Partially Nonclassical Method and Conformal Invariance in the Context of the Lie Group Method

Abstract

The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the ‘classical’ method, in which the invariant surface condition is not used, and from the ‘nonclassical’ method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the ‘classical’ and ‘nonclassical’ methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods ($\lambda$-formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example.

1. Introduction

In the Lie group method of finding solutions of partial differential equations (PDEs), the infinitesimal group generators are defined as solutions of ‘determining equations’ obtained from the condition of invariance of a PDE under the group transformations (see, for example, [1,2,3]). Knowing the infinitesimal invariance transformations for a given PDE allows solutions of the PDE (invariant solutions) to be defined. The group generators can also be used for defining the finite invariance transformations, which, in particular, allows new solutions of the PDE to be derived from the known ones. A generalization of the above outlined method (classical Lie group method) for finding invariant solutions has been introduced in the seminal paper of Bluman and Cole [4]. This generalization, named the nonclassical method, is based on the observation that the invariant surface condition, representing the first-order PDE, solutions of which are the invariant solutions, stands separately from the invariance requirement for the original PDE and so can be used to rearrange the differential polynomial obtained from the invariance requirement. This does not influence the possibility of finding invariant solutions using the group generators defined on the basis of the rearranged polynomial. At the same time, the group generators, obtained on the basis of the rearranged polynomial, may be different from those yielded by applying the classical method, and so may lead to new solutions of the PDE under consideration. However, the infinitesimal generators, obtained by applying the nonclassical method, cannot be used for finding the group transformations since rearranging the invariance condition polynomial spoils the group property on which the procedure of defining that polynomial is based.

Although applying the procedure of the nonclassical method should, in principle, enlarge the set of solutions for the infinitesimal group generators, as compared with that of the classical method, rearranging the differential polynomial results in a substantial complication, namely, the determining equations for the group generators become nonlinear. Thus, unlike the case of the classical method, the procedure of defining solutions by the nonclassical method is not completely algorithmic. In particular, Bluman and Cole, applying their method to the linear heat equation in [4], were not able to solve the determining equations. Therefore, the method seemed to be unpractical and had not been used until the late 1980s when several authors independently showed that the Bluman–Cole method, being applied to some nonlinear PDEs, can result in a solvable system of determining equations which yields new solutions of the PDEs, unobtainable via the classical Lie group method [5,6,7,8]. In particular, the relations between the nonclassical method and the direct method of Clarkson and Kruskal [9] have been established [6,7].

Those findings initiated numerous studies, applying both the nonclassical method and the direct method of Clarkson and Kruskal to PDEs arising in different areas of mathematical physics, as well to works devoted to finding connections between the nonclassical method and various direct methods (see reviews in [10,11,12]). Also a number of other approaches to finding solutions of PDEs, originating from the ideology of the nonclassical method and so can be considered as its generalizations and extensions, have been developed (see, e.g., reviews in [11,13,14]). As a result, the notion of nonclassical symmetry was further developed such that there are several types of non-Lie symmetries and each of them can be called a nonclassical symmetry. Therefore, in many papers, the term ‘Q-conditional symmetry’, proposed in [5,8], is used, instead of ‘nonclassical symmetry’, for the symmetries yielded by applying the original Bluman and Cole method. The terminology originates from the common notation for the invariant surface condition which, in the most extensively studied case of a single PDE with one dependent and two independent variables, is of the form

$$Q(x,t,u,u^{\left(1\right)})=\eta (x,t,u)-\xi (x,t,u)u_x-\tau (x,t,u)u_t=0$$

(1)

The term ‘Q-conditional’ implies that the relevant manifold M, Which, in the case of classical Lie symmetry, includes only a PDE under consideration, in the case of nonclassical symmetry (Q-conditional symmetry), includes the PDE, the invariant surface condition and its differential consequences.

In the present paper, the so-named ’partially nonclassical method’ is introduced. (The basic ideas of the method have been presented at the conference ICNAAM 2018, and the extended abstract has been published in [15]). It is worth remarking that the terminology may be misleading; in particular, the ’partial symmetries’ introduced in [16] have no relation to the ‘partially nonclassical method’ of the present paper. The idea of the ‘partially nonclassical method’ falls under the general idea of changing the relevant manifold M of the nonclassical method by skipping a part of the invariant surface conditions (if the system of PDEs is considered) and/or of their consequences. The idea was first introduced in [17], as applied to systems of PDEs. In the realization of that general idea, considered in [17], the invariant surface condition and its consequences are used not for all dependent variables of the system but only for a part of them. In the ’partially nonclassical method’, the general idea of reducing the relevant manifold M by deleting some of the relations, used in a common nonclassical method, is realized in application to a single PDE. In that realization, not all differential consequences of the invariant surface condition, but only a part of them, are used. Although the procedure of the method is described as applied to a single PDE, it is obviously applicable both to a single PDE and to a system of PDEs.

As it is indicated in [17], a change in the manifold for nonclassical symmetry by deleting some equations, in a general case, does not lead to new symmetries (see also a detailed discussion in the book in [18]). Nevertheless, this approach can be useful for finding nonclassical symmetries in the so-called no-go case, when the infinitesimal generator for one of the independent variables is identically zero ($\tau =0$, in the case of (1)). This was noted earlier in [15], as applied to a single PDE, and in the paper in [19], in which new nonclassical (Q-conditional) symmetries have been constructed in the no-go case for the diffusive Lotka–Volterra system. In the present paper, it is demonstrated that applying the ‘partially nonclassical method’ to a single PDE in the no-go case may yield new nonclassical symmetries.

In the present paper, the ‘partially nonclassical method’ is applied to the equation for the stream function of the boundary layer (BL) flow, to which the flat steady-state boundary layer equations (BLEs) can be reduced. The classical symmetries of the boundary layer equations have been considered by Ovsiannikov [20]. The nonclassical symmetries have been computed in [21], except for the no-go case. In addition to defining the nonclassical symmetries, in [21], the generalization of the nonclassical method, which allows similarity reductions and solutions of the BL equations to be defined, which are not obtainable by the classical and nonclassical Lie group methods, has been developed. The nonclassical symmetries of the stream function equation of the flat steady-state BL problem, in the case of one of the infinitesimal generators associated with an independent variable being identically zero, have been considered in [22]. Although some solutions of the BLE have been identified, in fact, the nonclassical method has not been applied (see more details in Section 3.3). Similarity reductions of the BLE, different from those obtained by classical and nonclassical methods in [20,21], have been identified in [23] using the direct method that may be considered as a generalization of the Clarkson and Kruskal method [9]. An extension of the method of [23] to equations with three independent variables, in particular, to the unsteady BLE, is presented in [24]. (There have been some wrong ideas about the interpretation from the point of view of group analysis of the method of [23]; the issue is analyzed in [22]).

