand wave behavior dates back to the sixth century BC when Pythagoras studied the properties of sound waves produced by a string in musical instrument [3]. During the scientific revolution, rapid advances in understanding waves were made due to the works of many prominent scientists such as Bynam et al. [4]. Furthermore, a revolutionary change in human perception of wave phenomena took place in the nineteenth century when Maxwell formulated his electromagnetic field theory [5].
In fact, mathematicians focused their attention in investigating the formation and motion of waves. Thus, extensive studies have been conducted to obtain exact solutions to both linear and nonlinear wave equations. In this context, Cajori and Farlow obtained the first exact solution to the linear wave equation [6, 7]. Further researches have been conducted to investigate the nonlinear wave equations where different approximate and numeric methods have been developed; see, for example, [8–15].
Nevertheless, the classical wave model lacks the ability to predict the behavior of wave phenomena arising in physical systems under certain circumstances. Such situations are frequently met when dealing with waves whose behavior exhibits diffusion phenomena which can only be described using the damped wave equation.
A damped wave in general is a wave whose amplitude of oscillation decreases with time. This damping is expressed mathematically by coupling an extra term to the classical wave equation. The additional term has the effect of controlling the speed of oscillation [16]. Thus, the damped equation model can describe effectively many physical systems involving processes that dissipate the energy stored in the oscillation. In order to understand these physical systems, the damped wave equations need to be investigated.
In fact, many authors have studied different classes of damped wave equations by investigating their solutions or by studying their asymptotic behavior [17–19]. Some other researchers adopted the numerical approach in tackling this type of equations [20]. However, studying damped wave equations in terms of Lie point symmetries admitted by them was not sufficiently addressed in literature. Indeed, attention was focused so far on classifying symmetries of classical wave equations. In this context, a lot of work has been published; see, for example, [21–23]. Ibragimov extended this kind of work to the damped wave equations. For example, he studied different forms of unperturbed damped wave equations by classifying their Lie symmetries [23]. Some other researchers implemented the symmetries of certain classes of damped wave equations to obtain analytical solutions [24].
Indeed, one of the main applications of symmetries is the construction of conservation laws for a given system. The link between conservation laws and symmetries was first introduced by Noether in 1918 [25, 26]. Although Noether’s theorem provides a very powerful method for obtaining conservation laws, it has a limitation in the sense that it is only applicable for variational PDEs as it requires the existence of a Lagrangian. Therefore, the multiplier approach has been developed for deriving conservation laws for variational and nonvariational PDEs. The procedure of obtaining conservation laws using the multiplier approach is explained in many references; see, for example, [27, 28].
The aim of this paper is to extend the investigation of damped wave equations of Ibragimov [22] by considering the ()-dimensional nonlinear damped wave equation given bywhere the equation will be classified in terms of Lie point symmetries it admits. These symmetries will be utilized in performing different similarity reductions and, where possible, exact solutions will be obtained. Furthermore, the obtained symmetries will be exploited to construct conservation laws for some cases of interest by applying the multiplier approach.
2. Symmetry Classification of the Damped Wave Equation
We will consider (1) which can be equivalently written in the following form:A symmetry of a differential equation in general is a transformation that keeps its family of solutions invariant. To obtain the symmetry algebra of (2), we take the infinitesimal generator of symmetry algebra of the formwhere the coefficients , , , and are functions of , , , and . Using the invariance condition, that is, applying the 2nd prolongation to (2), yields the following system of determining equations:Since we are interested in classifying Lie symmetries and corresponding solutions of (2), we start by writing (13) in the formFrom the above equation, two cases are considered; namely, and . We discuss each case separately as follows.
Case 1 (). Substituting in the overdetermined system, (4)–(13), leads to the following solution:The above solution is valid for arbitrary and arbitrary and consequently leads to the minimal subalgebra group of vector fields
Case 2 (). In this case we can reconsider (14) by writing in the formSince the left hand side of (16) depends only on , it can be put in the formAt this stage, we consider three possibilities; namely, , , and functionally depending on .
Case 2.1 (). In this case, (17) can be put in the formFrom the above equation it is instantly found that while . Substituting for in (10) and differentiating the resulting equation with respect to giveThus, three cases arise from (19); namely, (1) , , (2) , , and (3) , which we consider one by one.
