Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

Appropriate reduction of a dynamical system is one of the important approximation approaches when the system is too complicated to be solved analytically. The derivation of a reduced dynamical system which describes the asymptotic behavior of the solution has been achieved by using various singular perturbation methods developed for ordinary differential equations, partial differential equations, and difference equations [

While the above singular perturbation methods and the Lie symmetry group analysis had been developed independently, relations between them have been studied recently. It has been shown that the asymptotic dynamics derived with the singular perturbation methods can also be derived with the Lie group analysis when it is applied to singular perturbation problems of ordinary differential equations or difference equations [

This paper reports that, by extending the method appropriately, we can reduce partial differential equations to asymptotic dynamics. The main characteristic of the extension is to add some constraint equations to the equations originally considered in order to find partial Lie symmetry groups. The extended method is applied to one ordinary differential equation and two partial differential equations in the paper. While examples taken up here are quite simple in order to make it easy to catch the essence of the procedure, this method works for all the same types of singular perturbation problems in which secular terms are included in the naïve expansion of the solution.

The singular perturbation method with the Lie group analysis [

The method is extended in this study. The procedure is briefly summarized as follows. (1) Specify beforehand a nonperturbative solution on which we would like to see the effect of the perturbation. (2) Construct equations composed of the independent and dependent variables and their derivatives based on the nonperturbative solution specified in step (1). Those equations are referred to as constraint equations in what follows. (3) Find an approximate Lie symmetry group of the original equations on the manifold formed by the constraint equations constructed in the previous step. (4) For the approximate Lie symmetry group found in step (3), construct the differential equation that is satisfied by the group-invariant solution and make an appropriate approximation when necessary. As a consequence, this differential equation generates the reduced system, which describes the asymptotic behavior of the original system.

The procedure is almost the same as that presented before for ordinary differential equations and difference equations [

The method presented in this study is particularly meaningful when it is applied to partial differential equations. However, first, we apply the method to an ordinary differential equation, because the calculation is simpler and therefore serves to clarify the essence of the method.

Consider the following ordinary differential equation [

The two independent solutions of the nonperturbative equation,

Focusing on one of these solutions,

We find a Lie symmetry group admitted by (

For this Lie symmetry group, the group-invariant solution,

The original differential equation (

In this section the present method is applied to two partial differential equations. In general, in the case of partial differential equations, because of the increase of independent variables, it often happens that a partial differential equation admits no approximate Lie symmetry group suitable for the derivation of a system describing the asymptotic dynamics. However, it is in some cases possible to find an approximate Lie symmetry group suitable for deriving the asymptotic dynamic properties by adding some constraint equations constructed from the nonperturbative solution, thereby restricting the manifold on which we find a Lie symmetry group. The reduced systems obtained here are consistent with those derived with the singular perturbation methods presented before.

Consider a system of weakly nonlinear first-order partial differential equations,

Similar to the approach followed in the previous section, we find a Lie symmetry group such that it is not admitted by the entire manifold formed by the original equation; rather, it is admitted by a part of the manifold determined from a solution of the nonperturbative system. In the following, we focus on such a nonperturbative solution,

Let

The criterion for the invariance of the original differential equation (

Suppose

The group-invariant solution,

We consider a perturbed Klein-Gordon equation represented by

As in the previous examples, a specific nonperturbative solution

Let

The criteria for the invariance of the original system on the manifold formed by the constraint equations (

In summary, this paper presents an investigation of perturbation problems, which involves the application of a particular type of Lie group method to partial as well as ordinary differential equations. As a consequence, we have obtained the dynamics which describes the asymptotic behavior of the original system. The main characteristic of the present method is the addition of some constraint equations when searching for a Lie symmetry group, an approach which allows us to find a partial Lie symmetry group. The constraint equations are determined from the nonperturbative solution on which we try to investigate the effect of the perturbation. The reason why it is necessary to specify a nonperturbative solution beforehand for the partial differential equations is that, unlike for ordinary differential equations, the nonperturbative system has an infinite number of independent solutions. In terms of technique, those constraint equations are constructed in such a way that the arbitrary constants or functions included in the nonperturbative solution are eliminated as we have done in deriving (

Examples to which the method is applied here are singular perturbation problems in which secular terms appear in the naïve expansion of the solution [

Future work should involve a clarification of the relations between the present method and other types of singular perturbation methods presented thus far, such as the WKB or boundary layer problem methods. In addition, there are existing methods that have enabled the successful derivation of the asymptotic behavior for various other partial differential equations [

Let us search for the Lie symmetry group of (

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by JSPS KAKENHI Grant no. 26800208.