A new method to obtain either first- or second-order reductions for parametric polynomial ODEs

Accepted Manuscript A new method to obtain either first- or second-order reductions for parametric polynomial ODEs J. Ramírez, J.L. Romero, C. Muriel

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S0377-0427(19)30125-6 https://doi.org/10.1016/j.cam.2019.03.006 CAM 12177

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Journal of Computational and Applied Mathematics

Received date : 31 July 2018 Revised date : 13 December 2018 Please cite this article as: J. Ramírez, J.L. Romero and C. Muriel, A new method to obtain either first- or second-order reductions for parametric polynomial ODEs, Journal of Computational and Applied Mathematics (2019), https://doi.org/10.1016/j.cam.2019.03.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A new method to obtain either first- or second-order reductions for parametric polynomial ODEs J. Ram´ırez, J. L. Romero, C. Muriel Department of Mathematics, University of C´ adiz, 11510 Puerto Real, Spain.

Abstract For a nth-order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable, a new method to obtain either first- or second-order reductions, respectively, is introduced. The corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative. The method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation. A procedure to determine, in function of the parameters, if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided. The introduced method strictly generalizes several methods that have recently appeared in the literature. As applications of the method, this study considers first-order reductions for a generalized Emden-Fowler equation and second-order reductions for the travellingwaves equation of a fifth-order Korteweg-de Vries equation and for a Painlev´e-Chazy equation. As far as we know, some of the reported reductions are new. Several Maple programs to determine if a given polynomial equation admits any of the considered reductions and, if this is the case, to determine the reduced equations are also included. Keywords: Polynomial ordinary differential equations, first- and second-order reductions, exact solutions for partial differential equations in Mathematical Physics. 2010 MSC: 34A05, 34C14, 34C20, 34G20

1. Introduction A very important issue in applied sciences is the search of exact solutions for ordinary differential equations (ODEs) and partial differential equations (PDEs). For both types of equations, Lie theory and some of its variants have been the most useful tools to obtain these solutions. For a given ODE or PDE, that theory can be used to find another ODEs or PDEs (a reduced equation) whose solutions lead (directly or through similarity transformations) to solutions of the original equation. There is a huge amount of papers that have used the method of classical and nonclassical Lie symmetries [21] to Email address: [email protected], [email protected], [email protected] (J. Ram´ırez, J. L. Romero, C. Muriel)

Preprint submitted to Journal of Computational and Applied Mathematics

December 13, 2018

reduce known PDEs. In many cases no solution is known for the reduced equation and therefore no solutions of the original equation can be derived. Although additional reductions could be investigated by using the same or another methods (nonlocal symmetries, λ-symmetries [20], etc.), the search of these further reductions may lead to very difficult problems and some specific methods to solve them may become necessary. In this paper it is considered a nth-order ordinary differential equation of the form P (y, v, v 0 , . . . , v (n) ) = 0,

(1)

where P (y, v, v 0 , . . . , v (n) ) is either polynomial in the variables v, v 0 , . . . , v (n) with coefficients that are functions of the independent variable y or polynomial in the variables v 0 , . . . , v (n) with coefficients that are functions of y and v. For the class of equations that are polynomial in v, v 0 , . . . , v (n) , several ad hoc reduction methods have been investigated. Some of the known methods that are related to the study in this paper correspond to reductions that have the general form v 0 = ak (y)v k + · · · + a1 (y)v + a0 (y), where k is a positive integer and, in principle, a0 (y), . . . , ak (y) are arbitrary smooth functions. Some of these known methods are: 1. The tanh method [2], that uses the fact that the function v defined by v(y) = tanh (y) satisfies the equation v 0 = 1 − v 2 . 2. The modified extended tanh function method [5, 7, 8] and the modified extended direct algebraic 0 2 method [1], that use reduced equations Pm of thei form v = b + v , maybe after a previous rational change of variables of the form u = i=−h ci v , where b and the ci are constant, for −h ≤ i ≤ m. 3. The so-called simplest equation method [15, 16], that may be considered as a representative for some other methods to obtain reductions with ai (y) constant, 0 ≤ i ≤ k. 4. The general case in which the functions ai (y) are arbitrary smooth functions has also been previously considered (see [26] and the references therein). A second type of reduction methods that have also been studied in the literature considers reduced equations of the general form: v 00 = ak (y, v)(v 0 )k + · · · + a1 (y, v)v 0 + a0 (y, v). This type of reductions has been studied in [27]. For equations of the form (1), in this paper we introduce a new reduction method to obtain ODEs whose solutions are also solutions of (1). The method is applicable to obtain: (a) First-order reduced equations of the form v0 =

pM X

ai (y)v i/q ,

(2)

ai (y, v)(v 0 )i/q ,

(3)

i=pm

for the case that P is also polynomial in v. (b) Second-order reduced equations of the form 00

v =

pM X

i=pm

2

where q ∈ Z+ and pm , pM are integers (not necessarily positive) such that pm ≤ pM . In general, an arbitrary equation (1) does not necessarily admit reductions of the form described above; therefore the search of such reductions must be considered as an ansatz. For pm = 0, reductions (2) have previously been studied in [28]. Since in (2) and (3) pm may be a negative integer, the types of reductions in this paper are strictly more general than those considered in the above-mentioned literature; in fact, several of the reductions we have reported in the examples correspond to negative values for pm . When the given equation has exponents with integer parameters, it may be quite involved the determination of the admissible reductions. The procedure of Subsection 2.1 permits to deal with equations that have integer parameters in the exponents and performs a parametric study of the exponents that are relevant for determining possible reductions. It must also be remarked that two of the equations considered in the examples have parameters in the exponents of v within the corresponding equation (1). An important issue about the reductions (2) and (3) is that the integer numbers pm , pM and q have only to satisfy the mentioned conditions and therefore an infinite number of reduced equations of the types (2) or (3) could be investigated for a given equation of the form (1). In subsections 2.1 and 2.2 we provide a method to analyse the possible reductions for Eq. (1). This method lets to determine if the given equation may admit such reductions and, in case of admissibility, the possible values of pm , pM and q can be determined. The reduction method in this paper has been applied to obtain new reductions for several equations: 1. The method to obtain first-order reductions (2) is illustrated, in Subsection 3.1.1, by considering the equations of the form av m v 00 −bv 0 +cv k = 0, that corresponds to the travelling wave solutions for the partial differential equation wt = awm wxx + cwk , and appears as Eq. 1.1.9.19 in [25]. 2. In Subsection 3.1.2, it is considered the equation v 00 = (ay b v r + cy d v s )(v 0 )t , which appears as Eq. 2.6.2 in [24] and it is a generalization of the Emden-Fowler equation [6, 9]: u00 = Axn um . 3. The method to obtain second-order reductions (3) is first applied, in Subsection 3.2.1, to the equation for travelling-wave solutions of the fifth-order Korteweg-de Vries equation (23) [2]. 4. In Subsection 3.2.2 some second-order reductions for a Painlev´e-Chazy equation [3] have been obtained. Some of the reductions of the form (2) or (3) that we have derived for the four mentioned equations can be difficult to solve; however, several of them can be explicitly solved (separated equations, linear equations, etc.) or correspond to well-known types of equations: such as the equations of Bernoulli, Riccati, Chini, etc. In the Appendix, several Maple programs have been included for easing the calculations, that may be rather involved for complicated equations. Although some lines of the codes correspond to the equation av m v 00 − bv 0 + cv k = 0, which is used to illustrate the method in detail, these codes can easily be adapted to deal with any concrete equation of the form (1).

3

2. Preliminary analysis of possible reductions 2.1. Reductions to first-order ODEs 2.1.1. The role of the highest exponent for reductions of type (2) Let us assume, for a moment, that the solutions of a reduced equation of the form v 0 = d0 (y)v k0 + · · · + ds (y)v ks ,

(4)

where k0 , k1 , . . . , ks ∈ Q, k0 > k1 > · · · > ks and d0 (y), ds (y) 6= 0, are also solutions of (1). The (formal) substitution of v 0 and its derivatives up to order n in the left-hand side of Eq. (1) provides a generalized polynomial (i.e., a polynomial with rational exponents) in the variable v whose coefficients depend on the coefficients of P and on the coefficients di (y), 0 ≤ i ≤ s, and their derivatives. This polynomial will be denoted by P(v). The term with the highest exponent comes from the expansion of some of the terms of the polynomial P . It can be checked (see Section 3 in [28]) that the expansion of any of these terms of P according to the powers of v also provides a polynomial whose term with the highest exponent depends only on d0 (y), k0 and the coefficients of P , but not on the remaining terms in the right-hand side of (4). Therefore, in order to determine the highest exponent of v in P(v) it is sufficient to consider the expansion of the term that has the highest exponent in the (generalized) polynomial expression of v 0 in (4). Similarly, in order to determine the lowest exponent of v in P(v) it is sufficient to consider the expansion of the term which has the lowest exponent in (4). 2.1.2. Determination of the highest and the lowest rational exponents associated to (2) The previous discussion motivates the following procedure for the determination of the highest and the lowest values of the exponents in (2) that could lead to non-trivial reductions. We first consider a generic term of (4), written as f (y)v α , and substitute v 0 by f (y)v α in (1), where α is an unknown rational number and f = f (y) is an unknown smooth function; then v 00 is substituted by f 0 (y)v α +αf (y)v α−1 (f (y)v α ) and so on. After the pertinent substitutions, it is obtained a (generalized) polynomial expression R(v) in the variable v of the form R(v) = f1 (y)v r1 (α) + · · · + fm (y)v rm (α) ,

(5)

where the coefficients f1 (y), . . . , fm (y) depend on α, the coefficients of P and f and its derivatives. It will be assumed that the functions f1 (y), . . . , fm (y) are not zero. The form of the exponents in (5) is ri (α) = ai α + bi ,

(1 ≤ i ≤ m)

where ai , bi are integer numbers. Now our aim is to determine, for any α ∈ Q, the highest exponent u(α) and the lowest exponent l(α) in (5): u(α) = max {ri (α) : 1 ≤ i ≤ m},

l(α) = min {ri (α) : 1 ≤ i ≤ m}.

