Reductions of PDEs to first order ODEs, symmetries and symbolic computation

Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49

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Reductions of PDEs to first order ODEs, symmetries and symbolic computation J. Ramírez, J.L. Romero, C. Muriel∗ Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Spain

a r t i c l e

i n f o

Article history: Received 10 February 2014 Revised 22 September 2014 Accepted 21 April 2015 Available online 6 May 2015 Keywords: Reductions Modified mapping methods Generalized Abel equations Generalized elliptic equations

a b s t r a c t For ordinary differential equations which are polynomial in the dependent variable and its derivatives, two methods are provided to find, if possible, reductions to first-order ordinary differential equations. Computer codes for the underlying algorithms are also given. These techniques are applied to find new explicit solutions of relevant equations of Mathematical Physics. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The search of exact solutions for partial differential equations (PDEs) has motivated a huge amount of papers. Most methods try to find reductions of any given equation. If the reduced equation can be solved then its solutions lead to solutions (of special types) of the original equation. One of the most used method to obtain reductions of a given PDE is the method of classical and nonclassical Lie symmetries [1]. It happens very frequently that no solution is known for the reduced equation, which may be a PDE or an ordinary differential equation (ODE). In this case, one may try to obtain subsequent reductions by using the same or another methods (nonlocal symmetries, λ-symmetries, etc.). It also may happen that the search of these further reductions leads to problems very difficult to solve and no solutions, of the corresponding special type, are found for the original PDE. This is the case, for instance, when the reduced equation is an ODE which is not of a known type or lacks Lie point symmetries. Several authors have developed some ad hoc methods to find solutions of some nonlinear wave equations and some evolution equations. Some of these methods try to find solutions that depend on the known solutions of an ODE of some special type. In this paper we consider an nth order ODE of the form

P (y, v, v , . . . , vn)) = 0,

(1)

where P(y, v, v , . . . , vn) ) is a polynomial in the variables v, v , . . . , vn) whose coefficients are functions of the independent variable y. This class of equations has the following property: if we use for (1) a change of the dependent variable of the form v = R(y, w), where R is a rational function in the variable w whose coefficients are functions of y, then the resulting equation is of the form (1) too, but for the variable w.

∗

Corresponding author. Tel.: +34 956 012 708. E-mail addresses: [email protected] (J. Ramírez), [email protected] (J.L. Romero), [email protected] (C. Muriel).

http://dx.doi.org/10.1016/j.cnsns.2015.04.022 1007-5704/© 2015 Elsevier B.V. All rights reserved.

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In the present study, we develop a method to find solutions to (1) that, simultaneously, are solutions to one of the two following types of equations:

v = ak (y)vk + ak−1 (y)vk−1 + · · · + a1 (y)v + a0 (y),

(2)

(v )2 = ak (y)vk + ak−1 (y)vk−1 + · · · + a1 (y)v + a0 (y),

(3)