It is shown in the present paper that applying the ‘partially nonclassical method’ to the flat steady-state BLE yields new similarity reductions and exact explicit solutions even in the case where one of the infinitesimal generators associated with an independent variable is identically zero. In that case, distinct from the common nonclassical method, where determining equations for the nonzero group generator are much more complicated than the original BL equations, applying the partially nonclassical method yields solvable determining equations.

The second issue identified in the title of the paper, conformal invariance, is in some way related to the ‘partially nonclassical method’, but it is also of general value in the context of the Lie group method (for exploiting the conformal invariance for physical systems see, e.g., reviews [25,26]). It is observed that, although the classical Lie group method is based on the requirement of invariance of a PDE, applying the standard procedure of the classical method to the PDE yields transformations which do not leave the differential polynomial of the PDE invariant but modify it by a conformal factor. It is shown in Section 6 that this observation should be taken into account in the situations where the form invariance is required by the problem formulation. In addition, that observation leads to the formulation of the Lie group method for defining similarity reductions of a PDE, which unifies the classical, nonclassical and partially nonclassical methods ($\lambda$-formulation) and also opens the door to further extensions. What is of special significance about that formulation is that the corresponding computational procedure, distinct from some other generalizations of the Lie method, is algorithmic.

This paper is organized as follows. In the next Section 2, the formulation of the ‘partially nonclassical method’, preceded by those of classical and nonclassical methods, is presented. In Section 3, the application of the ‘partially nonclassical method’ to the flat steady-state BL equations is considered. The new solutions of the BLE based on the solution for the group generator, obtained in Section 3.3.2, are discussed in Section 4. In Section 5, the discussion of the issue of conformal invariance in the context of the Lie group method starts and it continues in Section 6. The $\lambda$-formulation of the Lie method is presented in Section 7. Some comments on the results are provided in the concluding Section 8.

2. Partially Nonclassical Method for Finding Similarity Reductions of PDEs

2.1. Classical and Nonclassical Methods

In the Lie classical method as applied to a PDE in two independent variables,

$$\Delta (x,t,u,u^{\left(k\right)})=0,$$

(2)

with $u^{\left(k\right)}$ denoting the derivatives of the unknown function u with respect to the x and t up to the order k. The one-parameter (a) Lie group of infinitesimal transformations in $(x,t,u)$ represented by the vector field

$$\mathbf{v}=\xi (x,t,u)\frac{\partial }{\partial x}+\tau (x,t,u)\frac{\partial }{\partial t}+\eta (x,t,u)\frac{\partial }{\partial u}$$

(3)

is considered. The generators, $\xi$, $\tau$ and $\eta$, of (3) are determined from the infinitesimal invariance requirement

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0}=0$$

(4)

where ${\mathbf{v}}^{\left(k\right)}$ is the usual kth prolongation of the transformation group. The notation (4) indicates that, in the polynomial in derivatives of u obtained by applying the invariance requirement ${\mathbf{v}}^{\left(k\right)}(\Delta )=0$, Equation (2) itself is used to eliminate one of the derivatives (usually, it is the highest x-derivative). Then, an overdetermined system of determining equations for the infinitesimal group generators $\xi$, $\tau$, $\eta$ can be obtained from the conditions of vanishing coefficients of the monomials. Having defined the group generators, the corresponding similarity reduction and therefore the invariant solutions may be obtained as solutions of the first-order quasi-linear equation that expresses the invariant surface condition

$$Q(x,t,u,u^{\left(1\right)})=\eta (x,t,u)-\xi (x,t,u)u_x-\tau (x,t,u)u_t=0$$

(5)

In the Bluman and Cole nonclassical method, the infinitesimal group generators are obtained from the requirement

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0,\left\{DQ\right\}=0}=0$$

(6)

where

$$\left\{DQ\right\}=\left\{D^{\left(i\right)}Q\right\},i=0,1,\dots ,k-1;D^{\left(0\right)}Q=Q$$

(7)

It means that, in the polynomial in derivatives of u obtained by applying the invariance requirement to the PDE, both Equation (2) and the invariant surface condition (5) with its differential consequences are used. Usually, the equation is used to replace the highest x-derivative by the expression obtained from (2) and the invariant surface condition is used to replace all t-derivatives by the expressions obtained from (5) and its differential consequences. As the result, the determining equations for the infinitesimal group generators and their solutions differ from those obtained by applying the classical Lie method, which may lead to similarity reductions unobtainable using the classical method. However, as distinct from the classical method, the group generators defined by applying the nonclassical method cannot be used for finding group transformations—they are used only for defining similarity reductions from (5).

2.2. Partially Nonclassical Method

The ‘partially nonclassical method’, differs from the nonclassical method in that, in the polynomial in derivatives of u obtained by applying the invariance requirement to the PDE, not all the relations from the set {equation, invariant surface condition, its differential consequences} are used, as follows:

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0,\left\{DQp\right\}=0}=0$$

(8)

where

$$\left\{DQp\right\}=\left\{D^{\left(i\right)}Q\right\},i=0,1,\dots ,\mathbf{j}-\mathbf{1},\mathbf{j}+\mathbf{1},\dots ,k-1$$

(9)

One may compare Equations (8) and (9) with Equations (6) and (7), defining the procedure of the nonclassical method. Technically, it means that, in the procedure defined by Equations (8) and (9), for example, not all t-derivatives are replaced by their expressions from the corresponding equations. It may seem that there is an ambiguity in the application of the ‘partially nonclassical’ method, namely, it is not clear what relations should be excluded from the list of the relations ($\Delta =0,Q=0,D^{\left(1\right)}Q=0$ and so on). The formulation of the method presented in Section 7 ($\lambda$-formulation) makes this algorithmic.

Note that it is possible to create the method, which uses the basic ideas of both the ‘partially nonclassical method’, as applied to systems of PDEs, and the method of [17]. It should provide more possibilities for applications of the symmetry analysis to systems of evolution equations.