(6)

For any α ∈ Q, the values l(α) and u(α) will be used to determine, in case of existence of a reduction of the form (2), the lowest and the highest possible exponents pm /q and pM /q, respectively. 4

Since l(α) = min {ri (α) : 1 ≤ i ≤ m} = −max {−ri (α) : 1 ≤ i ≤ m},

(7)

the study we provide below will be focused on the analysis of the mentioned highest exponent. The corresponding study for the lowest exponent would be completely analogous. Let us recall that any affine real function is convex and that its corresponding epigraph (i.e., the set of points lying on or above its graph) is convex. It is clear that the epigraph of u(α) is a convex set because it is the intersection of the epigraph of the affine functions r1 (α), . . . , rm (α). Hence, since a real function is convex if and only if its epigraph is a convex set, we conclude that u(α) is a convex piecewise affine function. Similarly,
it can be proved that the hypograph of l(α) i.e., the set of points lying on or below the graph of l(α) is convex and that l(α) is a concave piecewise affine function. As an example, let us consider the nonlinear equation

av 2 v 00 − bv 0 + cv 2 = 0,

(8)

where a, b, c are non-zero constants. This equation is a particular case of Eq. (17), with m = k = 2. In order to determine the functions l(α) and u(α) that correspond to Eq. (8), we substitute v 0 by
f (y)v α and v 00 by f 0 (v)v α + αf (y)v α−1 f (y)v α in the left-hand side of Eq. (8). Then we obtain the (generalized) polynomial R(v) = a αf 2 v 2α+1 + af 0 v α+2 − bf v α + cv 2 .

(9)

The list of exponents of v that appear in (9) is r1 (α) = 2,

r2 (α) = α,

r3 (α) = α + 2,

r4 (α) = 2α + 1.

(10)

The graphs of these four affine functions are the straight lines L1 , L2 , L3 and L4 , respectively, that are displayed in Fig. 1. The graphs of the piecewise affine functions u(α) and l(α) that correspond to Eq. (8) also appear in Fig. 1. The epigraph of u(α) and the hypograph of l(α) are convex sets; u(α) is a convex piecewise afine function and l(α) is a concave piecewise afine function.

5

L4

β 6

u(α)

L3

4

L2

u(α) u(α)

L1

L1

2

l(α ) -2

l(α )

L3

-1

1

2

3

α L2 L4

l(α )

-2

Figure 1: Graphs of the functions ri (α), i = 1, 2, 3, 4, that appear in (10). The graphs of the piecewise affine functions u(α) and l(α) are displayed with thick lines.

2.1.3. Analytical expression for u(α) We now develop a procedure for determining u(α) in an explicit form. For a better understanding of the mentioned procedure it is convenient to identify any exponent of the form r(α) = aα + b with any of the following two auxiliary objects: (a) The straight line L of equation β = aα + b, in the plane with coordinates (α, β). (b) The pair (a, b). Let L be the collection of the m lines associated to the m exponents in (5). If L ≡ β = aα + b and ≡ β 0 = a0 α + b0 belong to L, we will write L  L0 if the corresponding pairs (a, b) and (a0 , b0 ) satisfy (a, b) ≤l (a0 , b0 ), where ≤l denotes the lexicographical order in R2 (for this order (a, b)
 L = hL1 , . . . , Lm i ≡ (a1 , b1 ), . . . , (am , bm ) , L0

where Li ≺ Lj for 1 ≤ i < j ≤ m. Let us observe that if Li ≺ Lj then:

i If ai = aj the lines Li and Lj are parallel and ri (α) = ai α + bi < rj (α) = ai α + bj for α ∈ R. Therefore, among the lines in L with the same slope, only the line with the highest coefficient of type b is relevant for determining u(α). bj − bi ii If ai < aj then Li is not parallel to Lj and the abscissa αi,j = of the intersection point ai − aj between Li and Lj satisfies ri (α) = ai α + bi > rj (α) = aj α + bj for α < αi,j and ri (α) < rj (α) for α > αi,j .

Taking into account the preceding remarks, we will consider a set of lines L0 ⊂ L by choosing, among the lines with the same slope, the elements with higher b. The set L0 can be built as follows. Let i1 the highest index such that ai1 = a1 , which implies that ai1 < ai1 +1 . Let i2 the highest index such

6

that ai2 = ai1 +1 . We can proceed in this way, by constructing the indices i3 , . . . , ir−1 , and the index ir = n satisfying air = air−1 +1 . In terms of the pairs (ai , bi ), the set L can be written as 

L = (ai1 , b1 ), . . . , (ai1 , bi1 ), (ai2 , bi1 +1 ), . . . , (ai2 , bi2 ), . . . , (air , bir−1 +1 ), . . . , (air , bir ) . The ordered set L0 ⊂ L is then defined as 

L0 = (ai1 , bi1 ), (ai2 , bi2 ), . . . , (air , bir ) .

For the example considered above



 L = L1 , L2 , L3 ,L4 ≡ (0, 2), (1, 0), (1, 2), (2, 1) .  L0 = L1 , L3 , L4 ≡ (0, 2), (1, 2), (2, 1) .

(11)

Coming back to the general case, we now use L0 to determine the list of lines that delimit the epigraph of the piecewise affine function u(α), going from left to right. We set p1 = 1 and j1 = ip1 = i1 ; the first line that delimits that epigraph is Lj1 = Li1 . Now we determine the abscissas of the intersection points between Lj1 and Lip , p = p1 + 1, . . . , r: αj1 ,ip =

bip − bj1 aj1 − aip

and select the lowest of these numbers, which is denoted by α1 . It may happen that several of the lines Lip with p = p1 + 1, . . . , r have a common intersection point with Lj1 , at (α1 , rj1 (α1 )). We denote by ip2 the highest index of the lines with former
property, and we set j2 = ip2 . With this notation the former intersection point is αj1 ,j2 , rj1 (αj1 ,j2 ) . The line Lj2 will be the second line that delimits the epigraph of u(α). Similarly, we determine the abscissas of the intersection points between Lj2 and Lip , p = p2 + 1, . . . , r: bi − bj2 αj2 ,ip = p aj2 − aip

and select the lowest of these numbers, denoted by α2 . It may happen that several of the lines Lip with p = p2 + 1, . . . , r have a common intersection point with Lj2 at (α2 , rj2 (α2 )). We denote by ip3 the highest index of the lines with former property and denote j3 = ip3 . The line Lj3 will be the third line that defines the epigraph of u(α). By proceeding in this way, after a certain number s ≤ r of stages we get the index js = ir = m. The ordered list of straight lines that delimit the epigraph of the piecewise affine function u(α) is hLj1 , . . . , Ljs i.

(12)

The corresponding analytical expression for u(α) is   rj1 (α), α ≤ αj1 ,j2 , rj (α), α ∈]αjk−1 ,jk , αjk ,jk+1 ], u(α) =  k rjs (α), α > αjt−1 ,jt , 7

(k = 2, . . . , s − 1).