Case 2.1.1 (, ). Differentiating (8) with respect to givesSince , the above equation can be written asNotice that (21) generates three possibilities; , , and .
Case 2.1.1a (). Hence, (21) becomesSolving (22) and substituting the solution in the overdetermined system (see (4)–(13)) produce the following solution:with such that . The generators of the five-dimensional symmetry group are generated by vector fields given by
Case 2.1.1b (). In this case (21) takes the formFollowing the method adopted in Case 2.1.1a leads to the following solution of the overdetermined system equations (4)–(13):with and and . The corresponding algebra of the -symmetry group is spanned by
Case 2.1.1c (). To deal with this case, we first differentiate (21) with respect to to obtainIt is straightforward to solve the above equation to getSubstituting (27) in (4) yieldsHence, (27) becomesAlso, (21) can be recast asSolving the above equation immediately yieldswhere . After some more manipulation, the following solution of the overdetermined system equations (4)–(13) is obtained:with . Accordingly, the vector fields generating a five-parameter Lie group are given by
Case 2.1.2 (, ). In this case, . Hence, since both functions and are constant; then we expect this case to give the maximal set of Lie symmetries. After carrying out some calculations it turns out that the obtained solution of the determining system iswhere satisfies the differential constraint given byAs a result, this case admits the -group Lie symmetries given by along with an arbitrary symmetry
Case 2.1.3 (). It turns out that this case leads to finite six-dimensional symmetry algebra which is a subset of the algebra obtained in Case 2.1.2.
Case 2.2 (). In this case (17) takes the formwhich can be easily integrated over to giveAlso, solving (35) for yieldsUsing (37) in the overdetermined system, (4)–(13), and after manipulating further leads to the following solution:subject to , where . The symmetry algebra for this case is generated by the following five vector fields
Case 2.3 (). In this case, we write (17) asDifferentiating the above equation with respect to and solving immediately givewhere is an integration function. Substituting (40) in (4) givesConsequently, (40) becomesAfter some more manipulations, (42) reduces toUsing (41) in (17) giveswhich on integration over becomesFurthermore, simple calculations show that is a function of only. At this point, substituting (43) in (8) and differentiating the resulting equation with respect to we obtainAt this stage we consider two possibilities; namely, and .
Case 2.3.1 (). After some manipulations the following solution is obtained:where , provided that and . As a result, the symmetry algebra for this case is constructed, generating a five-parameter group .
Case 2.3.2 (). This case suggests that . Then, carrying out further calculations, one arrives at the following results:Substituting (48) in (8) yieldsIt is obvious that (49) leads to the following two equations:As a result, (51) gives rise to two possibilities. Namely, and .
Case 2.3.2a (, ). Substituting (50) and the value of in (48) givesConsequently, the symmetry algebra is extended by two extra symmetries: , where . Notice also that (45) in this case takes the form .
Case 2.3.2b (, ). The solution of the overdetermined system for this case takes the formwhich generates a five-dimensional symmetry group . Finally, our calculations have shown that the case with is not interesting as it does not generate extra symmetries.
At this stage, we conclude Section 2 where it is shown that (2) has at least four-dimensional symmetry algebra which is given by the group . In this section we have presented the particular forms of and that admit extra symmetries. Table 1 shows the extra symmetries obtained for each case arising in the classification.
3. Reduction of the Damped Wave Equation
In this section we briefly discuss solutions of considered damped wave equation (2) by reduction via symmetry algebra. In particular, we will use two-dimensional subalgebras to reduce the considered equation to an ODE. The procedure of reducing a PDE using its symmetries is standard and it is explained in detail in many references; see, for example, [29]. In the context of this paper, the procedure of performing the reduction will be shown in detail for the general form of (2) where the functions and are arbitrary. For particular forms of and , the reduced equations and the corresponding similarity variables are given in Table 2.
Notice that the commutation relations between elements of are . Thus, we have four subalgebras of two dimensions. We will perform reduction for each subalgebra as follows.
