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Abstract

Third-order nonlinear differential equations, Null forms, Symmetries, Darboux polynomials, Integrating factors and Jacobi last multipliers We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ordinary differentiable equations (ODEs). We establish an important interconnection between extended Prelle-Singer procedure and -symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, -symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

]Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

Mohanasubha, Chandrasekar, Senthilvelan and Lakshmanan]R. Mohanasubha, V. K. Chandrasekar, M. Senthilvelan and
M. Lakshmanan

1 Introduction

In a previous paper (Mohanasubha et al. Proc. R. Soc. A 2014) we have established a connection between extended Prelle-Singer procedure with five other analytical methods which are widely used to identify integrable systems described by second-order ODEs and brought out the interconnections between Lie point symmetries, -symmetries, Darboux polynomials, Prelle-Singer procedure, adjoint-symmetries and Jacobi last multiplier. We have also illustrated the interconnections by considering a nonlinear oscillator equation as an example. A natural question which arose was whether these interconnections exist in higher order ODEs as well. In this paper we consider this problem for third-order ODEs and come out with certain new and interesting results.

We start our investigation with the following question. Given any one of the quantities as a starting point in the family, consisting of multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, -symmetries, integrating factors and null forms can one obtain the rest of all quantities in this family? In the case of third-order ODEs exploring the interconnections between the above said quantities is difficult both conceptually as well as technically which will be clear as we proceed. In fact, we get an affirmative answer to our questions. We divide our analysis into two parts. In the first part, as we did in the case of second-order ODEs we try to interlink the quantities by introducing suitable transformations in the extended Prelle-Singer procedure. Once the interconnections are identified, in the second part, we demonstrate that starting from any one of the quantities in this family one can derive all the other quantities in a simple and unambiguous way.

In the first part, the interplay between various methods is brought out by establishing a road map between extended the Prelle-Singer procedure and the other methods, namely (i) Jacobi last multipliers, (ii) Darboux polynomials, (iii) Lie point symmetries, (iv) adjoint-symmetries and (v) -symmetries. In the above we have listed out the methods in chronological order. In other words JLM was the one invented first and the last one in this direction is the extended Prelle-Singer procedure. So naturally we try to accommodate other methods into extended Prelle-Singer procedure. In the later procedure one has three equations to integrate to obtain the null forms and and the integrating factor . To interconnect these quantities we introduce three new variables and , which are to be determined through the transformations, and so that one can rewrite the determining equations for and in terms of and , respectively. Doing so, we find that the determining equation for the function exactly coincides with the determining equation for the Darboux polynomials which in turn establishes a connection between the integrating factors and Darboux polynomials. Now properly combining the expressions which arise in the extended Prelle-Singer procedure with the -symmetry determining equation we obtain the following relation between and with (see Sec.3 for details), namely . This interlink is vital and new to the literature. Since the null forms and are now expressed in terms of and , this interlink can also be written in terms of and . Doing so we obtain an equation solely in terms of and , that is which in turn determines if is known or vice versa (Here is the function defining the third-order ODE, see Eq.(a) below). We assume that is known. In this case can be determined by solving this equation and from the latter can be fixed and so and can be determined. The Darboux polynomials can be determined either from the multipliers or from their own determining equations. This in turn establishes the interconnection between all these methods. One important result which we observe in this part is that for a given , the null forms and are not unique which can be seen from the above relation. Because of this, one may find multiple null forms for a given equation and one has to choose the correct form of and in order to obtain an independent integral. From the remaining forms of and one may obtain other useful information such as symmetries, integrating factors, Darboux polynomials and Jacobi last multiplier; however the resulting integrals turn out to be dependent ones. This feature is different from second-order ODEs where one has one-to-one correspondence between the null forms and -symmetries (). We illustrate this important point by considering three examples.