(13)

The analytical construction of the function l(α) can be obtained in a similar way, by using (7). For the example we have considered at the end of Subsection 2.1.2 and by using (11), we have: 1. α1,3 = 0, α3,4 = 1. 2. u(α) = 2 for α < 0, u(α) = α + 2 for α ∈ [0, 1], and u(α) = 2α + 1 for α > 1. 3. By using (7) and the described procedure, it can be checked that l(α) = 2α + 1 for α < −1, l(α) = α for α ∈ [−1, 2], and l(α) = 2 for α > 2. Remark: If Eq. (1) has at least a parameter in the exponents of v or any of its derivatives then the exponents in (5) may depend on these parameters, the determination of the functions u and l becomes more involved, and a parametric study is necessary to facilitate the classifications described in the presented procedure. In the Appendix we have included the Maple program findhulls that eases the search of the functions u and l when several parameters appear in the exponents of the initial equation; this program works exclusively with the list of exponents that appear in (5) or (16). 2.1.4. Existence of reductions of the form (4) We now analyse the existence of reductions of the form (4). The term d0 (y)v k0 , which has the highest exponent in (4), will be denoted by f (y)v α and leads to a polynomial expansion of the type (5) whose term with the highest exponent has a coefficient that can depend on α, some coefficients of the equation (1), and f (y) and its derivatives. The proposed method can be applied when the coefficient of the term with the highest exponent in (5) may be zero for some f (y) 6= 0, among other conditions. Therefore, we must analyse for α in each one of the intervals ] − ∞, αj1 ,j2 ], ]αj1 ,j2 , αj2 ,j3 ], . . . , ]αjs−2 ,js−1 , αjs−1 ,js ], ]αjs−1 ,js , +∞[ if the coefficient of the term with the highest exponent vanishes for some f 6= 0. The transition points αjk ,jk+1 , 1 ≤ k ≤ s − 1, must be considered independently, because at these points there are two (or more) terms with the same exponent; the resulting coefficient is the sum of the corresponding coefficients and it might be zero for some f 6= 0. It may happen that for a given α ∈ Q there are no reductions of the form (4) with k0 = α; this occurs when the coefficient with the highest exponent in R(v) is zero only for f (y) = 0. The study of the lowest exponent in (4) can be driven similarly. We denote by Am the set of the α ∈ Q which (in principle) may lead to reductions of the form (4) with ks = α. It must be mentioned that a reduction of the form (4) with k1 = α or ks = α, and α ∈ {0, 1} cannot a priori be discarded: (a) For α = 0, the procedure indicates that v 0 must be substituted by f (y) in (1), v 00 by f 0 (y), and so on. These substitutions lead to a polynomial whose coefficients depend on the derivatives of f and could be zero for some f 6= 0. (b) For α = 1, v 0 must be substituted by f (y) v, v 00 by f 0 (y) v + f (y)(f (y)v) = [f 0 (y) + f (y)2 ]v, and so on. After the pertinent substitutions, the coefficients of v k , k > 1, in the resulting polynomial expression (5) could also be zero for a non-zero f . 8

Finally, if we can determine a pair (αm , αM ) with αm ∈ Am ∩ Q, αM ∈ AM ∩ Q and αm ≤ αM then a reduction of the form (4) may exist. If     p1 p2 p 1 q2 p 2 q1 (αm , αM ) = , = , (14) q1 q2 q1 q2 q1 q2 is any admissible pair then it might exist a reduction of the form (2) by considering pm = p1 q2 , pM = p2 q1 and q = q1 q2 . Let us observe that the fractions pq11 and pq22 not necessarily are irreducible. This means, for instance, that if αm = 21 = 24 = · · · and αM = 1 = 22 = 33 = · · · then the pairs ( 12 , 22 ), ( 42 , 44 ), ( 36 , 66 ), . . . are admissible, have a common denominator and provide several possible values for pm , pM and q. The final step for obtaining a reduction of (1) of the form (2) may be described as follows. If in P (y, v, v 0 , . . . , v (n) ) we substitute v 0 , . . . , v (n) by the corresponding derivatives of the right-hand side of (2) we get a generalized polynomial P(v) in the variable v. Its coefficients depend on: (1) the coefficients of P , and (2) the ai (y), pm ≤ i ≤ pM , and their derivatives up to order n − 1. If we impose that the coefficients of P(v) be zero then we obtain an overdetermined system of algebraic and ordinary differential equations for the functions ai (y), pm ≤ i ≤ pM . If this system has a solution apm (y), . . . , apM (y) then any solution of the corresponding equation (2) is a solution of (1). In the Appendix we include a Maple program that lets to determine the reduction of type (2) for Eq. (17); with minor modifications the resulting program could be used for the study of other polynomial equations. For Eq. (8) we have AM = [0, 1] (because {0, 1} ⊂ [0, 1] and the coefficient of v α+2 can be zero for some f 6= 0) and Am = {−1, 0, 1} (in this case α = −1 is a transition point for l(α)). By using the mentioned program with m = k = 2, pm = −1, pM = 1 and q = 1 we obtain the reduction b c v 0 = − v −1 − y + c1 , a a which is included in Case 1 of Table 2. Remark: If the equation depends on some parameters in the exponents, the procedure should be adapted to the results of a parametric study of the equation. The equations in the examples provided in this paper show how this adaptation could be done. 2.2. Second-order reductions of the form (3) In this subsection the possible values of the exponents for second-order reductions of the type (3) are considered. What has been said for reductions of type (2) can be easily adapted to reductions of the type (3). In this case, for the study of the highest and lowest exponents in (3), instead of reductions of the form (4), we must consider a reduced equation of the form v 00 = f0 (y, v)(v 0 )k0 + · · · + fs (y, v)(v 0 )ks ,

(15)

where k0 , k1 , . . . , ks ∈ Q, k0 > k1 > · · · > ks and f0 (y, v), fs (y, v) 6= 0, whose solutions are also solutions of (1). Now, the conclusion would be that for obtaining the highest exponent in (3) it 9

is sufficient to consider the term f0 (y, v)(v 0 )k0 , that has the highest exponent in the (generalized) polynomial expression of v 00 in (15). Similarly, for determining the lowest exponent in (3) it is enough to consider the term fs (y, v)(v 0 )ks . The determination of the highest and the lowest rational exponents of a reduction of the form (3) is also very similar. In this case v 00 is substituted by f (y, v)(v 0 )α , v 000 by
fy (y, v)(v 0 )α + fv (y, v)(v 0 )α+1 + αf (y, v)(v 0 )α−1 f (y, v)(v 0 )α ,

and so on. As a consequence we obtain a generalized polynomial expression in v 0 of the form R(y, v) = f1 (y, v)(v 0 )r1 (α) + · · · + fm (y, v)(v 0 )rm (α) ,

(16)

where ri (α) = ai α + bi for 1 ≤ i ≤ n and ai , bi ∈ Z, whose coefficients should be identically zero. The analysis of these exponents is as for first–order reductions; in particular, the corresponding functions u(α) and l(α) can be determined by using the procedure described in Subsection 2.1. Two examples of second–order reductions appear in subsections 3.2.1 and 3.2.2. It must also be mentioned that the reductions of the form (15) with k0 = α or ks = α and α ∈ {0, 1, 2} cannot a priori be discarded.

(a) For α = 0, v 00 must be substituted in (1) by f (y, v), v 000 by fy +fv v 0 , and so on. These substitutions lead to a polynomial whose coefficients depend on the derivatives of v and might be zero for some f 6= 0.
(b) For α = 1, v 00 must be substituted by f (y, v) v 0 , v 000 by f (y, v)2 + fy (y, v) v 0 + fv (y, v)(v 0 )2 and so on. After the substitutions, the coefficients of v 0 in the resulting polynomial expressions could also be zero for a non-zero f . (c) For α = 2, the substitutions of v 00 by f (y, v) (v 0 )2 , v 000 by (2f 2 + fv )(v 0 )3 + fy (v 0 )2 , and so on, do also lead to a polynomial expression whose coefficients could be identically zero for a non-zero f.

When P is polynomial in the variables v 0 , . . . , v (n) , the reduction method tries to determine the coefficients ai (y, v), pm ≤ i ≤ pM , in such a way that any solution v = v(y) for (3) is also a solution for (1). It is clear that this condition holds if the coefficients, that depend on (y, v), of the resulting generalized polynomial in v 0 are identically zero. So, it is considered the system S of partial differential equations obtained by equating to zero these coefficients of the powers of v 0 , whose unknown functions are apm (y, v), . . . , apM (y, v). If this system S admits the particular solution apm , . . . , apM then any solution v = v(y) of the corresponding equation (3) does also solve (1). The mentioned overdetermined system can be studied either directly or by using Gr¨ obner bases. In the Appendix we include a Maple program that lets to determine reductions of type (3) for Eq. (22).

3. Some examples of reductions 3.1. Examples of reductions to first-order polynomial equations 3.1.1. A detailed account of the procedure for an equation with two parameters. In this subsection we consider a second-order equation with two parameters, in order to describe in full detail the method explained in Subsection 2.1 to analyse the possible reductions of type (2). 10

The family of equations we consider is av m v 00 − bv 0 + cv k = 0,

(17)

where k and m are integer numbers and a, b, c are constants. In order to consider only non-trivial equations, we also assume that a, b, c 6= 0. Eq. (17) corresponds to travelling wave solutions of the evolution equation wt = awm wxx + cwk , which appears as Eq. 1.1.9.19 in [25].
By substituting v 0 by f (y)v α and v 00 by f 0 (v)v α + αf (y)v α−1 f (y)v α in the left-hand side of Eq. (17) we obtain the (generalized) polynomial a αf 2 v 2α+m−1 + af 0 v α+m − bf v α + cv k .