where k is a positive integer and, in principle, ai (y), 0  i  k, are arbitrary smooth functions. This study generalizes other methods by several authors. For instance: 1. The tanh method [2] uses the fact that the function v defined by v(y) = tanh (y) satisfies the equation v = 1 − v2 , which is of the form (2).  2. The sech method [2] uses the fact that the function v defined by v(y) = sech (y) satisfies the equation v = −v 1 − v2 , which is of the form (3). 3. The modified extended tanh function method [3–5] and the modified extended direct algebraic method [6] use reduced equations  i of the form v = b + v2 (which are of the class (2)), maybe after a previous rational change of variables of the form u = m i=−h ci v , where b and the ci are constant, for − h  i  m. 4. The extended mapping and the modified mapping methods [7,8] consider equations of the form (v )2 = c4 v4 + c2 v2 + c0 (which are of the form (3)), where c0 , c2 , c4 are constants. 5. The Fan sub-equation method [9,10], the F-expansion method [11] and the Jacobi elliptic function method [12] use equations of the type (v )2 = c4 v4 + c3 v3 + c2 v2 + c1 v + c0 , where ci is a constant for 0  i  4. 6. In the so-called simplest equation method [13,14] several classes of known equations of type (2) or (3) (Riccati, Jacobi elliptical or Weierstrass elliptical equations) whose solutions also solve the given equation are searched. Let us remark that, for the two methods we introduce in this paper, k may be an arbitrary positive integer and the coefficients ai , 0  i  k, may be arbitrary functions. Although in the literature most of these previous methods are used to find travelling wave solutions to some partial differential equations, it is clear that all the methods in this paper can be applied, in principle, to arbitrary polynomial equations of type (1). The structure of the paper is as follows: In Section 2 we describe the two mentioned methods to obtain solutions of an equation of the form (1) that are also solutions of equations of the forms (2) or (3). In order to describe how the two methods work in practice, we use a 2nd-order ODE, which can be obtained from a reduction of the Gardner equation. In general, these methods require a large amount of calculations and the help of a computer program may be necessary. These methods are applied in Sections 3 and 4 to find new solutions for well-known PDEs of Mathematical Physics, starting from known reduced equations of order n  2 with unknown solutions. By using the first method we obtain new solutions for the generalized Burgers–Fisher equation, the generalized Fisher equation, a modified Korteweg-de Vries equation and a Schwarzian Korteweg-de Vries equation. The second method has been applied in Section 4 to obtain new solutions for three families of equations: the Korteweg-de Vries equation, the Boussinesq equation and the Kawahara equation. These equations admit reductions of the form (3) which, as far as we know, were unknown. 2. The basic methods 2.1. Method 1: Reductions to generalized Abel equations We first describe a method to find reduced equations of the form (2) for a given polynomial equation (1). Let us suppose that v = v(y) is an arbitrary solution of an (unknown) equation of the form (2). By derivation with respect to the variable y we obtain

v (y) = ak vk + · · · + a0 + v [kak vk−1 + · · · + a1 ] = ka2k v2k−1 + (2k − 1)ak ak−1 v2k−2 + · · · .

(4)

For k = 4, a case that will be considered later, the complete expression for v reads

v (y) = a4 v4 + a3 v3 + a2 v2 + a1 v + a0 + v [4a4 v3 + 3a3 v2 + 2a2 v + a1 ]     = 4a24 v7 + (7a3 a4 )v6 + 6a2 a4 + 3a23 v5 + (a4 + 5a1 a4 )v4     + a3 + 4a1 a3 + 2a22 + 4a0 a4 v3 + (a2 + 3a1 a2 + 3a0 a3 )v2 + a1 + a21 + 2a0 a2 v + (a0 + a0 a1 ).

(5)

Clearly, for an arbitrary k, v is a (2k − 1)th-degree polynomial in the variable v, whose coefficients are functions of y. In general, vn) is a [n(k − 1) + 1]th-degree polynomial in v. The basic idea of this first method is described in what follows. If we substitute the polynomial expressions of v , v , . . . , vn) in P(y, v, v , . . . , vn) ) we get a polynomial P (v) in the variable v whose coefficients are functions of the coefficients of P and of the ai (y) and their derivatives up to order n − 1, 0  i  k. If we impose that the coefficients of P (v) be null then we obtain an

J. Ramírez et al. / Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49

39

overdetermined system of algebraic and ordinary differential equations for the functions ai (y), 0  i  k. If this system has a solution a0 (y), . . . , ak (y) then any solution of the corresponding equation (2) is a solution of (1). Obviously, the first step in this method must try to determine a value of k for which (1) may admit a non-trivial reduction of the form (2). It is clear that the degree of P (v) increases as k increases, k > 1, and the leader coefficient of the polynomial P (v) may be of two different types: A. The leader coefficient of P (v) does not depend on the coefficients ai (y), 0  i  k. In this case the overdetermined system is incompatible and, in order to find some reduction, the value of k should be increased. This happens, for instance, for the equation v + v8 = 0 if we choose k = 4. By (5), P (v) = v8 + (4a24 )v7 + · · · . For any ai , 0  i  4, this polynomial is not null. B. The leader coefficient of P (v) is a polynomial in ak , whose coefficients may depend on y. In this case two subcases may be considered B.1 The leader coefficient is identically null only when ak is null. In this sub-case, for obtaining a possible reduction (2), the value of k should be decreased. As an example, for the equation v + v6 = 0 and k = 4 we get, by (5), P (v) = (4a24 )v7 + (7a3 a4 + 1)v6 + · · · . If P (v) is the null polynomial then necessarily a4 = 0. B.2 The equation obtained by equating the leader coefficient of P (v) to zero (whose unknown function is ak ) has non-trivial solutions. Any of these solutions might be used to determine the remaining coefficients of (2). As an example, let us consider the equation

