Lie Group Analysis of a Forced KdV Equation

Abstract

The Korteweg-de Vries (KdV) equation considered in this work contains a forcing term and is referred to as forced KdV equation in the sequel. This equation has been investigated recently as a mathematical model for waves on shallow water surfaces under the influence of external forcing. We employ the Lie group analysis approach to specify the time-dependent forcing term.

1. Introduction

Many studies in mathematical physics, engineering, life sciences, and in all other sciences use mathematical models to describe certain phenomenon. Such models are represented by the nonlinear evolution equations. The existence of analytical solutions to these equations becomes a basis for a better understanding of the dynamics involved. The KdV equation [1] being a nonlinear evolution equation models the nonlinear wave phenomena on the shallow water surfaces, and its various forms have been proposed depending upon the applications in diverse fields of science and engineering. To date, a lot of solution procedures, both analytical and numerical, have been developed to solve these types of differential equations. However, some of these approaches may involve approximation of solutions. The current work is based upon the Lie group theory approach as a solution procedure.

The Lie point or higher-order symmetries of a differential equation enable one to obtain the solutions invariant under a particular symmetry or a linear combination of symmetries. The invariant solutions are a basis to finding exact solutions or numerical solutions. In most real-life applications, the differential equations, which are used to model a certain situation, contain arbitrary functions of dependent variable or its derivatives and independent variables. Instead of assuming the forms of these model parameters, the method of group classification can be employed to systematically specify their forms. There are various approaches to solving a group classification problem, namely, the direct analysis or the approach based upon the equivalence transformations.

We consider the forced KdV equation 𝑢_𝑡+𝑐𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=𝐹(𝑡), (1) where 𝑢 represents the elevation of free water surface and 𝛼 and 𝛽 are arbitrary constants which depend upon the long wave speed, 𝑐. The arbitrary smooth function 𝐹(𝑡) is the forcing term. The variables 𝑡 and 𝑥 represent time and space, respectively. This equation was proposed in [2] and the Hirota's bilinear approach was used to obtain the multiple soliton solutions. However, it is worth mentioning that 𝐹(𝑡) remained unspecified. In [3] the generalized wave transformation was used to transform a forced KdV equation with time-dependent coefficients into a homogeneous equation, and the soliton solutions were obtained by making use of the solitary wave ansatz. Likewise, the time-dependent coefficients remained arbitrary. The symmetry-based approach is given in [4–6]; the investigations in these references include approximate symmetry classification, derivation of some conservation laws, and the construction of the solutions. As mentioned earlier, the arbitrary functions assume their forms in a systematic fashion via the method of group classification. This approach has been utilized on similar equations [7, 8] to the underlying equation.

This work is organized as follows. The next section deals with the generation of the determining equations for Lie point symmetries and includes the classifying relation for the forcing term. The functional forms of the forcing term are specified via the direct analysis of the classifying relation. In Section 3, some results of the Lie group analysis are utilized for symmetry reductions and exact solutions. Finally, we summarize our investigations in Section 4.

2. Lie Group Analysis

In Lie's algorithm (see [9–11] for more details), a vector field 𝑋=𝜉¹(𝑡,𝑥,𝑢)𝜕_𝑡+𝜉²(𝑡,𝑥,𝑢)𝜕_𝑥+𝜂(𝑡,𝑥,𝑢)𝜕_𝑢 (2) is a generator of Lie point symmetries of (1) if and only if 𝑋^[3]𝑢_𝑡+𝑐𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥−𝐹(𝑡)∣₍₁₎=0, (3) where 𝑋^[3]=𝑋+𝜁_𝑡𝜕_𝑢𝑡+𝜁_𝑥𝜕_𝑢𝑥+𝜁_𝑥𝑥𝑥𝜕_{𝑢𝑥𝑥𝑥} (4) is the third prolongation of the vector field 𝑋. The variables 𝜁's are given by the formulae 𝜁_𝑡=𝐷_𝑡𝜂−𝑢_𝑡𝐷_𝑡𝜉¹−𝑢_𝑥𝐷_𝑡𝜉²,𝜁_𝑥=𝐷_𝑥𝜂−𝑢_𝑡𝐷_𝑥𝜉¹−𝑢_𝑥𝐷_𝑥𝜉²,𝜁_𝑥𝑥𝑥=𝐷_𝑥𝜁_𝑥𝑥−𝑢_𝑥𝑥𝑡𝐷_𝑥𝜉¹−𝑢_𝑥𝑥𝑥𝐷_𝑥𝜉², (5) where 𝐷_𝑡=𝜕_𝑡+𝑢_𝑡𝜕_𝑢+⋯,𝐷_𝑥=𝜕_𝑥+𝑢_𝑥𝜕_𝑢+⋯ (6) are the total derivative operators [10].