(18)

The exponents of the powers of v in (18) are 2α + m − 1, α + m, α and k. We observe that some of these exponents depend on the parameters k and m of the equation. Since k and m are arbitrary parameters, the relative order between these exponents (according to the order specified in Subsection 2.1.3) is not fixed, it depends on m and k. The functions ri (α) = ai α + bi , 1 ≤ i ≤ 4, that these exponents determine are given by r1 (α) = k,

r2 (α) = α,

r3 (α) = α + m,

r4 (α) = 2α + m − 1.

(19)

The graphs of these functions are straight lines that will be denoted by L1 , L2 , L3 and L4 , respectively. They are graphically represented in Figure 1 for the values of the parameters m = k = 2.

 The order considered in Subsection 2.1.3 provides the ordered lists of lines: L = L1 , L2 , L3 , L4

 for m > 0 and L = L1 , L3 , L2 , L4 for m < 0. For m = 0 we have  L2 = L3 , and L0 = L1 , L2 , L4 . 0 0 The corresponding reduced lists of type L are: L = L1 , L3 , L4 for m > 0, and L = L1 , L2 , L4 for m ≤ 0. In Table 1 we show the ordered sequences of straight lines that define the functions u and l, depending on the values of the parameters k and m; it must be observed that, since k and m are integer numbers, the condition m > 0 is equivalent to m ≥ 1. Table 1 can be easily obtained by using the program findhulls in the Appendix.

We now analyse the existence of possible reductions. In this case Am and AM depend on the parameters m and k. It is clear that the only coefficient in (18) that may be zero for some f 6= 0 is the coefficient of v α+m . Therefore, if α + m is the highest exponent in (18) then the conditions α + m ≥ 2α + m − 1, α + m ≥ α and α + m ≥ k must be satisfied. This implies that α ≤ 1, m ≥ 0 and α ≥ k − m, respectively. As we have mentioned in Subsection 2.1.4, if α = αi,j (i.e., α is a trasition point and ri (α) = rj (α) for some pair i, j with 1 ≤ i < j ≤ 4) the coefficient of v ri (α) = v rj (α) is the sum of the coefficients of v ri (α) and v rj (α) in (18) and therefore other possibilities of vanishing the coefficient of the higher exponent of v in (18) appear. 11

Case m>0

m=0

m<0

Subcase k ≥1+m 1+m>k >1−m 1−m≥k k>1 k=1 k<1 k ≥1−m 1−m>k >1+m 1+m≥k

Upper lines hL1 , L4 i hL1 , L3 , L4 i hL1 , L3 , L4 i hL1 , L4 i hL1 , L4 i hL1 , L2 , L4 i hL1 , L4 i hL1 , L2 , L4 i hL1 , L2 , L4 i

Lower lines hL4 , L2 , L1 i hL4 , L2 , L1 i hL4 , L1 i hL4 , L2 , L1 i hL4 , L1 i hL4 , L1 i hL4 , L3 , L1 i hL4 , L3 , L1 i hL4 , L1 i

Table 1: For Eq. (17), ordered sets of straight lines that define the graphs of the functions u(α) and l(α), according to the values of the parameters k and m.

For the obtainment of reductions to equations of the type (17), we have only considered the cases where a, b, c, m 6= 0. Some examples of reductions that can be obtained by using the procedure of Subsection 2.1 are shown in Table 2. It is implicitly assumed that the reductions are only valid when the corresponding terms are defined; for instance, in the first reduction it is assumed that (−m + 1)a 6= 0. Case 1 2 3 4 5

Conditions k=m k = 1 − m, a k a2k − b ak + c = 0 k = 1 − m, a c (k + 1)2 = b2 m = 3, k = −2, 25a c + 2b2 = 0

Reduction b −m+1 − c y + c v 0 = (−m+1) 1 a av v 0 = ak v k b 1 v 0 = a(k+1) v k + y+c v 1 q 1 1 15c −2 v 0 = y+c v − 2 + 5c v ± b(y+c b v 1 1) v0 =

m = −1, k = 2, 9a c = b2

3c 2 b v

+

2y+c1 v y 2 +c1 y+c2

+

2b 3c(y 2 +c1 y+c2 )

Table 2: Some of the reductions that have been obtained, when the conditions in the second column are accom-

plished. In the third column c1 and c2 are arbitrary constants.

Let us observe that the reduction in Case 1 is a Chini equation that admits implicit solutions for m = 21 . The reduction of Case 3 is a Bernoulli equation. It should also be observed that the reduction of Case 4 corresponds to a negative value of pm ; in fact pm = −4, pM = 2, q = 2. Therefore this reduction cannot be obtained by using the previous methods considered in the introduction. The reduced equation that corresponds to this Case 4 admits the Lie point symmetry X = 3(y +c1 )∂y +v∂v and has solutions that can be expressed in an implicit form. The reduction in Case 5 is a Riccati equation. As far as we have been able to check, the reductions in Table 2 do not appear in [25].

12

3.1.2. Reductions for a 3-parameter polynomial ODE In this section we consider the equation v 00 = (a y b v r + c y d v s )(v 0 )t ,

(20)

where r, s, t ∈ Z, r ≥ s and a, b, c, d are real constants. This equation corresponds to Eq. 2.6.2 in [24]; it also appears in Section 44.5 of [30]. Eq. (20) with t = 0 appears independently in that references. For Eq. (20), the substitutions of v 0 by f v α and of v 00 by f 0 (v)v α + αf (y)v α−1 (f (y)v α ) lead to
αf 2 v 2α−1 + f 0 v α − f t ay b v tα+r + cy d v tα+s .

The exponents of v in this last expression determine the functions ri (α): r1 = α,

r2 = 2α − 1,

r3 = tα + r,

r4 = tα + s.

(21)

Since r, s and t are arbitrary integer parameters, the relative order between these exponents (according to the order specified in Subsection 2.1.3) is not fixed: it depends on r, s and t. The straight lines that correspond to the exponents (21) will be denoted by L1 , L2 , L3 and L4 , respectively. The abscissas
of the intersection points between these straight lines numbers αi,j such that ri (αi,j ) = rj (αi,j ) are: α1,2 = 1, α1,3 =

s r+1 s+1 r , α1,4 = , α2,3 = , α2,4 = . 1−t 1−t 2−t 2−t

Let us observe that if r = 1 − t then α1,2 = α1,3 = α2,3 = 1. Similarly, if s = 1 − t then α1,2 = α1,4 = α2,4 = 1. If s = r = 1 − t then the abscissas αi,j are coincident. For t = 2 the straight lines L2 , L3 and L4 are parallel. Similarly, for t = 1 the lines L1 , L3 and L4 are parallel. We will consider independently the cases t = 0 and t 6= 0. A.- For t = 0 the functions u(α) and l(α) are determined through the ordered sets of straight lines shown in the following table:

Case r=s

r>s

Subcase r≥2 r=1 r≤0 s≥2 r≥1≥s 0≥r

Upper lines hL3 , L2 i hL3 , L2 i hL3 , L1 , L2 i hL3 , L2 i hL3 , L2 i hL3 , L1 , L2 i

Lower lines hL2 , L1 , L3 i hL2 , L3 i hL2 , L3 i hL2 , L1 , L4 i hL2 , L4 i hL2 , L4 i

Table 3: For Eq. (20) with t = 0, ordered sets of lines that define the graphs of the functions u(α) and l(α),

according to the values of the parameters r and s.

13

For equation (20) with t = 0 some of the reduced equations we have been able to obtain appear in the last columns of Tables 4 and 5. These tables show the relationships between the parameters that must be satisfied for the existence of the corresponding reductions. Table 4 corresponds to an arbitrary d and Table 5 corresponds to the values of d given in its fifth column. In Table 4, c1 and c2 are arbitrary constants. It is implicitly assumed that the presented solutions are only valid when the coefficients that appear in their expressions are defined. For instance, for the first reduction, it is assumed that s 6= −1, s 6= −3 and c/(s + 1) ≥ 0. N

s

r

b

a(s +

−1

4a2 + c d2 = 0 a01 = −a21 + ay b , a00 = −a0 a1 + cy d

Any

1

2 3

Any Any

2(d + 1) 2(d + 1)

4

−3

2s − 1 2s − 1

5

0

1

Any

6

0

3

−2(d + 3)

7

1

1

Any

8

2

3

2(d + 1)

d 2

3)2

−2

1

−1

Conditions = d(s + d + 3)

a(d + s + 2)2 = s c2 a(d + 1)2 = s c2

272 ac2 = 2 (d(d + 3)(2d + 3))2 a01 = −a21 + cy d + ay b , a00 = −a0 a1 2a(2d + 5)2 = c2

Reduction q 2c d2 s+1 v 0 = ± s+1 y v 2 − v0 = v0 = v0 =

d −1 s+3 y v y −1 v

c d+1 v s + d+s+2 y c d+1 v s d+1 y √ d (y + c1 )−1 v ± −c y 2 v −1

v 0 = a1 v + a0 v 0 = − d(d+3)(2d+3) y −d−3 v 2 27c d+3 −1 3c + 3 y v + 4d+6 y d+1 v 0 = a1 v + a0 v0 =

c d+1 v 2 + (d + 3)y −1 v 4d+10 y y −d−3 − (d+2)(d+3)(2d+5) c

Table 4: For Eq. (20) with t = 0 and d arbitrary, some of the reductions that have been obtained, when the conditions in the fifth column are accomplished.