Next we move on to the question that given any one of the system quantities how to derive the remaining quantities in this family. Using the interconnections which we identified above, and starting from any one of the quantities, we establish road maps to connect to all the other quantities. After a detailed analysis we find that it is sufficient to consider any one of the following three cases as starters, namely (i) Lie point symmetries, (ii) integrating factors or (iii) Darboux polynomials. Then the remaining quantities can be uniquely determined. The other three possibilities, namely (iv) -symmetries, (v) null forms and (vi) JLM are only subcases of the previous three cases, respectively. The exact road map in all the cases is explicitly demonstrated in Sec.4. To illustrate the ideas introduced in this paper we consider three examples.

Since some of the methods applicable for the third-order ODEs are not much discussed in the literature as in the case of second-order ODEs, to begin with we review briefly the six different analytical methods in the next section (Sec.2) in order to be self contained.

2 Methods

In this section we briefly recall the six analytical methods with reference to the third-order ODEs which are widely used in the contemporary literature to derive one or more of the following quantities, namely (i) multipliers, (ii) integrating factors, (iii) symmetries and (iv) integrals. We present the methods in chronological order.

a Jacobi last multiplier method

The Jacobi last multiplier method was introduced by Jacobi in the year 1844 (Jacobi, 1844,1886), but laid dormant for a long time. Nucci et al. have demonstrated the applicability of this method in exploring non-standard Lagrangians associated with certain second-order nonlinear ODEs (Nucci & Leach, 2008). The attractive feature of the method is that if we know two independent Jacobi last multipliers and of the given equation, then their ratio yields a first integral for the given equation.

We consider a third-order ODE,

(2.1)

where is a function of and . We rewrite the third-order ODE (a) equivalently as a system of following three first-order ODEs in the absence of explicit -dependence

(2.2)

where and are suitably chosen functions of and . The analysis can be extended in principle to the general case with -dependence included, though it becomes more involved now (see for example, Clebsch, 2009). We consider two integrals and of (a) whose total differentials are given by

(2.3)

The equations of motion which satisfy these two conditions can be written as

(2.4)

where we have defined for simplicity the expressions inside the parenthesis as and . We assume that these equations be the ones coming out from the original equation (a) after multiplying by an integrating factor . Then comparing Eqs.(a) and (2.4), we have

(2.5)

Interestingly one may also observe from (2.4) that

(2.6)

Substituting (2.5) in (2.6) and expanding it we can get the following equation for the multiplier , namely

(2.7)

Eq.(2.7) can be simplified to yield

(2.8)

where the differential operator . Choosing , and , Eq.(2.8) can be written as

(2.9)

Substituting the given equation in (2.9) and solving the resultant equation one can obtain the multiplier associated with a given third-order ODE (Jacobi, 1844,1886; Nucci & Leach, 2008). Note that there can be more than one multiplier for a given ODE (a). We further remark here that when the function in (a) also depends on explicitly, the above operator will be replaced by the total differential operator .

An important application of multipliers is that one can determine the integrals associated with the given equation by evaluating their ratios. If and are two multipliers, then it is straightforward to check from (2.9) that one can identify an integral as (Nucci, 2005). So if we have sufficient number of multipliers we can obtain the necessary integrals to prove the integrability of (a).

b Darboux polynomials approach

Darboux polynomials method was developed by Darboux in the year 1878 (Darboux, 1878). It provides a strategy to find first integrals. Darboux showed that if we have Darboux polynomials, where is the order of the given equation, then there exists a rational first integral which can be expressed in terms of these polynomials. In the case of third-order ODEs, we can get eight Darboux polynomials for a given equation (Darboux, 1878).

The Darboux polynomials determining equation is given by the following expression

(2.10)

where is the total differential operator and is the cofactor (Darboux, 1878). Note also that for a given , and an integral of (a), the quantity , where is arbitrary, is also a solution of (2.10) for the same cofactor. Using this fact and solving Eq.(2.10), we can obtain the Darboux polynomials and the cofactors , see Mohanasubha et al. (2014a) for further details on the method.

c Lie symmetry analysis

Lie symmetry analysis is one of the powerful methods to investigate the integrability property of the given ODE of any order where one first explores the symmetry vector fields associated with it. The Lie symmetry vector fields can then be used to derive integrating factors, conserved quantities and so on (Olver 1993).