3. Application to the Boundary Layer Equations

3.1. Boundary Layer Equations

The steady-state flat boundary layer equations have the form [27]

$$\begin{array}{ccc}& & u_x+v_y=0\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & u{u}_x+v{u}_y=U^{\left(e\right)}{U}_x^{\left(e\right)}+u_{yy}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & u(x,y)\to U^{\left(e\right)}\left(x\right)\mathrm{as}y\to \infty \hfill \end{array}$$

(12)

where x and y are coordinates, u and v are the x- and y-velocity components and $U^{\left(e\right)}$ is the external flow velocity. By introducing a streamfunction $\psi$,

$$u={\psi }_y,v=-{\psi }_x$$

(13)

the problem, for a given $U^{\left(e\right)}$, is reduced to one equation for $\psi$ 

$$\begin{array}{ccc}& & {\psi }_{yyy}+{\psi }_x{\psi }_{yy}-{\psi }_y{\psi }_{xy}-\Theta \left(x\right)=0\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & \Theta \left(x\right)=-U^{\left(e\right)}{U}_x^{\left(e\right)}\hfill \end{array}$$

(15)

The invariant surface condition in the $(\psi ,y,x)$ variables is

$$Q=\eta (y,x,\psi )-\xi (y,x,\psi ){\psi }_y-\tau (y,x,\psi ){\psi }_x=0$$

(16)

3.2. The Case of $\tau \ne 0$

In this case, it can be set without loss of generality $\tau =1$ and then the invariant surface condition takes the form

$$Q=\eta (y,x,\psi )-\xi (y,x,\psi ){\psi }_y-{\psi }_x=0$$

The procedure of the nonclassical method is defined by

$${\mathbf{v}}^{\left(3\right)}(\Delta ){|}_{\Delta =0,Q=0,D^{\left(1\right)}Q=0,D^{\left(2\right)}Q=0}=0$$

The partially nonclassical method is applied with the procedure defined by

$${\mathbf{v}}^{\left(3\right)}(\Delta ){|}_{\Delta =0,D^{\left(1\right)}Q=0,D^{\left(2\right)}Q=0}=0$$

(17)

which means that the invariant surface condition itself is not used, and only its differential consequences are used.

Among the possible solutions of determining equations yielded by applying (17), there are several solutions that appear among those produced by applying the common nonclassical method (listed in [21]) and there is one which yields a similarity reduction not defined in [21]. The solution is

$$\tau =1,\xi ={\varphi }^{\prime }\left(x\right),\eta =B(y-\varphi \left(x\right));\Theta \left(x\right)=A-B^{2}x$$

(18)

where A and B are arbitrary constants and $\varphi \left(x\right)$ is an arbitrary function. The solution (18) represents a missing case of the analysis of [21] (as noticed by the anonymous reviewer) and so it is also obtainable by the common nonclassical method. Thus, as could be expected on the basis of the analysis of [17], in the case of $\tau =1$, the procedure of the partially nonclassical method defined by (17) does not produce similarity reductions that could not be obtained by the common nonclassical method. Nevertheless, the solution (18), independently of how it has been identified, deserves consideration since it provides a new similarity reduction and a new exact explicit solution of the boundary layer equations.

Using the infinitesimal generators defined by (18) results in the new similarity reduction of the boundary layer Equation (14) defined by

$$\begin{array}{ccc}& & \psi =Bx(y-\varphi \left(x\right))+F\left(z(y,x)\right);z(y,x)=y-\varphi \left(x\right);\Theta \left(x\right)=A-B^{2}x\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & F^{\prime \prime \prime }+Bz{F}^{″}-B{F}^{\prime }-A=0\hfill \end{array}$$

(20)

An explicit solution of the ordinary differential Equation (20) can be defined in the form

$$F=c_0+\frac{z}{2}\left(-\frac{2A}{b^2}+c_{1}z-c_2{e}^{-\frac{1}{2}{b}^2{z}^2}\right)-\frac{c_2\sqrt{2\pi }}{4b}\left(1+b^2{z}^2\right)\mathrm{Erf}\left(\frac{bz}{\sqrt{2}}\right);b=\sqrt{B}$$

(21)

where $c_0$, $c_1$ and $c_2$ are arbitrary constants. Thus, Equations (19) and (21) define a new exact explicit solution of the steady-state boundary layer equations.

For a solution of the BLE equations to be consistent, the condition (12) is to be satisfied. Calculating the velocity $u=\partial \psi /\partial y$ yields

$$u-\frac{A}{b^2}+b^{2}x+c_{1}z-c_2\left(e^{-\frac{1}{2}{b}^2{z}^2}+b\sqrt{\frac{\pi }{2}}z\mathrm{Erf}\left(\frac{bz}{\sqrt{2}}\right)\right)$$

(22)

It is evident that, for the condition (12) to be satisfied by (22), the terms containing z should compensate each other when $z\to \infty$, which is achieved by setting

$$c_1=c_2\left|b\right|\sqrt{\frac{\pi }{2}}$$

(23)

Then, the expression for u takes the form

$$u=-\frac{A}{b^2}+b^{2}x-c_2\left(-\left|b\right|\sqrt{\frac{\pi }{2}}z+e^{-\frac{1}{2}{b}^2{z}^2}+b\sqrt{\frac{\pi }{2}}z\mathrm{Erf}\left(\frac{bz}{\sqrt{2}}\right)\right)$$

(24)

and $U^{\left(e\right)}\left(x\right)$ is given by

$$U^{\left(e\right)}=-\frac{A}{b^2}+b^{2}x$$

(25)

It is readily checked that (24) satisfies (14) with $\Theta \left(x\right)$ defined by (15) and (25) so that the solution is consistent.

3.3. The Case of $\tau =0$

3.3.1. Nonclassical Method

In this case, it can be set to $\xi =1$ without loss of generality so that the invariant surface condition takes the form

$$Q=\eta (y,x,\psi )-{\psi }_y=0$$

(26)

In the procedure of the nonclassical method, the invariance requirement, which results in a polynomial in $\psi$ and its derivatives, representing a source for determining equations for the group generator $\eta$, is supplemented by the PDE, the invariant surface condition and its differential consequences (in the case of the BLE, only the first one is needed) as follows:

$${\mathbf{v}}^{\left(3\right)}(\Delta ){|}_{\Delta =0,\mathbf{Q}=\mathbf{0},D^{\left(1\right)}Q=0}=0$$

(27)

More specifically, the system consists of the PDE (14) and the following relations:

$$\begin{array}{ccc}& & {\psi }_{yyy}=-{\psi }_x{\psi }_{yy}+{\psi }_y{\psi }_{xy}+\Theta \left(x\right)\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& & {\psi }_y=\eta \hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & {\psi }_{yy}={\psi }_y{\eta }_{\psi }+{\eta }_y\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & {\psi }_{yx}={\psi }_x{\eta }_{\psi }+{\eta }_x\hfill \end{array}$$