Reductions 5 and 7 are linear equations; reductions 1, 2 and 4 are Bernoulli equations; reductions 6 and 8 are Riccati equations and reduction 3 is of separated variables. Let us observe that for some values of the parameter s reductions 1, 2 and 3 may have a negative value for pm and therefore cannot be obtained by using the previously known methods mentioned in the introduction. For some specific values of r, s, b and d and with the condition 108c4 = 625a3 , we have also obtained 1 2 2 13 c 13 3 the reductions to Abel equations shown in Table 5, where b1 = ( 25 c ) , b2 = ( 10 ) and b3 = 19 ( 10 c ) : N 9 10 11 12

s 4 4 4 4

r 5 5 5 5

d 1 −3 −4 −8

b 2 − 10 3 − 14 3 −10

Condition 108c4 = 625a3 108c4 = 625a3 108c4 = 625a3 108c4 = 625a3

Reduction v 0 = −b1 yv 3 − b2 v 2 5 4 2 v 0 = −b1 y − 3 v 3 − b2 y − 3 v 2 + 23 y −1 v − b3 y − 3 7 5 1 v 0 = b1 y − 3 v 3 + b2 y − 3 v 2 + 13 y −1 v + b3 y − 3 v 0 = b1 y −5 v 3 + b2 y −3 v 2 + y −1 v

Table 5: For Eq. (20), some special reductions that correspond to the case t = 0 and the indicated specific values of the parameters of the equation.

14

As far as we know, the reductions in Tables 4 and 5 are not included in Table 22 of [24]. B.- For t 6= 0 the functions u(α) and l(α) are determined through the ordered sets of straight lines shown in the following tables, for the cases r = s and r > s respectively:

Case r=s

r=s

r=s

Subcase t ≥ 3, t + r ≥ 2 t ≥ 3, t + r = 1 t ≥ 3, t + r < 1 t = 2, r ≥ 0 t = 2, r = −1 t = 2, r ≤ −2 t = 1, r ≥ 1 t = 1, r = 0 t = 1, r ≤ −1 t ≤ 0, t + r ≥ 2 t ≤ 0, t + r = 1 t ≤ 0, t + r ≤ 0

Upper lines hL1 , L3 i hL1 , L3 i hL1 , L2 , L3 i hL1 , L3 i hL1 , L2 i hL1 , L2 i hL3 , L2 i hL1 , L2 i hL1 , L2 i hL3 , L2 i hL3 , L2 i hL3 , L1 , L2 i

Lower lines hL3 , L2 , L1 i hL3 , L1 i hL3 , L1 i hL2 , L1 i hL2 , L1 i hL3 , L1 i hL2 , L1 i hL2 , L1 i hL2 , L3 i hL2 , L1 , L3 i hL2 , L3 i hL2 , L3 i

Table 6: For Eq. (20), ordered sets of straight lines that define the graphs of the functions u(α) and l(α) that correspond to the case t 6= 0 and r = s.

Case r>s

r>s

r>s

r>s

Subcase t > 2, t + r ≤ 0 t > 2, t + r > 0, t + s < 2 t > 2, t + s ≥ 2 t = 2, s ≥ −1 t = 2, s < −1, r ≥ 0 t = 2, r < 0 t = 1, s ≥ 0 t = 1, s < 0, r ≥ 0 t = 1, r < 0 t < 1, t + r ≤ 0 t < 1, t + r > 0, t + s < 2 t < 1, t + s ≥ 2

Upper lines hL1 , L2 , L3 i hL1 , L3 i hL1 , L3 i hL1 , L3 i hL1 , L3 i hL1 , L2 i hL3 , L2 i hL3 , L2 i hL1 , L2 i hL3 , L1 , L2 i hL3 , L2 i hL3 , L2 i

Lower lines hL4 , L1 i hL4 , L1 i hL4 , L2 , L1 i hL2 , L1 i hL4 , L1 i hL4 , L1 i hL2 , L1 i hL2 , L4 i hL2 , L4 i hL2 , L4 i hL2 , L4 i hL2 , L1 , L4 i

Table 7: For Eq. (20), ordered sets of straight lines that define the graphs of the functions u(α) and l(α) that correspond to the case t 6= 0 and r > s.

Let us recall that, since r, s, t ∈ Z, the condition t > 1 is equivalent to the condition t ≥ 2. In the 15

next table we include some of the reductions for Eq. (20) we have been able to find for some special relationships between the parameters. N 1 2

t Any 1

s 1−t Any

b Any −1

Conditions a01 = (ay b + cy d )at1 − a21 a=d

3

−1

Any

4

−1

2d+15 d−6

2

−3

1

d−3 2

272 a = −(d + 3)(d − 6)2

5

−1

0

6

−1

0

2

−1

1

2

−3

64a = −(d + 3)(d − 1)2

8

−1

Any

3s − 4

9

−1

Any

3 2s

10

Any

Any

7

r 1−t 0 3 2s

−1

−1

(s+1)(t−1) t−2

3 2 (d

+ 1)

−3 3 − 6s − 23 s − 3 d(t−1)−(t−2) t−2

a2 (d + 1)3 = 2c3 s2

2a2 = c(d + 3)

a = −(d + 3)(d + 2)2 d = −2s − 1, a(s − 1)2 = −c3 d = −s − 3, a2 s = 2c3

c 1−t ( s+1 ) = (2 − t)( ad )2−t

Reduction v 0 = a1 v c v 0 = y d ( s+1 v s+1 + c1 ) q s 2c d+1 v 0 = ± d+1 y 2 v2 q d d+3 3 v 0 = 2−s y −1 v + 3 c(d−6) y 3 v d−6 d+3 q 2c d+1 v 0 = y −1 v ± d+3 y 2 q d+1 c −1 v ± 2 2 v 0 = −d+1 y 4 d+3 y v 0 = −(d + 2)y −1 v + v 0 = y −1 v +

c d+2 (d+2)(d+3) y

c 1−2s v s−1 1−s y s

s

v 0 = y −1 v + ac y − 2 −1 v 2 1

d

1+s

1−t y 2−t v 2−t v 0 = ( a(2−t) d )

Table 8: For Eq. (20), some of the reductions that have been obtained, when the corresponding conditions on

the parameters are accomplished.

In Table 8, c1 denotes an arbitrary constant. The reductions in Table 8 do not appear in Table 25 of [24], where some known solutions for (20) are shown. The reductions correspond to either separated variables equations, linear equations or Bernoulli equations. Let us observe that pm may be negative for some values of the parameters. 3.2. Examples of reductions to second-order equations 3.2.1. Second order reductions for a fifth-order Korteweg-de Vries equation As an example of reduction to second-order equations of the type (3) we consider the equation that corresponds to travelling wave solutions for the fifth-order Korteweg-de Vries equation ut + uxxxxx + auuxxx + bux uxx + cu2 ux = 0,

(22)

where a, b, c and d are real constants and a b c 6= 0. The equations in the family (22) have been widely studied by several authors [2, 11, 12]. Eq. (22) constitutes a family of fifth-order KdV-like equations with three parameters. Several well-known equations that correspond to (22) for different sets of values of the parameters (see [27] and the references therein) are: 1. The fifth-order equation of the hierarchy of KdV-like equations derived by P. Lax [18] in 1968. 2. An equation introduced by K. Sawada and T. Kotera [29] in 1975, for finding N-soliton solutions for KdV-like equations. This equation was also proposed by P.J. Caudrey, R.K. Dodd and J.D. Gibbon [4]. 16

3. A completely integrable equation introduced by D. J. Kaup [14] and B.A. Kupershmidt [17]. 4. Although the equations of Lax, Caudrey-Dodd-Gibbon-Sawada-Kotera and Kaup–Kupershmidt are completely integrable, it must be mentioned that the Ito equation [13] is also included in the family (22) but it is not completely integrable. The search of travelling-wave solutions of the form u(x, t) = v(y), where y = x + d t, leads to the following ODE for v = v(y): v (5) + avv (3) + bv 0 v 00 + cv 2 v 0 + dv 0 = 0.

(23)

Some first-order and second-order reductions for this ODE have been obtained in [28] and [27], respectively. In this section we study if (23) admits reductions of type (3). By following the method indicated in Subsection 2.2, we substitute in the left-hand side of (23) v 00 by f (y, v)(v 0 )α and the derivatives v 000 , v (4) and v (5) by the corresponding expressions obtained by considering v as a function of y. After the substitution of v 0 by W in the resulting equation we obtain the (generalized) polynomial α(2α − 1)(3α − 2)f 4 W 4α−3 + 6α(2α − 1)fy f 2 W 3α−2 +α(12α − 1)fv d2 W 3α−1 + α(af
2 v + 3fy2 + 4f fyy )W 2α−1 2α + (avf + f α + (6α + 1)fy fv + (8α + 3)f yv W y yyy )W
f2α+1 2 + (3α − 2)fv + (4α)f fvv W + (bf + avfv + 3fyyv )W α+1 +(cv 2 + d)W + 3fyvv W α+2 + fvvv W α+3 .