We consider a third-order ODE of the form (a). The invariance of Eq.(a) under an one parameter group of Lie point symmetries, corresponding to infinitesimal transformations,

(2.11)

where and are the coefficient functions of the associated generator of an infinitesimal transformation and is a small parameter, is specified by

(2.12)

Here and are the first, second and third prolongations, respectively, of the infinitesimal point transformations (2.11) and are defined to be

(2.13)

In the above over dot denotes total differentiation with respect to . Substituting the known expression in (2.12) and solving the resultant equation we can get the Lie point symmetries associated with the given third-order ODE. The maximum number of admissible Lie point symmetries for a third-order ODE (a) is seven (Olver 1993). The associated vector field is given by .

One may also introduce a characteristics and rewrite the invariance condition (2.12) in terms of a single variable in the form

(2.14)

One can deduce the coefficient functions and associated with the Lie point symmetries from out of the class of solutions to (2.14) which depends only on and , and also has a linear dependence in .

d Adjoint-symmetries

In general, for systems of one or more ODEs, an integrating factor is a set of functions, multiplying each of the ODEs, which yields a first integral. If the system is self-adjoint (that is the adjoint of the linearized symmetry condition is the same as the linearized symmetry condition) then its integrating factors are necessarily solutions of its linearized system. Such solutions are also the symmetries of the given system of ODEs. If a given system of ODEs is not self-adjoint, then its integrating factors are necessarily solutions of the adjoint system of its linearized system. These solutions are known as adjoint-symmetries of the given system of ODEs.

The adjoint ODE of linearized symmetry condition (2.14) can be written as (Bluman & Anco, 2002)

(2.15)

where is the total derivative operator which is given by . Solutions of the above equation are known as adjoint-symmetries. The adjoint-symmetry of the Eq.(a) becomes an integrating factor of (a) if and only if satisfies the adjoint invariance condition

(2.16)

Once we know the integrating factors, we can find the first integrals. Multiplying the given equation by these integrating factors, and rewriting the resultant equation as a perfect differentiable function,

(2.17)

we can identify the first integrals.

e -symmetries

All the nonlinear ODEs do not necessarily admit Lie point symmetries. Under such a circumstance one may look for generalized symmetries (other than Lie point symmetries) associated with the given equation. One such generalized symmetry is the -symmetry. Let be a -symmetry of the given ODE for some function . The invariance of the given ODE (a) under -symmetry vector field is given by , where is given by . Here , and are first, second and third - prolongations respectively whose explicit expressions are given by (Muriel & Romero, 2001, 2008, 2009)

(2.18)

Expanding the invariance condition, we find

(2.19)

where the infinitesimal prolongations are as given above. If we put , we can get the Lie prolongation formula. Solving the invariance condition (2.19) we can obtain the explicit forms of and . If the given ODE admits Lie point symmetries, then the -symmetries can be derived without solving the -prolongation condition. In this case the -symmetries can be deduced from the relation

(2.20)

where is the total differential operator and . The associated vector field is given by . The method of finding the integrals from -symmetries can also be extended to the third-order ODEs as in the case of second-order ODEs.

f Extended Prelle-Singer method

In a series of papers Chandrasekar, Senthilvelan and Lakshmanan have developed a stand-alone method, namely extended Prelle-Singer procedure, to investigate the integrability of the given ODE which may be of any order, including coupled ones. In the following, we recall briefly the extended Prelle-Singer procedure which is applicable for third-order ODEs (Chandrasekar et al. 2006).