v = bv3 +

1 yv + k1 , 3

(6)

where b  0 and k1 are arbitrary constants, which will appear in the third subsection within our study of the Gardner equation (9). The first step in the procedure is to determine the value of k we must use to find a first-order reduction. By using (4) and by substitution into (6) we obtain in the left hand side a polynomial whose degree is 2k − 1 and in the right hand side a polynomial of degree 3. It is clear that the degrees of both polynomial must be balanced. If k < 2 then P (v) is a 3rd-degree non-null polynomial for any ai , 0  i  1. For k > 2, P (v) is a polynomial whose degree is 2(k − 1) + 1 > 3, its leader term is ka2k and therefore, necessarily, ak = 0. Finally, if k = 2, P (v) is a 3rd polynomial whose leader coefficient is 2a22 − b. If we impose that the leader  coefficient of P (v) must be null then necessarily 2a22 − b = 0 and a2 = ± 2b . Any of these values of a2 can be used to determine a0 (y) and a1 (y) in order P (v) be the null polynomial. This is done in the third subsection of this Section. Once the value of k has been determined, the coefficients ai , 0  i  k, in (2) must be determined by solving the overdetermined system of differential and algebraic equations obtained by equating to zero the coefficients of the successive powers of v in P (v). Obviously, this overdetermined system may be compatible or incompatible. If the original equation depends on some parameters, that system might be compatible only for some special values of the parameters, or some relationships between them. In some cases we will introduce a change of dependent variable of the type

v = c0 (y) +

1 . w(y)

(7)

(7) to obtain different reductions of the equation: if we introduce v = c0 (y) + 1/w and its derivatives into equation (1) then we obtain another equation of the same type for the function w. In this case, the method provides an overdetermined system whose unknown functions are a0 (y), . . . , ak (y), c0 (y). There are several available methods to deal with overdetermined systems of ODEs. Maybe the most known methods use Gröbner bases. In the Appendix we include the MAPLE code we have used for the Gardner equation. This program tries to find, if possible, the parameters of equation (2) for which (1) can be reduced to (2). With appropriate modifications, this code can be used in a similar manner to deal with other equations. 2.2. Method 2: Reductions to generalized elliptic equations In this subsection we describe a method to find reduced equations of the form (3) for a given polynomial equation (1). It must be said beforehand that any reduced equation of the form (2) can also be written in the form (3): if we have a non-trivial reduction (2) then (v )2 = (ak vk +  + a0 )2 is a non-trivial reduction of the form (3). Therefore, to consider reductions that cannot be obtained with the first method, wewill only consider for (3) the case in which the right hand side R(y, v) is not a perfect-square polynomial in v and, therefore, R(y, v) is not a polynomial function.  Now we suppose that v = v(y) is a solution of some (unknown) equation of the form (3), written in the form v = R(y, v). By    derivation with respect to y and the substitution of v by R(y, v) it can be checked that v (y) can be expressed in the form

 v (y) = [S2 (y, v) + T2 (y, v) R(y, v)]R1/2−1 (y, v),

where S2 (y, v) and T2 (y, v) are polynomial in v whose coefficients depend on the ai (y), 0  i  k, and their first-order derivatives. Assuming that the coefficients ai (y) admit derivatives of any order, it can be checked, by induction on the order of the derivative, that vj) (y), j  2, can be expressed in the form

 vj)(y) = [Sj (y, v) + Tj (y, v) R(y, v)]R1/2−j+1 (y, v),

(8)

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where Sj (y, v) and Tj (y, v) are polynomial functions in v whose coefficients depend on the ai (y), 0  i  k, and their derivatives up to order (j − 1). By substituting the values of vj) given in (8) into the polynomial equation (1), by obtaining a common denominator and by selecting the corresponding numerator we get an expression in the form