(31)

Applying (27) means that, in the differential polynomial expressing the condition of invariance of the PDE (14), the derivatives ${\psi }_{yyy}$, ${\psi }_y$, ${\psi }_{yy}$ and ${\psi }_{yx}$ are replaced by the expressions given by (28)–(31) (of course, the lower order derivatives in the expressions for the higher order derivatives are also replaced). As a result, one arrives at the following relation:

$$\begin{array}{c}\Theta \left(x\right){\eta }_{\psi }+{\eta }^3{\eta }_{\psi \psi \psi }+3{\eta }_y{\eta }_{y\psi }+{\eta }^2\left(3{\eta }_{\psi }{\eta }_{\psi \psi }-{\eta }_{x,\psi }+3{\eta }_{y\psi \psi }\right)\hfill \\ +\eta \left(3{\eta }_{\psi \psi }{\eta }_y+3{\eta }_{\psi }{\eta }_{y\psi }-{\eta }_{yx}+3{\eta }_{yy\psi }\right)+{\eta }_{yyy}\hfill \\ +{\psi }_x\left(\eta {\eta }_{y\psi }+{\eta }_{yy}\right)=0\hfill \end{array}$$

(32)

Thus, applying (27) yields the overdetermined system of equations for $\eta$, which is more complicated than the original Equation (14), and no analytical solutions of that system are available.

The case of $\tau =0$ is treated in [22] and it is claimed that the ‘remarkable class of nonclassical symmetries’ has been identified. Nevertheless, the analysis of [22] is not what is named the nonclassical method. The condition of invariance of the original equation, which, with the use of the invariant surface condition and its differential consequences, results in the relation (32), is not used. Instead, Bäcklund transformations, relating the original equation with some other equations, have been found by combining the invariant surface condition and its differential consequences with the original equation. In particular, the celebrated von Mises transformation (see, e.g., [27]) of the BL equations to the nonlinear heat equation has been recovered. The Bäcklund transformations have been used for constructing solutions of the BL equations from solutions of the other equations and several solutions of the BL equations have been identified that way (the physical meaning of the solutions is not discussed). It is worth clarifying again that only solutions of the overdetermined system of equations for $\eta$ defined by (32) can be named ‘nonclassical symmetries’ and so only the corresponding similarity reductions can be named ‘nonclassical symmetry reductions’. ‘Thus, in fact, the nonclassical method has not been applied in [22].

3.3.2. Partially Nonclassical Method

As distinct from the common nonclassical method, the properly specified partially nonclassical method may provide solvable determining equations. Consider, for example, a specification of the partially nonclassical method, in which the relation (26) itself is not used, and only the equation and the first differential consequence of (26) are used, as follows:

$${\mathbf{v}}^{\left(3\right)}(\Delta ){|}_{\Delta =0,D^{\left(1\right)}Q=0}=0$$

(33)

It means that computations proceed such that, in the differential polynomial expressing the condition of invariance of the PDE (14), the derivatives ${\psi }_{yyy}$, ${\psi }_{yy}$ and ${\psi }_{yx}$ are replaced by the expressions given by (28), (30), (31) while ${\psi }_y$ is not replaced. As a result, the relation, providing determining equations for $\eta$ takes the form

$$\begin{array}{c}{\psi }_y^3{\eta }_{\psi \psi \psi }+{\psi }_x{\psi }_y{\eta }_{y\psi }+{\psi }_x{\eta }_{yy}+{\psi }_y^2\left(3{\eta }_{\psi }{\eta }_{\psi \psi }-{\eta }_{x\psi }+3{\eta }_{y\psi \psi }\right)\hfill \\ +{\psi }_y\left(3{\eta }_{\psi \psi }{\eta }_y+3{\eta }_{\psi }{\eta }_{y\psi }-{\eta }_{yx}+3{\eta }_{yy\psi }\right)+{\eta }_{yyy}+3{\eta }_y{\eta }_{y\psi }+\Theta \left(x\right){\eta }_{\psi }=0\hfill \end{array}$$

(34)

from which the determining equations are obtained as follows:

$$\begin{array}{ccc}& & {\eta }_{\psi \psi \psi }=0\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & {\eta }_{y\psi }=0\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & {\eta }_{yy}=0\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& & 3{\eta }_{\psi }{\eta }_{\psi \psi }-{\eta }_{x\psi }+3{\eta }_{y\psi \psi }=0\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& & 3{\eta }_{\psi \psi }{\eta }_y+3{\eta }_{\psi }{\eta }_{y\psi }-{\eta }_{yx}+3{\eta }_{yy\psi }=0\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & {\eta }_{yyy}+3{\eta }_y{\eta }_{y\psi }+\Theta \left(x\right){\eta }_{\psi }=0\hfill \end{array}$$

(40)

It follows from Equations (35)–(37) that $\eta$ has the form

$$\eta ={\psi }^2{A}_2\left(x\right)+\psi A_1\left(x\right)+A_{00}\left(x\right)+y{A}_{01}\left(x\right)$$

(41)

Next, substituting (41) into Equation (38) yields two equations for the functions $A_2\left(x\right)$ and $A_1\left(x\right)$ that are readily solved to give

$$A_2\left(x\right)=\frac{1}{6(x-x_0)};A_1\left(x\right)=\frac{c_2}{6(x-x_0)}$$

(42)

Then, it follows from the remaining two Equations (39) and (40) that

$$A_{01}\left(x\right)=\frac{c_1}{6(x-x_0)};\Theta \left(x\right)=0$$

(43)

Combining Equations (41)–(43) we obtain the following expressions for $\eta$:

$$\eta =\frac{c_{1}y+c_2\psi -{\psi }^2}{6\left(x-x_0\right)}+A_{00}\left(x\right)$$

(44)

where $x_0$, $c_1$ and $c_2$ are arbitrary constants and $A_{00}\left(x\right)$ is an arbitrary function.