(24)

From (24), and by using the notations in subsections 2.1 and 2.2, we obtain

 L = (0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (2, −1), (2, 0), (2, 1), (3, −2), (3, −1), (4, −3) , and consequently

 L0 = (0, 1), (1, 3), (2, 1), (3, −1), (4, −3) ;

i.e., among the eleven exponents r1 (α), . . . , r11 (α) that appear in (24), the relevant functions for determining u(α) are: r1 (α) = 1,

r5 (α) = α + 3,

r8 (α) = 2α + 1,

r10 = 3α − 1,

r11 (α) = 4α − 3.

(25)

Eq. (23) has no parameters in the exponents and the determination of the functions u(α) and l(α) is quite less involved than for former equations. By following the procedure of subsections 2.1 and 2.2, it can be checked that these functions are given by:   α ∈] − ∞, −2[,  1, 4α − 3, α ∈] − ∞, 1[, α + 3, α ∈ [−2, 2], u(α) = l(α) = 1, α ∈ [1, ∞[.  4α − 3, α ∈]2, ∞[,

For α ∈]2, ∞[ the highest exponent in (24) is 4α − 3 and the coefficient of W 4α−3 is zero only for α = 21 and α = 32 ; these values do not belong to the interval ]2, +∞[ and therefore there are no 17

reductions for αM ∈]2, ∞[. The coefficient of W α+3 in (24) is fvvv which might be zero for some nonzero f . Therefore, in principle, there may exist reductions with αM ∈ [−2, 2]. Finally, the coefficient of W in (24) is zero only for c = d = 0. Summing up, the possible values of αM are in the interval [−2, 2]. For α ∈] − ∞, 1[ the lowest admissible exponent is 4α − 3. As we have mentioned before, the coefficient of W 4α−3 is zero only for α = 21 and α = 23 , which are in the interval ] − ∞, 1[ and, in principle, there may exist reductions for αm = 12 , 23 , apart from αm ∈ {0, 1, 2}, which are possibilities that always might be admissible. Since Am = {0, 12 , 23 , 1, 2} and it must hold αm < αM , necessarily αM ∈ [0, 2] ∩ Q. We have found two reductions for (23) when the constants are related as follows: (a) For 100c = 9a2 , 10b = 13a, d = 0, v 00 =

3 2

(b) For 5c = −9a2 , 2b = 17a, d = 0, v 00 = −

  1  2 a 3 0 4  a  23 (v ) 3 − v (v 0 ) 3 . 5 5

 a 2 2 3  a  13 0 4 3 (v ) 3 + 3 v (v 0 ) 3 . 2 10 10

The reductions we have found correspond to αm = 23 , αM = 43 , pm = 2, pM = 4, and q = 3. We have not been able to find reductions for αm = 12 (with pm = 1 and qm = 2). The found reductions correspond to stationary solutions of (23) because d = 0. Since the independent variable does not appear in any of these two reductions, the equations can also be reduced to first-order equations which admit solutions that can be implicitly expressed through complicated expressions. 3.2.2. Reductions for a Painlev´e-Chazy equation In this subsection we consider a Painlev´e-Chazy equation. This equation was introduced by P. Painlev´e [22, 23]. J. Chazy [3] and R. Garnier [10] studied many properties of its solutions, in the context of extensions of the Painlev´e theory to third-order ODEs. The Chazy equation that is considered in this paper is v 000 = a (v 0 )−1 (v 00 )2 + b(v) v 0 v 00 + c(v) (v 0 )3 , (26) where a 6= 0 is a real constant and b(v), c(v) are non-zero smooth functions. Since b(v) and c(v) are arbitrary functions, it has no sense the search of reductions of the form (2). We will search for reductions of the form (3). The substitutions of v 00 and v 000 by f (y, v)(v 0 )α and its derivative, respectively, in (26), lead to (a − α)f 2 (v 0 )2α−1 − fy (v 0 )α + (f b − fv )(v 0 )α+1 + c (v 0 )3 .

(27)

In this case there are no parameters in the exponents of v 0 . It is easy to check that the piecewise affine functions u(α) and l(α) are given by:    2α − 1, α ∈] − ∞, 1[, 3, α ∈] − ∞, 2[, α, α ∈ [1, 3[, u(α) = l(α) = 2α − 1, α ∈ [2, ∞[,  3, α ∈ [3, ∞[. 18

Let us analyse the set AM of possible values for αM . As we have metioned in Section 2.2, {0, 1, 2} ⊂ AM . Since u(α) = 3 for α ∈]−∞, 2[ and the coefficient of (v 0 )3 in (27) is c 6= 0, we have AM ∩]−∞, 2] = {0, 1, 2}. The transition point α = 2 is already included in {0, 1, 2}. For α ∈]2, ∞[, u(α) = 2 α − 1 and the coefficient of (v 0 )2α−1 in (27) is zero only for α = a. Therefore, in principle, there may exist nontrivial reductions if a ∈]2, ∞[, with αM = a. Hence, AM = {0, 1, 2} if a < 2 and AM = {0, 1, 2, a} if a ≥ 2. We now analyse the set Am of possible values for αm . As before, {0, 1, 2} ⊂ Am . For ] − ∞, 1[ l(α) = 2α − 1 and the coefficient of (v 0 )2α−1 is zero for α = a. Therefore it may be αm = a for a < 1. The transition point α = 1 is already in the set {0, 1, 2}. For α ∈]1, 3[ we have l(α) = α and the coefficient of (v 0 )α is fy , that can be zero for some f 6= 0; hence ]1, 3[⊂ Am . The transition point α = 3 is another possible point for αm ; i.e., 3 ∈ Am . For α ∈]3, ∞[ we have l(α) = 3 and the coefficient of (v 0 )3 in (27) is c 6= 0. Therefore, Am = {0, a} ∪ [1, 3] for a ≤ 1 and Am = {0} ∪ [1, 3] for a > 1. Some of the reductions we have been able to find are listed below: (a) v 00 = g (v 0 )2 , being g = g(v) any function that satisfies the equation
g 0 − (a − 2)g + b g = c;

(28)

for this reduction αm = αM = 2.  0  1 g 0 2 1 g 0 0 a 00 (b) v = (v ) + v + (v ) , where c1 is an arbitrary constant, and g(v) must 1−a g y + c1 y + c1 0 0 )2 −gg 00 satisfy the equations b = (2a−3)g . This means that g = g(v) must be any c = 2(g (a−1)g , (a−1)g 2 particular solution of the linear equation g 0 = 3)2 c = (a − 1)b2 + (3 − 2a)b0 .

(a−1)b 2a−3 g

and the compatibility condition is (2a −

For this reduction, the values of αm and αM depend on a as follows: 1. If a =

p0 q0

< 1 then αm = a, 1 =

2. If 1 < a = 3. If 2 < a = (c)

v 00

p0 q0 p0 q0

1 = g + (1 − a)y

q0 q0

< 2 then αm = 1 = then αm = 1 = "

q0 q0

2q0 q0 , pm = p0 , pM = 2q0 and q = q0 . 0 = 2q q0 , pm = q0 , pM = 2q0 and q = q0 .

, αM = 2 = q0 q0

, αM = 2

,2=

2q0 q0 ,

αM = a, pm = q0 , pM = p0 and q = q0 .

#
(g 0 )2 − (2 − a) g + (1 − a)y g 00 0 2 c1 0 0 a (v ) + v + 0 1−a (v ) , where c1 is an ar(2 − a)g 0 (g ) (2a−3)g 00 g0 equation g 00 =

bitrary constant and g = g(v) must satisfy the equations b =

and c =

a(g 00 )2 −g 0 g 000 . (g 0 )2

b 0 Therefore, g(v) can be any particular solution of the linear 2a−3 g . The compatibility condition and the possible values for αm and αM are as for reduction (b). 0

0

(d) v 00 = g (v 0 )2 + h (v 0 )a , where g = g(v) and h(v) must satisfy b = (2 − a)g + hh , and c = g 0 − g hh . Hence, g(v) and h(v) may be particular solutions of Eq. (28) and the linear equation h0 = [b + (a − 2)g]h, respectively. The possible values for αm and αM are: 1. If a =

p0 q0

< 2 then αm = a, αM = 2 =

2. If 2 < a =

p0 q0

then αm = 2 =

2q0 q0 ,

2q0 q0 ,

pm = p0 , pM = 2q0 and q = q0 .

αM = a = 19

p0 q0 ,

pm = 2q0 , pM = p0 and q = q0 .