We assume that the ODE (a) admits a first integral with constant on the solutions, so that the total differential gives

(2.21)

where the subscript denotes partial differentiation with respect to that variable. Rewriting equation (a) in the form and adding the null terms and to the latter, we obtain that on the solutions the 1-form

(2.22)

Hence, on the solutions, the 1-forms (2.21) and (2.22) must be proportional, provided (2.22) is a total differential. To ensure this we multiply (2.22) by the function which acts as the integrating factor for (2.22) so that we have on the solutions that

(2.23)

Comparing now equations (2.21) with (2.23) we end up with a set of four relations which relates the integral(), integrating factor() and the null term():

(2.24)

In order to determine and , we impose the compatibility conditions , , , , and . Then we obtain the following determining equations,

(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)

where . Solving the above system of over determined equations we can obtain the unknown functions and . From the known expressions, and , we can determine the integrals which appear on the left hand side of Eq.(2.24) by straightforward integration.

Integrating Eq.(2.24) we find

(2.31)

where

We may note that every independent set in (2.31) defines a first integral.

3 Interconnections

In the previous section we have discussed six specific analytical methods which are used to derive integrating factors, symmetries of various kinds, null forms and integrals associated with the third-order ODEs. A question which we raise now is what is the interconnection between these various methods, that is given any one of the quantities in the family, say multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, -symmetries or integrating factors and null forms, can one obtain the rest of the quantities in this family. To answer this question we can explore the hidden interconnections that exist between these functions and interlink all these methods. To achieve this task we introduce certain transformations in the extended Prelle-Singer procedure which in turn connect globally all the above mentioned quantities. The details are given below (It may be noted that one can establish the same interconnection by taking any one of the other methods as the starting point).

With the aid of the following transformations on the null forms of the PS method, see Eq.(2.25),

(3.32)

where and are two unknown functions and is the total differential operator, Eqs.(2.26) and (2.25) respectively can be rewritten in the form

(3.33)

and

(3.34)

We introduce another transformation on the integrating factor as

(3.35)

where is a function to be determined, in (2.27) so that the latter can be rewritten in a compact form in the new variable as

(3.36)

One may note that Eq.(3.36) is nothing but the Darboux polynomial determining equation (2.10) with the cofactor .

We mention here that once we know the functions and the integrating factor can be determined within the Prelle-Singer procedure itself. But to connect the integrating factors to Darboux polynomials these transformations are essential. More importantly the connection between the null forms and -symmetries can be unearthed through the function which appears in the transformations (3.32) and (3.35), as we see below.

a Connection between -symmetries and null forms

Now we investigate how these expressions, namely and are interconnected with -symmetries. In the case of second-order ODEs the -symmetry is nothing but the null form with a negative sign (Muriel & Romero, 2009). This one-to-one correspondence came from the result that the -determining equation in the PS procedure differs only by a negative sign from that of the -symmetry determining equation (Mohanasubha et al. 2014). However, in the case of third-order ODEs we have two null forms and which have to be connected to a single function as demonstrated below.

Let be a first integral of (a) then is an integrating factor, see Eq.(2.24). The total derivative gives

(3.37)

Let also be a first integral of for some function , then

(3.38)

where is given by . Here and are the first and second - prolongations, respectively, whose explicit expressions are given in the first two expressions of (2.18). Expanding Eq. (3.38) we get the following expression for corresponding to ,

(3.39)

where and are the first and second Lie point prolongations which are given in Eq.(2.13). For the vector field the above expression (3.39) reduces to

(3.40)

Now we connect this expression which comes out from the -symmetry analysis with the null forms in the extended PS procedure. In this regard we can deduce the following expressions from the last three equations of (2.24),

(3.41)

Using the above in Eq. (3.40) we can obtain an equation which connects the null forms and with in the form

(3.42)

Eq. (3.42) is essentially a redefinition of the -function determining equation (3.40). Thus in the case of third-order ODEs the null forms, and , are connected with the -symmetries through a differential relation. This interconnection is brought out for the first time in the literature. We mention here that while deriving the relation (3.42) we assumed that . When we have (vide Eq.(3.40)) and in this case one of the null forms vanishes. This result is also consistent with the extended PS procedure.