4. Solutions of the BL Equations Originating from (44)

Invariant solutions of Equation (14) are obtained as solutions of the first-order equation

$$\frac{\partial \psi }{\partial x}=\frac{c_{1}y+c_2\psi -{\psi }^2}{6\left(x-x_0\right)}+A_{00}\left(x\right)$$

(45)

The constant $c_2$ can be eliminated by the transformation $\psi \to \psi +c_2/2$, subsequently including the new x-dependent terms into the arbitrary function $A_{00}\left(x\right)$. Next, the constant $x_0$ is of no importance; it represents the leading edge coordinate where the initial conditions are set and so can be taken as zero. Thus, we have to solve the equation

$$\frac{\partial \psi }{\partial x}=\frac{c_{1}y-{\psi }^2}{6x}+A_{00}\left(x\right)$$

(46)

The solutions of Equation (46) must satisfy the condition at infinity which, in accordance with the second relation of Equations (12), (15) and (43), is

$$u\to U_0\mathrm{as}y\to \infty$$

(47)

where $U_0$ is a constant. Equation (46) contains two arbitrary elements, $c_1$ and $A_{00}\left(x\right)$, and so the following three cases are to be considered separately.

4.1. Case I: $c_1=0$; $A_{00}\left(x\right)\ne 0$

In this case, the solution of Equation (46), after redefining the arbitrary function as $A_{00}\left(x\right)=6xF{\left(x\right)}^2$, is represented in the form

$$\psi =6xF\left(x\right)tanh\left(F\left(x\right)\left(y+G\left(x\right)\right)\right)$$

(48)

where $F\left(x\right)$ and $G\left(x\right)$ are arbitrary functions. Substituting (48) into the original Equation (14) defines $F\left(x\right)$ as $F\left(x\right)=k{x}^{-2/3}$ so that solution (48) takes the form

$$\psi =6x^{1/3}tanh\left(\frac{ky}{x^{2/3}}+G\left(x\right)\right)$$

(49)

Calculating the u-velocity as $u=\frac{\partial \psi }{\partial y}$ yields

$$u=6k^2{x}^{-1/3}{\mathrm{sech}}^2\left(\frac{ky}{x^{2/3}}+G\left(x\right)\right)$$

(50)

4.2. Case II: $c_1\ne 0$; $A_{00}\left(x\right)=0$

Solutions of Equation (46) that can satisfy the condition at infinity (46) are obtained for $c_1>0$. Setting $c_1=k^2$, we obtain the following solution:

$$\psi =\frac{k\sqrt{y}\left(B\left(x\right)I_{-2/3}\left(z\right)+I_{2/3}\left(z\right)\right)}{I_{-1/3}\left(z\right)+B\left(x\right)I_{1/3}\left(z\right)},z=k\frac{y^{3/2}}{9x}$$

(51)

where $I_n\left(z\right)$ are the modified Bessel functions of the first kind. Substituting (51) into the original Equation (14) yields $B^{\prime }\left(x\right)=0$ so that the solution takes the form

$$\psi =\frac{k\sqrt{y}\left(B_0{I}_{-2/3}\left(z\right)+I_{2/3}\left(z\right)\right)}{I_{-1/3}\left(z\right)+B_0{I}_{1/3}\left(z\right)},z=k\frac{y^{3/2}}{9x}$$

(52)

where $B_0$ is an arbitrary constant. Note that, equivalently, the solution can be represented in terms of Airy functions.

4.3. Case III: $c_1\ne 0$; $A_{00}\left(x\right)\ne 0$

In this case, upon setting $c_1=k^2$ and redefining the constant k and arbitrary function $A_{00}\left(x\right)$ by the relations $k=6q^{3/2}$ and $A_{00}\left(x\right)=6q^2{x}^{-1/3}F\left(x\right)$, the solution of Equation (46) is represented as follows:

$$\psi =\frac{6q{x}^{1/3}\left(B{i}^{\prime }\left(z\right)+G\left(x\right)A{i}^{\prime }\left(z\right)\right)}{Bi\left(z\right)+G\left(x\right)Ai\left(z\right)},z=q\frac{y}{x^{2/3}}+F\left(x\right)$$

(53)

where $Ai\left(z\right)$ and $Bi\left(z\right)$ are the Airy functions, with primes denoting differentiation with respect to the argument, q is an arbitrary constant and $F\left(x\right)$ and $G\left(x\right)$ are arbitrary functions. Substituting (53) into the original Equation (14) yields $G^{\prime }\left(x\right)=0$ so that the solution takes the form

$$\psi =\frac{6q{x}^{1/3}\left(B{i}^{\prime }\left(z\right)+G_{0}A{i}^{\prime }\left(z\right)\right)}{Bi\left(z\right)+G_{0}Ai\left(z\right)},z=q\frac{y}{x^{2/3}}+F\left(x\right)$$

(54)

where $G_0$ is an arbitrary constant.

4.4. Physical Interpretation of Solutions

For all the solutions (49), (52) and (54), the longitudinal velocity $u=\frac{\partial \psi }{\partial y}$ vanishes at infinity, $u\to 0$ as $y\to \infty$, which satisfies the condition (47) with $U_0=0$. The zero value of the u-velocity at infinity means that the solutions can describe either jets or the flows induced by a movement of a rigid plate in its own plane. We will restrict ourselves to considering flows for which the transversal velocity component $v=0$ at $y=0$, which means that either a jet flow or the flow along an impermeable surface is considered. It follows from the relation $v=-\frac{\partial \psi }{\partial x}$ that $\psi =\mathrm{const}$ along the x-axis and so, without loss of generality, we can impose the condition

$$\psi =0\mathrm{at}y=0$$

(55)

For the solution, defined by Equation (49), applying the condition (55) yields $G\left(x\right)=0$ so that the solution takes the form

$$\psi =6x^{1/3}tanh\left(\frac{ky}{x^{2/3}}\right)$$

(56)

which coincides with that of Schliehting’s solution for a two-dimensional jet emerging from an infinitely small slit into the surrounding fluid (see [27]).

For solution (52), calculating the value of $\psi$ at the x-axis yields

$$\psi =6B_{0}k{\left(3x\right)}^{1/3}\frac{\Gamma \left(\frac{2}{3}\right)}{\Gamma \left(\frac{1}{3}\right)}\mathrm{at}y=0$$

(57)

while the value of u at $y=0$ is found to be proportional to $B_0^2$. Thus, the condition $v=0$ at the surface $y=0$, which requires $B_0=0$, also implies vanishing of the u-velocity. As the result, the solution satisfying conditions of vanishing of both u and v at the surface $y=0$ takes the form

$$\psi =k\sqrt{y}\frac{I_{2/3}\left(z\right)}{I_{-1/3}\left(z\right)},z=k\frac{y^{3/2}}{9x}$$

(58)

The limit of solution (58) as $x\to 0$ is

$$\psi \to k\sqrt{y},u\to \frac{k}{2\sqrt{y}}\mathrm{as}x\to 0$$

(59)

The remarkably simple solution (58) can be interpreted as a two-dimensional jet impinging on the leading edge of the rigid impermeable surface of a body, such that the surface is along the x-axis being infinitely long downstream and the leading edge is at $x=0$, while the body extends in the negative y-direction. The slit, from which the jet emerges at $x=0$, is assumed to be infinitely small so that, in order to have a finite rate of the flow volume discharged per unit height of the slit, and a finite flux of momentum, a fluid velocity in the slit should be infinite (as it is in Equation (59)). The jet spreads outwards in the downstream direction, whereas the velocity of the jet decreases in the same direction due to the influence of (internal) friction. Profiles of the u-velocity at different x are shown in Figure 1.