Let us observe that former cases do not impose a priori restrictions on the parameter a. Therefore in this study the cases a = 1 − n1 , with n ∈ Z \ {0, −1} are included. Some of these cases were studied by J. Chazy [3]. Furthermore, a may be a negative number and therefore the reduction (d) can have a term in v 0 with negative or rational exponent. The reductions (a) and (d) admit solutions that can be implicitly expressed. The reduction of Case (b) admits the symmetries   Z v g 1/(a−1) ∂v X1 = g 1/(1−a) ∂v , X2 = (y + c1 )∂y + g 1/(1−a) and admits solutions that can be implicitly defined.

Conclusions For an ordinary differential equation of the form (1) that is either polynomial in the variables v, v 0 , . . . , v (n) with coefficients that are functions of the independent variable y or it is polynomial in the variables v 0 , . . . , v (n) with coefficients that are functions of y and v, and may have exponents with integer parameters, a new method to obtain either first- or second-order reductions has been introduced; the reduced equations are the forms (2) or (3), respectively. A procedure to determine, in function of the parameters, if the given equation may admit such reductions and, if this is the case, to restrict the possible variations of the exponents in the reduced equations is also provided. The introduced method strictly generalizes several methods that have recently appeared in the literature. The method is applied to obtain first-order reductions for a generalized Emden-Fowler equation and second-order reductions for the travelling-waves equation of a fifth-order Korteweg-de Vries equation and for a Painlev´e-Chazy equation. Some of the reported reductions cannot be obtained by the previous reduction methods mentioned in the introduction. Several Maple programs to determine if a given polynomial equation admits any of the considered reductions and, when applicable, to determine the reduced equations, are included in the Appendix.

Acknowledgements The authors acknowledge the financial support of the Junta de Andaluc´ıa research group FQM377 and of the University of C´adiz (Plan Propio de Investigaci´ on PR2017-090). The authors also appreciate the useful comments of the anonymous referees.

Appendix In this appendix we describe the complete procedure we have followed, by using Maple, for obtaining the reductions of Eq. (17). We will also comment the changes that could be done in the code in order to obtain the reductions for the remaining equations.

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The Maple codes we have included in this appendix have been designed in order to be easy to understand and use. They are not necessarily the most efficient possible codes. Obviously, by using different software the efficiency to perform the necessary calculations for reducing a given equation might change. In order to obtain for Eq. (17) the generalized polynomial (18) (that corresponds to (5), by means of the substitution of v 0 by f (y)v α in Eq. (17)) we could use the following Maple code : with(PDEtools): ec:=a*v(y)^m*diff(v(y),y$2)-b*diff(v(y),y)+c*v(y)^k; simplify(dsubs(diff(v(y),y)=f(y)*v(y)^alpha,ec),power,symbolic);

For reductions of the type (3) the third line should be substituted by simplify(dsubs(diff(v(y),y$2)=f(y,v(y))*diff(v(y),y)^alpha,ec),power,symbolic); A program to determine the possible exponents in the reduced equation The procedure to study equations with or without parameters in the exponents of (1) is the following: 1. If the equation has some parameters, and we wish to assign fixed values for some of them, we start by providing a set of values to the pertinent parameters of the equation. 2. We store in the list lisini the exponents of type a α + b corresponding to (5) or (16). In the program we use al to represent α. 3. We use the list lis to store the different exponents that are in listini; i.e., we do not consider possible repetitions in the initial list that may appear for especific values of the parameters. 4. We determine the numbers α where two of the exponents determined by the elements of lis are coincident. These numbers are stored in the list lispun. 5. Checking the certainty of the comparisons. (a) We introduce the list of hypothesis or assumptions that we assume with respect the parameters (m > 0, m ≤ k,...). (b) The new list, initially empty, lisdes is created to store (after the introduction of a set of assumptions) the comparisons between parameters that MAPLE is not able to deal with. The comparisons are arranged with respect 0. (c) It is introduced a routine, called isok, that determines, for a given expression expr, if expr > 0, expr = 0 and expr < 0 are simultaneously false or at least one of them produces F AIL. (d) It is checked that the comparisons between coefficients of type a or b may be ensured. If any of them is not certain, then it is stored in lisdes. (e) It is checked if the comparisons between the elements of lispun may be ensured. If some of them is not certain then it is stored in lisdes. 21

6.

7.

8. 9.

10.

11.

(f) If, after the mentioned checkings the list lisdes is empty, then the results can be considered as valid. In the contrary a further analysis is required by providing more precise assumptions. We classify the exponents in lispun in groups in such a way that the elements of each group are the elements of lispun with the same value of a. Each group is ordered increasingly according to the coefficients b of the exponents of the group. The ordered groups are stored in the list liscat. For each group in liscat, we select the exponent with the highest coefficient b. These exponents are stored in the list lisup. It must be taken into account that if two or more exponents have the same a then only the exponents with the highest and the lowest b are relevant for the determination of u(α) and l(α), respectively. For each group in liscat, the exponent with the lowest coefficient b is selected. These exponents are stored in the list lisdown. Another routine, called sellist, takes a list li of exponents of the type aα + b (as lisup or lisdown) and it has as auxiliary variables: (a) a counter cont (initialized as 1), (b) the number nel of elements of the list li, (c) a list lisE that is initialized with the first element of li where the exponents that determine the dominant functions will be stored, (d) two auxiliary variables tem y minx and a global variable poinE which is a list (initialy empty), and it is used to store the points where the functions defined by the exponents in listE are coincident. The routine continues with a while loop that is executed whereas cont< nel.   (a) We store in the auxiliary variable tem the list of pairs A[ii], ii where ii varies from cont+1 up to nel and A[ii] is the unique solution of the equation rcont (α) = rii (α). (b) The pairs of tem are sorted increasingly according to the first coordinate. We choose the first component of the first element of the ordered list and it is assigned to the variable minx. (c) We include minx in the list poinE. (d) We select the elements in tem whose first component is minx and select the highest second component. (e) This highest second component is assigned to the variable cont. Let us recall that lisup is sorted according invreasing values of a and listdown is sorted according decreasing values of a. (f) We add the exponent whose index is cont in the list li to the list lisE. End while loop, end routine. Utilization of the routine for the determination of u(α). We call the routine with the list lisup and save the result in lineup. The value of poinE is assigned to pointup. The indices (in the original list lisini) that are in lineup are stored in the new list lineuppos. Utilization of the routine for the determination of l(α). We first define the list poinE as an empty list. We call the routine with the list lisdown and save the result in linedown. The value of poinE is assigned to pointdown. The indices (in the original list lisini) that are in linedown are stored in the new list linedownpos.

The Maple code to perform the tasks required in the previous procedure is the following: 22

(*Program findhulls*) restart; with(ListTools); lisini:= simplify([k, al, lis:= [op({op(lisini)})]; lispun:= [op({op(map(xx->

with(combinat);#Provide here the possible values of the parameters al+m, 2*al+m-1]); #exponents, in an arbitrary order # this removes possible repetitions, it can change in the initial order solve(op(1, xx)= op(2, xx), al), choose(lis, 2)))})];# Intersection points

isok:= proc (expr) local tem:={is(expr>0),is(expr=0),is(expr<0)}; tem= {false} or {FAIL} subset tem end proc assume(m> 1, k> m+1);#checking of the validity of orderings. lisdes:= {}; for zz in choose(lis, 2) do for xx in 0, 1 do tam:=coeff(op(1, zz)-op(2, zz), al, xx); if isok(tam) then lisdes:= lisdes union {simplify(numer(normal(tam)), size)} end if end do end do for zz in choose(lispun, 2) do tam:=op(1, zz)-op(2, zz); if isok(tam) then lisdes:= lisdes union {simplify(numer(normal(tam)), size)}) end if end do; print(’unequalities to be reconsidered’, lisdes) # print unsolved conditions with the provided assumptions. liscat:=map(sort,[Categorize((x,y)-> coeff(x,al)=coeff(y,al),lisini)],(x,y)->is(coeff(x,al,0) op(-1, x), liscat),(x,y)->is(coeff(x,al) op(1, x), liscat),(x,y)->is(coeff(x,al)>coeff(y,al))); sellis:= proc (li::list) local cont:=1, nel:=nops(li), lisE:=[op(1, li)], tem, minx; global poinE:= []; while cont< nel do tem:= sort([seq([solve(op(ii, li)= op(cont, li), al), ii], ii = cont+1 .. nel)],(x,y)->is(op(1,x) op(2, xx), select( xx-> op(1, xx)= minx , tem))); lisE:= [op(lisE), op(cont, li)] end do end proc; lineup:= sellis(lisup); pointup:= poinE; lineuppos:= map( xx-> SelectFirst( zz-> zz= xx, lisini, output= indices), lineup); linedown:= sellis(lisdown); pointdown:= poinE; linedownpos:= map( xx-> SelectFirst( zz-> zz= xx, lisini, output= indices), linedown);