For practical purpose we can rewrite the relation (3.42) in terms of a single variable, say in , as follows. Using (3.33) we can express in terms of and substituting the latter in the second expression in (3.32) we find in terms of . The resultant expression reads

(3.43)

Now replacing the null functions, and , which appear in (3.42) by , we find

(3.44)

Substituting the known expressions and and in (3.44) and solving the resultant equation we can obtain which in turn unambiguously fixes the null forms and through the relations given in Eq.(3.32). In other words one can get the null forms and from by finding the function also. In this sense Eq.(3.44) may also treated as a “second bridge” which connects -symmetries with null forms. If we already know the null forms and , Eq.(3.42) yields the -symmetries in a straightforward manner. In this sense one can determine (i) from and and (ii) and from . The expressions (3.42) and (3.44) interconnect the PS procedure and the symmetry analysis.

b Connection between Lie point symmetries and null forms

The relation between Lie point symmetries and -symmetries has already been established earlier (vide Eq.(2.20)). Substituting this in (3.42) we can obtain an expression that relates Lie point symmetries with the null forms in the following manner:

(3.45)

where is the characteristics.

For the sake of completeness, in the following, we recall the following interconnection that is already known in the literature.

c Connection between Jacobi last multiplier and Darboux polynomials/integrating factors

Using the expression (3.35) we can relate the Darboux polynomials with the integrating factor and in fact the denominator of (3.35) is nothing but the Darboux polynomials.

By comparing Eqs.(2.10) and (2.9) we can relate the Darboux polynomials with Jacobi last multiplier as

(3.46)

Using this relation, we can find the Jacobi last multiplier from the Darboux polynomials. So the integrating factor in the PS method is the product of the function and the Jacobi last multiplier , that is .

d Connection between adjoint-symmetries and integrating factors

We rewrite Eqs.(2.25),  (2.26) and (2.27) as a single equation in one variable, for example in . Then the resultant equation reads

(3.47)

Comparing the adjoint of the linearized symmetry condition equation (2.15) and (3.47) one can conclude that the integrating factor is nothing but the adjoint-symmetry , that is

(3.48)

Thus the integrating factor turns out to be the adjoint-symmetry of the given third-order nonlinear ODE.

e Connection between Lie point symmetries and Jacobi last multipliers

The connection between Lie point symmetries and JLM is known for a long time in the form

(3.49)

where

(3.50)

where , and are three sets of Lie point symmetries (see below) of the third-order ODE, , and , , are their corresponding first and second prolongations, respectively, and the inverse of becomes the multiplier of the given equation (Nucci, 2005).

f Comparison between interconnections for second- and third-order ODEs

In the above said interconnections, some of the interconnections are common to both second- and third-order nonlinear ODEs except for their orders, while the others are different. Such common and differing connections are listed below.
1) Common features:
The common connections are the ones between (i) Lie point symmetries and -symmetries, (ii) Lie point symmetries and Jacobi last multiplier, (iii) adjoint-symmetries and integrating factors, and (iv) Darboux polynomials and Jacobi last multiplier.
2) Differing features:
(i) The uncommon relation is the connection between -symmetries and null forms. In the cases of second- and third-order ODEs the connection between -symmetries and null forms are entirely different. In the case of second-order ODEs the -symmetry is nothing but the null form with a negative sign. This one-to-one correspondence came from the result that the -determining equation in the PS procedure differs only by a negative sign from that of the -symmetry determining equation (Muriel & Romero, 2009). In other words there is an one-to-one correspondence between the -symmetries and null form . However, in the case of third-order ODEs, we have two null forms ( and ) which have to be connected to the single function . Our analysis shows that these two null forms are connected with the -symmetry through a single expression.
(ii) The connection between characteristics and null forms is also different in the case of second- and third-order ODEs. In the case of second-order ODEs, there exists only one null form which is directly connected with the characteristics, while in the case of third-order ODEs, the equation which connects the characteristics and null forms contains both the null forms.

g Flow chart of the interconnections

The above interconnections are clearly depicted in Fig.1. This may be compared with that of the second-order ODEs given as Fig.1 in Mohanasubha et al. (2014).