For solution (54), applying condition (55) yields

$$B{i}^{\prime }\left(F\left(x\right)\right)=-G_{0}A{i}^{\prime }\left(F\left(x\right)\right)$$

(60)

which implies that $F\left(x\right)=F_0$, where $F_0$ is a constant, and the constant $G_0$ is related to $F_0$ by

$$G_0=-\frac{B{i}^{\prime }\left(F_0\right)}{A{i}^{\prime }\left(F_0\right)}$$

(61)

Introducing (61) into the expression for $u=\frac{\partial \psi }{\partial y}$ results in the following:

$$u=6x^{-1/3}{F}_0\mathrm{at}y=0$$

(62)

The solution (54), with $G_0$ defined by (61), can be interpreted either (in the spirit of the previously discussed solution) as a jet flow, which approaches a rigid impermeable surface, moving in its own plane according to the law (62), or as a flow induced by the moving surface.

5. Lie Group Method and Conformal Invariance

Let us start a discussion of the issue using, as an example, application of the Lie classical method to the Burgers equation

$$u_t+u{u}_x+u_{xx}=0$$

(63)

The classical symmetry group of Equation (63) is

$$\begin{array}{c}\tau =c_0+2c_{1}t+c_2{t}^2\hfill \\ \xi =c_4+c_{1}x+c_{3}t+c_{2}xt\hfill \\ \eta =c_3-c_{1}u+c_2(x-ut)\hfill \end{array}$$

(64)

Consider the $c_1$ subgroup

$$\mathbf{v}=x\frac{\partial }{\partial x}+2t\frac{\partial }{\partial t}-u\frac{\partial }{\partial u}$$

Group transformations corresponding to (65) are

$$\tilde{x}=e^{a}x,\tilde{t}=e^{2a}t,\tilde{u}=e^{-a}u$$

Introducing transformations (65) into Equation (63) results in the following transformation of the differential polynomial of the equation:

$${\tilde{u}}_{\tilde{t}}+\tilde{u}{\tilde{u}}_{\tilde{x}}+{\tilde{u}}_{\tilde{x}\tilde{x}}=e^{-3a}\left(u_t+u{u}_x+u_{xx}\right)$$

(65)

It means that applying the infinitesimal invariance condition to Equation (63), with the subsequent use of the equation in the resulting expression, yields transformations which do not leave the differential polynomial of the equation invariant but modify it by a conformal factor. Since both $u(x,t)$ and $\tilde{u}(\tilde{x},\tilde{t})$ are solutions of the equation, the conformal invariance, instead of strict invariance, of the differential polynomial of the equation does not invalidate the procedure of defining solutions but may be of importance for some other applications of the Lie group method (see Section 6).

To elucidate the reason why we, seeking transformations that leave the equation invariant, arrive at the conformal transformations, let us consider an arbitrary PDE

$$\Delta (x,t,u,u^{\left(k\right)})=0$$

with the only restriction that one of the derivatives (usually the highest derivative) enters $\Delta$ linearly. As a matter of fact, it is a preassumption of applying the classical method since eliminating the derivative, which has a multiplier dependent on other derivatives, would enormously complicate matters. The reason for obtaining conformal invariance instead of strict invariance (the latter is implied when one applies the infinitesimal invariance condition) becomes evident from the following observation. The use of the equation for eliminating the highest (or other entering linearly) derivative from a polynomial, obtained by applying the infinitesimal invariance requirement

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0}=0$$

(66)

is equivalent to subtracting the equation with a proper multiplier (the coefficient of that derivative in the polynomial) from the polynomial. Thus, (66) is equivalent to

$${\mathbf{v}}^{\left(k\right)}(\Delta )-\lambda (x,t,u)\Delta =0$$

(67)

where the multiplier $\lambda (x,t,u)$ is the coefficient of the derivative that is eliminated. Now, if one looks at the definitions of the strict invariance

$$\begin{array}{c}\tilde{\Delta }(\tilde{x},\tilde{t},\tilde{u},{\tilde{u}}^{\left(k\right)})=\Delta (x,t,u,u^{\left(k\right)}),\frac{D\tilde{\Delta }\left(a\right)}{Da}=0\Rightarrow {\mathbf{v}}^{\left(k\right)}(\Delta )=0\hfill \end{array}$$

(68)

and the conformal invariance

$$\begin{array}{c}\tilde{\Delta }(\tilde{x},\tilde{t},\tilde{u},{\tilde{u}}^{\left(k\right)})=e^{\lambda (x,t,u)a}\Delta (x,t,u,u^{\left(k\right)})\hfill \\ \frac{D\tilde{\Delta }\left(a\right)}{Da}=\lambda (x,t,u)\tilde{\Delta }\left(a\right)\Rightarrow {\mathbf{v}}^{\left(k\right)}(\Delta )=\lambda (x,t,u)\Delta \hfill \end{array}$$

(69)

it becomes evident that using the equation for eliminating the linearly entering derivative from the relation, obtained by imposing the infinitesimal invariance requirement (the procedure described by (66)), is equivalent to imposing the conformal invariance requirement.

For example, considering again the application of the Lie method to the Burgers Equation (63), we find that, in the relation obtained from the invariance requirement, the monomial with the highest derivative is

$$\dots +{\mathbf{u}}_{\mathbf{xx}}\left(\frac{\partial \eta }{\partial u}-\frac{\partial \xi }{\partial x}\right)+\dots =0$$

(70)

Then, the conformal factor $\lambda$, which is the coefficient of $u_{xx}$, is calculated using (64), which yields

$$\lambda =\frac{\partial \eta }{\partial u}-\frac{\partial \xi }{\partial x}=-3(c_1+c_{2}t)$$

(71)

For the $c_1$ subgroup ($c_2=0$) it corresponds to (65).