Remark on the initial data and outputs of the program In what follows, we provide a short discussion on the use of the program. For Eq. (17), we have defined lisini as the list [k, al, al + m, 2 ∗ al + m − 1], which corresponds to (18). 1. With the assumptions m < 0, k ≥ 1 − m, the program provides the data that appear in the seventh row of Table 1. 2. With the assignation m = 0 and the assumption k > 1, the program provides the fourth row in Table 1. 3. It may happen that different assumptions on the parameters of the equation provide the same results; in this case they are summarised in the same line of Table 1. For instance, the program findhulls indicates that the assumptions m > 0, k ≥ 1 + m are not conclusive because the comparison −m + 1 <> 0 remains undetermined. In order to obtain the ordered lists of lines that appear in the first row of Table 1, two cases have been considered independently: m > 1

23

and m = 1. The analysis of these two cases provides the same results, that are shown in that row. For Eq. (23), the values of u(α), l(α) that appear in Subseccion 3.2.1 have been obtained by defining the list lisini as [1, al, al + 1, al + 2, al + 3, 2 ∗ al − 1, 2 ∗ al, 2 ∗ al + 1, 3 ∗ al − 2, 3 ∗ al − 1, 4 ∗ al − 3], which is a consequence of (24). Program to obtain reductions of type (2) The following program corresponds to Eq. (17) with m = 3 and k = −2. This case appears in the fourth row of Table 2. The variables pp, pm, qq correspond to pM , pm , q, respectively. with(DEtools); with(PDEtools); with(PolynomialTools); m:= 3; k:= -2; ec:= a*v(y)^m*diff(v(y),y$2)-b*diff(v(y),y)+c*v(y)^k qq:= 2; pp:= 2; pm:= -4; aux1:= numer(normal(simplify(eval(dsubs(diff(v(y),y)= sum(a[i](y)*v(y)^(i/qq), i= pm .. pp), ec), v(y)= V^qq), power, symbolic))): aux2:= [op(CoefficientList(aux1, V)), a*b*c <> 0]; answ:= casesplit(aux2, parameters = [a, b, c], ctl= 50)

As we have commented above, to obtain a reduction as a function of the parameters may be rather involved. A preliminary study to determine the parametric regions where the same sets of lines define u(α) or l(α) has been done. For any fixed set of parameters, we have used this program to find the reductions, in case of existence. If for several sets of parameters the form of the found reduction is similar, then we have tried to check if a more general reduction is valid. In this case we have substituted the several particular reductions by the more general one. Reductions 1, 2 and 3 in Table 2 have been obtained in this way. However, in other cases, we have only found reductions for some specific values of the parameters: reductions 4 and 5 in Table 2 are of this type. A code to check the first reduction in Table 2 is simplify(eval(dsubs(diff(v(y),y)=b/(a*(1-m))*v(y)^(1-m)-c*y/a+c1,ec),k=m));

Program to obtain second-order reductions of type (3) The following program corresponds to second-order reductions for Eq. (22). restart; with(DEtools); with(PDEtools); ec:= diff(v(y),y$5)+a*v(y)*diff(v(y),y$3)+b*diff(v(y),y)*diff(v(y),y$2)+c*v(y)^2*diff(v(y),y)+d*diff(v(y),y); pm:= 2; pp:= 4; qq:= 3; pru:= sum(a[k](y, v(y))*(diff(v(y), y))^(k/qq), k= pm .. pp); aux1:=simplify(convert(eval(dsubs(diff(v(y),y$2)=pru, ec),[diff(v(y),y)=V^qq, v(y)=w]),diff), power,symbolic): aux1:=[coeffs(collect(numer(normal(aux1)), V), V), a*b*c <> 0]: answ:= casesplit(aux2, parameters = [a,b,c,d], ctl = 200);

24

References [1] L. M. B. Assas, New exact solutions for the generalized Kawahara and modified Kawahara equation using the modified extended direct algebraic method, International Journal of Management Science and Engineering Management 4 (4) (2009) 294–301, doi:10.1080/17509653.2009.10671082. [2] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R. S. Martino, J. C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, Journal of Symbolic Computation 37 (2004) 669–705, doi:10.1016/j.jsc.2003.09.004. [3] J. Chazy, Sur les ´equations diff´erentielles du troisi`eme ordre et d’ordre sup´erieur dont l’int´egrale g´en´erale a ses points critiques fixed, Acta Mathematica 34, (1911) 317–385. [4] P. J. Caudrey, R. K. Dodd, and J. D. Gibbon, A new hierarchy of Korteweg-de Vries equations, Proceedings of the Royal Society London A, 351, no. 1666, pp. 407 -422, (1976). [5] S. El-Wakil, M. Abdou, Modified extended tanh-function method for solving nonlinear partial differential equations, Chaos, Solitons & Fractals 31 (5) (2007) 1256–1264, doi: 10.1016/j.chaos.2005.10.072. [6] R. Emden, Gaskugeln, Anwendungen der mechanischen W¨ armentheorie auf Kosmologische und metheorologische Probleme, Leipzig (1907). [7] E. Fan, Travelling wave solutions for nonlinear equations using symbolic computation, Computers & Mathematics with Applications 43 (6) (2002) 671–680, doi:10.1016/S0898-1221(01)00312-1. [8] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A 277 (4) (2000) 212–218, doi:10.1016/S0375-9601(00)00725-8. [9] R. H. Fowler, The solutions of Emden’s and similar differential equations, Month. Not. Roy. Astron. Soc. 91 (1930), 63–91. [10] R. Garnier, Sur les ´equations diff´erentielles du troisi´eme order dont l’int´egrale g´en´erale est uniforme et sur une classe d’´equations nouvelles d’ordre sup´erieur dont l’int´egrale g´en´erale a ses ´ points critiques fixes, Annales scientifiques de l’E.N.S. 3 s´erie 29 (1912) 1–126. [11] U. Goktas, W. Hereman, Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations, J. Symbolic Computation 11 (2008), 1–31. [12] M. Hickman, W. Hereman, J. Larue, U. Goktas, Scaling invariant Lax pairs of nonlinear evolution equations, Applicable Analysis 91 (2012) 381–402. [13] M. Ito, An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders. J. Phys. Soc. Jpn. 49, (1980) 771-778 [14] D. J. Kaup On the inverse scattering problem for cubic eigenvalue problems of the class Ψxxx + 6QΨx + 6RΨ = λΨ. Stud. Appl. Math. 62 (1980) 186-216 [15] N. A. Kudryashov, Exact solitary waves of the Fisher equation, Physics Letters A 342 (1) (2005) 99–106, doi:10.1016/j.physleta.2005.05.025. 25

[16] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons & Fractals 24 (5) (2005) 1217 – 1231, doi:10.1016/j.chaos.2004.09.109. [17] B. A. Kupershmidt A super Korteweg–de Vries equation: an integrable system Physics Letters 102A No. 5–6 (1984) 213–215. [18] P. D. Lax Integrals of nonlinear equations of evolution and solitary waves Commun. Pure Appl. Math. 21 (1968) 467-490 [19] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, The NIST Handbook of mathematical functions, NIST- Cambridge Univ. Press, (2010). [20] C. Muriel, J. L. Romero, New methods of reduction for ordinary differential equations, IMA J. Appl. Math. 66 (2) (2001) 111–125. [21] P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, 1993. [22] P. Painlev´e, Memoire sur les ´equations diff´erentielles dont l’int´egrale g´en´erale est uniforme. Bulletin de la Soci´et´e Math´ematique de France 28 (1900) 201–261. [23] P. Painlev´e, Sur les ´equations diff´erentielles du second ordre et d’ordre sup´erieur dont l’int´egrale g´en´erale est uniforme. Acta Mathematica 25 (1902) 1–85, doi:10.1007/BF02419020. [24] A. G. Polyanin, V. F. Zaitsev, Handbook of exact solutions for ordinary differential equations, second edition Chapman and Hall/CRC (2003). [25] A. G. Polyanin, V. F. Zaitsev, Handbook of nonlinear partial differential equations Chapman and Hall/CRC (2004). [26] J. Ram´ırez, J. L. Romero, C. Muriel, Reductions of PDEs to first order ODEs, symmetries and symbolic computation, Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49 doi:10.1016/j.cnsns.2015.04.022. [27] J. Ram´ırez, J. L. Romero, C. Muriel, Reductions of PDEs to second order ODEs and symbolic computation, Applied Mathematics and Computation 291 (2016) 122–136 doi:10.1016/j.amc.2016.06.043. [28] J. Ram´ırez, J. L. Romero, C. Muriel, Two new reductions methods for polynomial differential equations and applications to nonlinear PDEs, Journal of Computational and Applied Mathematics 333 (2018) 36–50 doi:10.1016/j.cam.2017.10.025. [29] K. Sawada, T. Kotera, A method for finding N-Soliton solutions of the KdV equation and KdV-like equation Prog. Theor. Phys. 51 (1974) 1355-1367 [30] D. Zwillinger, Handbook of ordinary differential equations, Academic Press, (1997).

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