P (y, v) + Q(y, v) R(y, v), where P and Q are polynomial in v whose coefficients depend on the coefficients of P, the ai and their derivatives up to order (n − 1), where 0  i  k. The basic idea of this second method is as follows. If we impose that the coefficients of P and Q be identically null then we obtain an overdetermined system of algebraic and differential equations for the functions ai (y), 0  i  k. If this system has a solution then any solution of the corresponding equation (3) is a solution to (1). The first step in this method is to determine a possible value of k for which a non-trivial reduction (3) may be found. This can be done as it has been discussed for the first method. We could also fix a value of k and try to determine if there exists some reduction for that k. In the Appendix we include the MAPLE code for a program that tries to find reductions of the form (3) for the Korteweg-de Vries (KdV) equation, by using this method and a bilinear transformation of the form (7). This code is an example that can be easily modified to deal with other equations. 2.3. Applications to the Gardner equation The Gardner equation, also known as the combined KdV-mKdV equation [15], is:

ut + (2au − 3bu2 )ux + uxxx = 0,

(9)

where a and b are constants and mKdV stands for the modified KdV equation. This equation was first derived rigorously within the asymptotic theory for long internal waves in a two-layer fluid with a density jump at the interface. The competition among dispersion, quadratic and cubic nonlinearities constitutes the main interest. This equation, like the KdV equation, is completely integrable with a Lax pair and inverse scattering transformations. Soliton solutions exist only for b > 0. Gardner equation is widely used in various branches of Physics, such as plasma physics, fluid physics, quantum field theory, etc. The equation plays a prominent role in ocean waves: it describes internal solitary waves in shallow seas, and admits quite interesting tanh and elliptic cn type solutions (see, for instance, [16] and the references therein for details). Equation (9) admits the classical symmetry



X=

2a2 x + t 3 9b



∂ x + t ∂t +

−3bu + a ∂u . 9b

(10)

By using the corresponding similarity variables y =

1 a2 √ 3 (x − 3b t ) t

and u =

a 3b

+

1 √ 3

t

v(y), equation (9) can be reduced to the

equation

3v − (9bv2 + y)v − v = 0,

(11)

where y is the new independent variable. Although this equation lacks Lie point symmetries, it can be reduced by integration to the equation

v = bv3 +

1 yv + k1 , 3

(12)

where k1 is an arbitrary constant. A. Reductions by using the first method Equation (12) is in fact equation (6) and we have shown, in the first subsection, that for k = 2, equation (12) might admit a reduction of the form v = a2 (y)v2 + a1 (y)v + a0 (y). The substitution into (12) of the corresponding expression (4) gives us a 3rd-degree polynomial P (v) which must be identically null. Therefore the coefficients of P must vanish:

3a2 a1 + a2 = 0,

−2a22 + b = 0, −6a0 a2 −

3a21

−

3a1

(13)

+ y = 0, −a0 a1 − a0 + k1 = 0.

In this case the resulting system is quite simple and can be solved directly: a2 = ± compatibility of the system) necessarily k1 =

1 6a2 .

Therefore, when k1 =

equation

1 2 y

bv + . v = ± √ 3 2b

1 6a2

=

± √1 3 2b



b 2,

a1 = 0, a0 =

y 6a2 ,

where (for the

we have found the first-order reduced

(14)

Let us observe that, due to the fact that a0 (y) is not a constant function, the reduced equation we have found cannot be obtained by using the previous methods cited in the introduction. Obviously, this reduction is also valid for equation (11). Equation (14) is a Riccati equation (of the type 1.2.2.4 in [18]) whose solutions can be expressed in terms of the Airy functions Ai and Bi:

√ 6   2 c1 Ai (ξ ) + c2 Bi (ξ ) v= √ , √ 3 3 b c1 Ai(ξ ) + c2 Bi(ξ )



−y

ξ = √3

6



.

(15)

J. Ramírez et al. / Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49

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B. Reductions by using the second method We now search reductions of the form

(v )2 = a4 (y)v4 + a3 (y)v3 + a2 (y)v2 + a1 (y)v + a0 (y),

(16)

for equation (12). In order to obtain a reduction of the form (16) which is not of the form (3), we assume that

S=



a4 (y)v4 + a3 (y)v3 + a2 (y)v2 + a1 (y)v + a0 (y)

is not a polynomial in v (i.e. the right hand side of (16) is not a perfect-square polynomial in v). By calculating v and by substituting the resulting expression into (6) we obtain

a4 v4 + a3 v3 + a2 v2 + a1 v + a0 + S





2 3





(4a4 − 2b)v3 + 3a3 v2 + 2a2 − y v + (a1 − 2k1 ) = 0,

(17)