6. ‘Lie Invariance’ versus Form Invariance — Derivation of the Lorentz Transformations

In this section, the conditional term ‘Lie invariance’ is used for the notion of invariance in applications of the Lie group method to differential equations. It means that the transformations of dependent and independent variables of a PDE $\Delta =0$ satisfy the condition of validity of the equation in both original and transformed variables as follows:

$$\tilde{\Delta }=0\mathrm{if}\Delta =0$$

(72)

where $\tilde{\Delta }=0$ is the equation in the transformed variables. Evidently, condition (72) is satisfied both in the case when the differential polynomial $\Delta$ of the equation is form-invariant and in the case when it is conformally invariant. To show in what situations the difference between conformal invariance, yielded by applying the standard Lie group method, and form invariance could be of conceptual significance, we will consider the application of the Lie group method to the derivation of the Lorentz transformations of special relativity.

The basic principles of the special relativity theory are the relativity principle and the principle of universality of light propagation in inertial frames. Combined together, they lead to a statement that the times and coordinates of events vary from frame to frame in such a way that the equation of light propagation

$$c^{2}d{t}^2-(d{x}^2+d{y}^2+d{z}^2)=0$$

(73)

remains invariant. The transformations between the reference frames should satisfy a number of physical requirements, such as associativity, reciprocity and so on, which are all covered by the requirement that the transformations form a group. In more general terms, the form of the equation of propagation of light defines the form of the spacetime metric, that is to say the form of the functions $g_{ik}(t,x,y,z)$ in the expression for the interval $ds$ between two events. Upon introducing the four-dimensional coordinates $(x^0,x^1,x^2,x^3)$ instead of $(t,x,y,z)$, the expression for the interval is written in the form

$$d{s}^2=g_{ik}{x}^i{x}^k$$

(74)

where $g_{ik}$ are functions of the spacetime coordinates (as usual, the summation over repeated indices is assumed). The equation of propagation of light corresponds to the zero value of the interval $d{s}^2=0$ so that Equation (73) defines the spacetime metric of special relativity, the Minkowski metric. The relativity principle requires (see, e.g., [28]) that the time and coordinate transformations between different frames leave the mathematical form of the functions describing a physical process, in particular, the components of the metric tensor $g_{ik}$, unchanged. It means that the metric should be form-invariant under the transformations. This requirement acquires a more general meaning in the context of general relativity and cosmology (see, e.g., [29,30,31]).

To elucidate the issue, let us consider the application of the Lie group method to the derivation of the time and coordinate transformations between inertial frames (Lorentz transformations) based on invariance of the equation of propagation of light and the relativity principle. Consider two arbitrary inertial reference frames K and $K^{\prime }$ in the standard configuration with the y- and z-axes of the two frames being parallel while the relative motion is along the common x-axis. The space and time coordinates in K and $K^{\prime }$ are denoted, respectively, as $\{X,Y,Z,T\}$ and $\{x,y,z,t\}$. The velocity of the $K^{\prime }$ frame along the positive x direction in K is denoted by v. Symmetry arguments lead to the conclusion that distances in directions normal to the direction of relative motion do not transform, $y=Y$ and $z=Z$, so it is sufficient to apply the invariance requirement to the equation

$$c^{2}d{t}^2-d{x}^2=0$$

(75)

or to the metric defined by the left-hand side of Equation (75).

A one-parameter group of transformations, with the group parameter a, is considered

$$x=f(X,T;a),t=q(X,T;a)$$

(76)

According to the Lie method, the infinitesimal transformations corresponding to (76) are introduced by

$$x\approx X+\xi (X,T)a,t\approx T+\tau (X,T)a$$

(77)

The expressions in (77) are substituted into the left-hand side of (75) and the resulting relation is linearized with respect to a, which yields

$$\begin{array}{c}{c}^{2}d{t}^2-d{x}^2=c^{2}d{T}^2-d{X}^2\hfill \\ +a\left(2c^{2}d{T}^2{\tau }_T-2d{X}^2{\xi }_X+2dTdX\left(c^2{\tau }_X-{\xi }_T\right)\right)\hfill \end{array}$$

(78)

where subscripts denote differentiation with respect to the corresponding variable.

From this point, we can proceed in two different ways. First, we can impose the ‘Lie invariance’ requirement in the sense of applications of the Lie group method to differential equations, that is to say, impose the requirement of validity of Equation (75) in both coordinate systems

$$c^{2}d{t}^2-d{x}^2=0\mathrm{if}{c}^{2}d{T}^2-d{X}^2=0$$

(79)

and eliminate $d{T}^2$ from Equation (78) using the second relation of (79), which yields

$$c^{2}d{t}^2-d{x}^2=2a\left(d{X}^2\left({\tau }_T-{\xi }_X\right)+dTdX\left(c^2{\tau }_X-{\xi }_T\right)\right)$$

(80)

For the first relation of Equation (79) to be valid, the right-hand side of Equation (80) should vanish, which, in view of the arbitrariness of the differentials $dX$ and $dT$, is achieved if

$$\begin{array}{ccc}& & {\tau }_T-{\xi }_X=0\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& & c^2{\tau }_X-{\xi }_T=0\hfill \end{array}$$

(82)

The determining Equations (81) and (82) reduce to the Laplace equation for the group generators

$${\tau }_{TT}-c^2{\tau }_{XX}=0,{\xi }_{TT}-c^2{\xi }_{XX}=0$$

(83)

and so $\tau$ and $\xi$ are not defined uniquely.

The second possible way to proceed from relation (78) is to impose, instead of the condition (79), the condition of form invariance of the metric. It requires the a part in (78) to vanish, which results in the determining equations of the form

$$\begin{array}{ccc}& & {\tau }_T=0\hfill \end{array}$$

(84)

$$\begin{array}{ccc}& & {\xi }_X=0\hfill \end{array}$$

(85)

$$\begin{array}{ccc}& & c^2{\tau }_X-{\xi }_T=0\hfill \end{array}$$

(86)

Solutions of the overdetermined system (84)–(86) are easily found as

$$\xi =-bT+c_1,\tau =-\frac{b}{c^2}X+c_2$$

(87)

where b, $c_1$ and $c_2$ are arbitrary constants. Having the infinitesimal group generators defined by (87), the finite group transformations can be found via solving the Lie equations with proper boundary conditions. Before this, the common kinematic restrictions that one event is the spacetime origin of both frames and that the x and X axes slide along another can be imposed to make the constants $c_1$ and $c_2$ vanishing (space and time shifts are eliminated). The constant b can be eliminated by redefining the group parameter as $\widehat{a}=ab/c$ (‘hats’ will be omitted in what follows).