(a) Reduction under the Subalgebra . Solving the characteristic system, it is straightforward to find the following similarity variables:In the light of these similarity transformations, (2) is easily reduced to the formThe second level of reduction is performed using the symmetry . Hence, the symmetryis inherited by (55). This inherited symmetry is used to obtain the new similarity variables. Thus, again, we solve the characteristic system to obtain the following similarity variables:Using the new similarity variables, (55) is reduced to
(b) Reduction under the Subalgebra . Following the same procedure as in case (a) we reduce (2) again into (58) but with the following similarity variables: and .
(c) Reduction under the Subalgebra . The damped wave equation, that is (2), in this case is reduced toThe corresponding similarity variables are and .
(d) Reduction under the Subalgebra . In this case (2) is reduced towhere and .
Since reductions performed are valid for arbitrary functions and , we can use them to obtain exact solutions for all particular forms of and arising from the classification. For example, if we consider the case where and , then we can obtain exact solution for damped wave equation (2) by first substituting for in (59) to obtainThen, solving (61) we getFinally, in order to express the solution in terms of the original variables of (2), we transform each similarity variable in (62) to its corresponding variable given in case (c) above; that is, we substitute for and in solution (62). This substitution gives rise to a static exact solution of (2), given byFurthermore, we can perform more reductions by considering combination of symmetries. For instance, if we use the subalgebra , then (2) is reduced intowhere and . In particular, if we choose and then (64) becomesThe solution of (65) is given by . This solution is equivalent to the following solution of (2):Furthermore, if we choose and then (64) becomesSolving (67) we obtain , which gives rise to the following exact solution of (2):Exact solutions for different forms of the studied equation are provided in Table 3.
4. Conservation Laws
A conservation law of a system of PDEs in general is a divergence expression which vanishes on solutions of the system. Keeping in mind that (2) does not possess a Lagrangian, we implement the alternate multiplier approach to construct conservation laws for some interesting cases that have been obtained in Section 2. This powerful approach can be summarized in two steps as follows.(i)Determine a set of conservation laws multipliers so that a linear combination of the PDEs with the multipliers yields a divergence expression.(ii)Find the corresponding flux/densities to construct the conservation laws.
A conserved form of a partial differential equation is represented by a two-form differential operator for which is a three-form operator which vanishes on the solutions of the PDE ( is the total exterior derivative). The multiplier method to determine these is detailed in [28], inter alia.
In the remaining part of this paper we will briefly give the multipliers and the corresponding conservation laws resulting from the multiplier approach. The cases will be presented in the same order followed in Section 2 where for each of the cases and denotes the conserved fluxes in and directions, respectively, while denotes the conserved densities.
Case 1. This general case admits nine distinct multipliers. In particular, the multipliers and are admitted. That is, in terms of real valued functions, we get and . Notice that we have taken into account that all ’s and ’s are interchangeable for this general case. For the remaining cases we will show the extra multipliers and their corresponding conserved vectors.
Case 2. This case admits the multiplier . The corresponding conserved flow is given byFurthermore, the multiplier is also admitted. It gives rise to the following conserved current:
Case 3. We have two multipliers for this case. The first one is . The corresponding conserved vector is given byThe second multiplier is with corresponding conserved vector
Case 4. This case admits two multipliers. The first one is . It leads to the following conserved flow:The second multiplier is . It gives rise to the following conserved flow:
Case 5. This case admits two multipliers given byfrom which the conservation law is too tedious to construct.
Case 6. We have four multipliers given byIn particular, the fourth multiplier gives rise to the following conserved flow:Notice that the last case admits the additional multipliers and .
Case 7. This case admits six multipliers given by
5. Conclusion
In this paper, we carried out a symmetry classification of the ()-dimensional nonlinear damped wave equation. It has been shown that the minimal subalgebra admitted by the equation is four-dimensional. In some interesting cases, the subalgebra can be extended by one to three extra symmetries. Reductions of the studied equation have also been performed using two-dimensional subalgebras where some exact solutions have been obtained. Furthermore, the multipliers have been provided for most of the cases discussed in the paper while conservation laws have been constructed for some cases of interest.
The study presented in this paper paves the way for further investigation regarding nonlinear damped wave equations arising in mathematical physics or other scientific fields. In particular the () nonlinear damped wave equation could also be considered.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.