Figure 1: Flow chart connecting Prelle-Singer procedure with other methods for third-order ODEs

4 Illustration

a Example 1:

We consider the following third-order nonlinear ODE, namely

(4.51)

Equation (4.51) is a sub-case of the general form of a scalar third-order ODE which is invariant under the generators of time translation and rescaling (Feix et al. 1997; Polyanin & Zaitsev 2003). A sub-case of equation (4.51) has been considered by both Bocharov et al. (1993) and Ibragimov & Meleshko (2005) in the form . They showed that it can be linearized to a linear third-order ODE through a contact transformation. The equation has been considered by Euler & Euler (2004), who showed that it can be linearized to a free particle equation through a Sundman transformation. Equation (4.51) was studied by Chandrasekar et al. (2005) from the integrability point of view using Prelle-Singer procedure. Here we consider Eq.(4.51) from the perspective of deriving multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, -symmetries, null forms, integrating factors, sequentially and demonstrate the effectiveness of exploring the interconnections. From these quantities we also derive integrals and the general solution of this equation for the sake of completeness.

We begin our analysis with Lie point symmetries. Equation (4.51) admits a set of three parameter Lie point symmetries of the form

(4.52)

The characteristics associated with the above vector fields is given by

(4.53)

Using the relation we can derive the -symmetries associated with the above vector fields which in turn read

(4.54)

Once we know the -symmetry we can find the null forms and . To obtain them we determine the function using the relation (3.44). We obtain

(4.55)

We can fix the allowed forms of the function from the known expression of through the relation (3.33). The resultant expressions read

(4.56)

Using the functions and we can obtain the null forms and respectively, vide Eq.(3.32), as

(4.57)

and

(4.58)

Substituting the null forms and in Eq.(2.27) and solving the resultant equation we find the integrating factors for (4.51) in the form

(4.59)

The functions and also become adjoint-symmetries of (4.51) as well which can be directly verified by substituting them in (2.15). Using the expressions for and we can fix the Darboux polynomials for the given equation by recalling the relation . The Darboux polynomials turn out to be

(4.60)

The polynomials and share the common cofactor . From the property that the ratio of Darboux polynomials which share the same cofactor is the first integral, we can find another Darboux polynomial which shares the cofactor as the above. Doing so, we have found two more Darboux polynomials satisfying the equation as

(4.61)

We can find more number of Darboux polynomials using the property of Darboux polynomial that the combination of Darboux polynomials is also a Darboux polynomial. By multiplying an integral with the Darboux polynomial we can obtain more Darboux polynomials. The ratios of the Darboux polynomials and lead to the first integrals and (vide Eq.(4.62)), respectively.

The JLM associated with the given equation can be obtained from the Darboux polynomials (4.60) by recalling the relation .

Once we know the null forms and integrating factors we can construct the associated integrals of motion by evaluating the integrals given in (2.31). Our analysis shows that

(4.62)

It is a straightforward matter to verify that , and are the three independent integrals for the given Eq.(4.51). From the knowledge of , and we get the general solution in the form

(4.63)

The above said results are tabulated in Table 1.

b Example 2:

We consider another interesting example (Bluman & Anco, 2002)

(4.64)

The adjont symmetries of this equation were first worked out by Bluman and Anco (2002). Subsequently the extended Prelle-Singer method was applied to this equation by Chandrasekar et al. (2005) and they have derived the null forms, integrating factors and integrals through this method. The exact expressions are given in Table 2. The procedure given in the previous example may be followed for this example as well to derive the quantities displayed in Table 2. Here also because of nonuniqueness of with , one may find two independent integrals and from the same -function. The vector field provides one independent integral and the vector field gives the two independent integrals and . The vector field provides only a dependent integral.