Then the Lie equations take the forms

$$\frac{dx\left(a\right)}{da}=-ct\left(a\right),\frac{d\left(ct\left(a\right)\right)}{da}=-x\left(a\right);x\left(0\right)=X,t\left(0\right)=T$$

(88)

The initial value problem (88) is readily solved to give

$$x=Xcosha-cTsinha,ct=cTcosha-Xsinha$$

(89)

To complete the derivation of the transformations, the group parameter a is to be related to the velocity v using the condition

$$x=0\mathrm{for}X=vT$$

(90)

which yields

$$a={tanh}^{-1}\frac{v}{c}\mathrm{or}a=\frac{1}{2}ln\frac{1+v/c}{1-v/c}$$

(91)

Substitution of (91) into (89) results in the Lorentz transformations

$$x=\frac{X-(v/c)cT}{\sqrt{1-v^2/c^2}},ct=\frac{cT-(v/c)X}{\sqrt{1-v^2/c^2}}$$

(92)

The reason why the ‘Lie invariance’, expressed by the condition (79) of validity of the equation of light propagation in both coordinate systems (in both frames), is not sufficient for defining the Lorentz transformations is evident in light of the discussion in Section 5. The equation of light propagation remains valid even in the case when the differential polynomial of the equation is not invariant but conformally invariant. Conformal invariance is more general than form invariance, required by the relativity principle, and therefore it does not straightforwardly lead to the Lorentz transformations. In order to derive the Lorentz transformations using ‘Lie invariance’, some additional conditions are to be imposed (see discussion in [32]).

7. The $\mathit{\lambda }$-Formulation of the Classical, Nonclassical and Partially Nonclassical Methods

The discussion of Section 5 leads to a computationally convenient formulation of the Lie group method which encompasses the classical, nonclassical and partially nonclassical methods. The procedure of the nonclassical method is expressed by the relations

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0,\left\{DQ\right\}=0}=0$$

(93)

$$\left\{DQ\right\}=\left\{D^{\left(i\right)}Q\right\},i=0,1,\dots ,k-1;D^{\left(0\right)}Q=Q$$

(94)

which implies that Equation (2), invariant surface condition (5) and all differential consequences of the latter are used to eliminate some of the derivatives (in the case of equations with two independent variables, it is usually t-derivatives, like $u_t$, $u_{tx}$ and so on) from the polynomial obtained by applying the invariance requirement, Similarly to what was stated for the classical method procedure, one can equivalently represent the nonclassical method procedure as

$${\mathbf{v}}^{\left(k\right)}(\Delta )-\lambda \Delta -{\lambda }_{0}Q-{\lambda }_1{D}^{\left(1\right)}Q-\dots -{\lambda }_{k-1}{D}^{(k-1)}Q=0$$

(95)

with properly specified ${\lambda }_i(x,t,u)$.

Next, the procedure of the ‘partially nonclassical’ method is expressed by

$${\mathbf{v}}^{\left(k\right)}(\Delta ){|}_{\Delta =0,\left\{DQp\right\}=0}=0,$$

(96)

$$\left\{DQp\right\}=\left\{D^{\left(i\right)}Q\right\},i=0,1,\dots ,\mathbf{j}-\mathbf{1},\mathbf{j}+\mathbf{1},\dots ,k-1$$

which is the same as that for the nonclassical method but with an important distinction: while in the nonclassical methods all the relations, including the equation, invariant surface condition and its differential consequences, are used to eliminate derivatives (the t-derivatives) from the differential polynomial obtained from the infinitesimal invariance condition, in the ‘partially nonclassical’ method, only part of the relations is used and so some of the derivatives, which could be eliminated, remain in the polynomial. The procedure in (96) can be equivalently represented by the relation of the form (95) where, however, as distinct from that for the nonclassical method, some ${\lambda }_i$ are set to zero.

Thus, a unified framework that can be termed as the ‘$\lambda$-formulation’ for the classical, nonclassical and partially nonclassical methods for finding similarity reductions of PDEs is represented by the relation (95), where it is implied that the functions ${\lambda }_i(x,t,u)$ are included into a set of unknown variables, on equal footing with the group generators. It provides more freedom for the forms of the group generators and, at the same time, the procedure of obtaining the determining equations for the group generators remains completely algorithmic. Similar to the procedures for the classical and nonclassical methods, the determining equations are obtained from the condition of vanishing the coefficients of all the monomials in the differential polynomial defined by applying the infinitesimal invariance requirement. Since the coefficients include, besides the group generators, the functions ${\lambda }_i$, the resulting system of determining equations becomes even more overdetermined, which makes the procedure more convenient from a computational point of view. The additional advantage of the ‘$\lambda$-formulation’ is that one does not need to apply the classical, nonclassical and ‘partially nonclassical’ methods separately, the different method procedures simply correspond to different sets of possible overdetermined systems of determining equations. Moreover, the $\lambda$-formulation opens the way for other generalizations of the classical and nonclassical methods (in order not to overload this paper, we do not further consider this issue).

8. Concluding Comments

The ‘partially nonclassical’ method, in general, enables finding similarity reductions of PDEs in the ‘no-go’ case, when the infinitesimal generator, associated with one of the independent variables, is zero (for equations with one dependent and two independent variables, it is the $\tau =0$ case), which is usually the ‘dead-end’ of the computational procedure of the standard nonclassical method. It is demonstrated in this paper that applying the ‘partially nonclassical method’, even to such a well-studied physical system as flat steady-state BL equations, yields new similarity reductions and solutions.

The issue of conformal invariance in the context of the Lie group method raised in this paper, although not being straightforwardly related to the Lie group machinery, as applied to finding solutions of nonlinear PDEs, could be of importance in some other aspects. First, the observation that applying the classical Lie method computational scheme yields transformations that do not leave the differential polynomial of the equation invariant but modify it by a conformal factor is of importance in the applications where one needs to find conditions for a differential polynomial to be form-invariant (for example, in the contexts of general relativity and cosmology). The analysis in the present paper shows that in applying the infinitesimal Lie group technique for the purpose of finding transformations of variables that leave the differential polynomial form-invariant, one should not follow the traditional computational scheme of the classical Lie group method, which includes eliminating one of the derivatives from the expression obtained by imposing the infinitesimal invariance condition.

Next, the analysis of the reasons why the procedure, designed to incorporate the invariance requirement into the method, results not in strict invariance but in conformal invariance, leads to the $\lambda$-formulation, which provides a framework and a computational scheme unifying the classical, nonclassical and ‘partially nonclassical’ methods.