The invariance conditions (3) are separated with respect to the powers of the derivatives of 𝑢, and this yields the determining equations, which are a system of linear partial differential equations of homogeneous type in 𝜉¹, 𝜉², and 𝜂. It is easy but tedious to generate the determining equations manually. Nowadays there are many computer software packages for symbolic computation, which have been developed to find symmetries interactively or automatically. However, we are yet to develop the software package which solves the group classification problem whether complete or partial group classification.

The coefficients of the generator of Lie point symmetries (2), namely, 𝜉¹, 𝜉², and 𝜂, satisfy the determining equations 𝜉¹_𝑢=0,𝜉²_𝑢=0,𝜂_𝑢𝑢=0,𝜉¹_𝑥𝜂=0,_𝑥𝑢−𝜉²_𝑥𝑥=0,3𝜉²_𝑥−𝜉¹_𝑡𝜉=0,₁𝐹^′(𝑡)+𝜉¹_𝑡−𝜂_𝑢𝐹(𝑡)−(𝑐+𝛼𝑢)𝜂_𝑥−𝛽𝜂_𝑥𝑥𝑥+𝜂_𝑡(𝜉=0,𝑐+𝛼𝑢)²_𝑥−𝜉¹_𝑡−𝜉²_𝑡+𝛽𝜉²_𝑥𝑥𝑥−𝛼𝜂−3𝛽𝜂_𝑥𝑥𝑢=0, (7) where the subscripts denote partial differentiation with respect to the indicated variables, and a “prime” represents total derivative with respect to 𝑡.

The manipulation of (7) leads to the coefficients of Lie point symmetry generator given by 𝜉¹=𝑘₁𝑡+𝑘₂,𝜉²=13𝑘₁(𝑡),1𝑥+𝑎𝜂=3𝛼3𝑎^′(𝑡)−2𝑘₁(𝑐+𝛼𝑢), (8) where 𝑘₁, 𝑘₂ are arbitrary constants, and 𝑎(𝑡) is an arbitrary function which satisfies the classifying relation 5𝛼𝑘₁𝐹(𝑡)+3𝛼𝑘₁𝑡+𝑘₂𝐹^′(𝑡)=3𝑎^′′(𝑡). (9) Assume that 𝐹(𝑡) is an arbitrary smooth function of 𝑡; then from the classifying (9) we obtain 𝑘₁=0,𝑘₂=0,𝑎(𝑡)=𝑘₃𝑡+𝑘₄, (10) where 𝑘₃ and 𝑘₄ are arbitrary constants. Thus, we have a two-dimensional principal Lie algebra, which is spanned by the operators 𝑋₁=𝜕_𝑥,𝑋₂=𝛼𝑡𝜕_𝑥+𝜕_𝑢. (11) Our goal is to obtain the functional forms of the forcing term, 𝐹(𝑡), for which the principal Lie algebra is extended. Therefore, the analysis of the classifying relation (9) considering the cases 𝑎^′′(𝑡)=0 and 𝑎^′′(𝑡)≠0 yields the various forms of 𝐹(𝑡), and their corresponding extensions of the principal Lie algebra are given in Table 1. It is worth mentioning that some of the obtained symmetry classification results are comparable with those found in [4, 5]. It is, however, noted that in [6] a more general case is considered.

Table 1 

Classification results.

3. Symmetry Reductions and Exact Solutions

It can be seen from Table 1 that the symmetry Lie algebra is three-dimensional for the first three cases and four-dimensional in the last case. We consider Case 3 to illustrate the procedure involved in performing similarity reductions. Since the symmetry Lie algebra is three-dimensional, we look for solutions invariant under the linear combination of the operators 𝑋₁=𝜕_𝑥,𝑋₂=𝛼𝑡𝜕_𝑥+𝜕_𝑢,𝑋₃=3𝛼(𝑛+𝑡)𝜕_𝑡+𝛼𝑥𝜕_𝑥−2(𝑐+𝛼𝑢)𝜕_𝑢, (12) which are the symmetries of the equation 𝑢_𝑡+𝑐𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=𝐹₀(𝑛+𝑡)^−5/3. (13) In order to obtain all the possible invariant solutions, the most systematic procedure is to determine the optimal system of one-dimensional subalgebras [10, 11] for (13). We follow the approach given in [10] by firstly computing the commutators of the symmetry Lie algebra (12) and thereafter obtaining the adjoint representations (the calculations are summarized in Tables 2 and 3, resp.).

Table 2 

Table of commutators.

Table 3 

Table of adjoint representations.

We use Table 3 to simplify the linear combination of operators (12) given by Γ=𝑎₁𝑋₁+𝑎₂𝑋₂+𝑎₃𝑋₃ (14) for some constants 𝑎₁, 𝑎₂, and 𝑎₃.

Firstly, we let 𝑎₃≠0 (take 𝑎₃=1). The operator (14) becomes Γ=𝑎₁𝑋₁+𝑎₂𝑋₂+𝑋₃. (15) We eliminate 𝑎₂𝑋₂ by acting Ad(𝑒^𝑎2𝑋2) on (15) and obtain Γ^′=𝑎^′₁𝑋₁+𝑋₃ (16) for some constant 𝑎^′₁. Likewise, in order to eliminate 𝑎^′₁𝑋₁, we act on (16) by Ad(𝑒^{(𝑎′1/𝛼)𝑋1}) to get Γ^′′=𝑋₃.

Next we let 𝑎₃=0 (𝑎₂≠0). We take 𝑎₂=1, and from (14) we have Γ=𝑎₁𝑋₁+𝑋₂. (17) If 𝑎₁≠0 (i.e., 𝑎₁>0 or 𝑎₁<0 ), then we obtain Γ=𝜆𝑋₁+𝑋₂, where 𝜆=±1.

Finally, we let 𝑎₂=𝑎₃=0. The operator (14) reduces to Γ=𝑋₁ for 𝑎₁=1.

Therefore, an optimal system of one-dimensional subalgebras is given by {𝑋₁,𝑋₂,𝜆𝑋₁+𝑋₂,𝑋₃}.

Next we utilize the optimal system to construct the invariant solutions of (13). However, the invariance under space translation, that is, 𝑋₁=𝜕_𝑥, is trivial hence, it is not considered. The other cases are presented as follows.

Case 1. Invariance under 𝑋₂: the corresponding characteristic system is given by 𝑑𝑡0=𝑑𝑥=𝛼𝑡𝑑𝑢1, (18) the solution of which leads to the invariants 𝐶₁=𝑡,𝐶₂𝑥=𝑢−.𝛼𝑡 (19) Therefore, the invariant solution takes the form 𝑢()(𝑡)+𝑥𝑡,𝑥=𝑓,𝛼𝑡 (20) where 𝑓(𝑡) satisfies the reduced equation 𝑐−𝐹𝛼𝑡(𝑛+𝑡⁰)^5/3+𝑓𝑡+𝑑𝑓𝑑𝑡=0. (21) Now solving (21) for 𝑓(𝑡) and substituting into (20), we obtain the exact solution 𝑢()=1𝑡,𝑥𝑡𝑓₀+𝐹₀3+3𝑛2())𝑛+𝑡(𝑛+𝑡^1/3+𝑥−𝑐𝛼, (22) where 𝑓₀ is an arbitrary constant.

Case 2. The invariance under 𝜆𝑋₁+𝑋₂(𝜆≠0) yields the invariant solution 𝑢()(𝑡)−𝑥𝑡,𝑥=𝑓,𝛼𝑡+𝜆 (23) where 𝑓(𝑡) is an arbitrary smooth function of its argument. Upon substitution of (23) into (13) and solving the resulting ordinary differential equation (ODE), we have 𝑓(𝑡)=1𝑓𝛼𝑡+𝜆⁰−𝑐𝑡+3𝐹₀(𝜆−3𝑛𝛼−2𝑡𝛼)2(𝑛+𝑡)^2/3, (24) for an arbitrary contant 𝑓₀. Therefore, we obtain the exact solution 𝑢()=1𝑡,𝑥𝑓𝛼𝑡+𝜆⁰−𝑐𝑡−𝑥+3𝐹₀(𝜆−3𝑛𝛼−2𝑡𝛼)2(𝑛+𝑡)^2/3. (25)

Case 3. Invariance under 𝑋₃: in this case, the invariant solution assumes the form 𝑢()=𝑓(𝑧)𝑡,𝑥(𝑛+𝑡)^2/3−𝑐𝛼, (26) where 𝑧=𝑥(𝑛+𝑡)^−1/3 is the similarity variable. The function 𝑓(𝑧) is an arbitrary function which satisfies the third-order ODE 𝑑3𝛽³𝑓𝑑𝑧³+3𝛼𝑑𝑓𝑑𝑧+2𝑓+𝑧𝑑𝑓𝑑𝑧−3𝐹₀=0. (27)

4. Conclusion

In this work, the KdV equation with a forcing term was investigated using the Lie symmetry approach. The direct analysis of the classifying equation was employed to obtain the functional forms of the forcing term, which include power law and linear and constant time dependence. The three-and four-dimensional symmetry Lie algebras were obtained, respectively, for these forms of the forcing term. The optimal system of one-dimensional subalgebras of the Lie algebra of the invariant equation with the forcing term having the power law nonlinearity was obtained. As a result, for the same invariant equation, exact solutions were derived and the symmetry reduction was performed in the case where exact solutions were not obtained.

Acknowledgment

M. Molati thanks the North-West University, Mafikeng Campus, for the Postdoctoral Fellowship.