If we equate to zero the coefficients of vi , 0  i  4, and Svj , 0  j  3, then the resulting system is incompatible because the equations a2 = 0 and 2a2 − 23 y = 0 are incompatible. Therefore, if S is not a polynomial in v, equation (6) does not admit reductions of the form (16). When k1 = ± √1 , equation (12) obviously admits the reduction 3 2b

(v )2

b y 1 2 y = v4 + v2 + 2 3 18b

because, according with (14), 2b v4 + 3y v2 +

(18) 1 2 y 18b

= (± √1 (b v2 + 3y ))2 . 2b

3. Solutions for some equations that can be reduced to generalized Abel equations In order to illustrate the potentiality of the methods we have introduced, in this section we study some well-known PDEs of Mathematical Physics that admit reductions to polynomial equations of the type (1), which have unknown solutions. We use the first method considered in Section 2 to obtain reductions of the form (2) and, by means of their solutions, to obtain new solutions for the original PDEs. 3.1. The generalized Burgers–Fisher equation The generalized Burgers–Fisher equation is an equation of the form

ut − uxx + k1 uk2 ux − k3 u(1 − uk2 ) = 0,

(19)

where k1 , k2 , k3 are constant. The original Burgers equation was introduced in [23]. Equation (19) has important applications in various fields of financial mathematics, gas dynamic, traffic flow, applied mathematics and physics applications (see [24] and the references therein). This equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect and diffusion transport. The travelling wave solutions of the equation are of the form u(t, x) = v(y) where y = x + k4 t, being k4 a constant; the resulting equation is

k4 v − v + k1 vk2 v − k3 v(1 − vk2 ) = 0,

(20)

where the independent variable is y. 1

In order to apply the methods described in Section 2, we use the change of the dependent variable given by v = w k2 to transform equation (20) into a polynomial equation in w, w and w :

−k2 ww + (k2 − 1)w2 + k2 w(k1 w + k4 )w + k3 k22 w2 (w − 1) = 0.

(21)

Clearly, the order of this equation can be reduced by using the Lie point symmetry y . However, the general solution of such first-order reduced equation cannot be found by the standard procedures. By using the first method described in Section 2, we have investigated some reductions of (21). Most of them are only admissible when some relationships between the constants hold. We have found two classes of reductions: A. When k3 = −k1 (k1 + k4 ) equation (21) admits the reduction

w = k2 (k1 + k4 )w, whose solutions are of the form w = c1 exp (k2 (k1 + k4 )y). −k B. For k3 = (k +11 )2 (k1 + k4 (k2 + 1)) and 1/k2 a negative integer we have obtained the reduction 2

w =

    k2 w k1 + k2 k4 + k4 − 1 c1 w k2 exp y + k1 (w − 1) , k2 + 1 k2 + 1

(22)

where c1 is an arbitrary constant. For this equation, the solutions we are able to find do also depend on the value of some constants:

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B.1 When k2 = 1 and c1  0 equation (22) admits solutions that can be expressed in terms of Bessel functions:

w=

k1 Jγ1 (ξ )c2 

where γ1 = − k

k1 , 1 +2k4











γ1 + k4 ξ J1−γ1 (ξ ) γ2 − (k1 + k4 )ξ J−γ2 (ξ )c2  2γ1 + γ2      k1 Jγ1 (ξ )c2  γ1 − J−γ1 (ξ ) −γ1

4 γ2 = − k12k ,ξ = +2k4

√

2 c1 k1 k1 +2k4



exp( 14 (k1 + 2k4 )y) and c2 is an arbitrary constant.

B.2 When k1 + k2 k4 + k4 = 0, equation (22) admits solutions that are implicitly given by the quadrature

k2 + 1 k2

dη

w

k1 (η2 − η) + c1 η

1− k1

= y + c2 ,

2

where c2 is a constant. It must be observed that for an arbitrary k2  0 any solution to (22) does also solve (21), although (22) is not of the form (2) (with k a positive integer). Since not all the coefficients of the reduced equations we have found are constant, most of the corresponding solutions cannot be found by using the methods we have described in the introduction. As far as we know, the obtained solutions are new. 3.2. The generalized Fisher equation The generalized Fisher equation is the equation

ut − uxx + k1 u(1 − uk2 )(k3 − uk2 ) = 0,

(23)

where k1 , k2 , k3 are constants. The original Fisher equation, introduced by Fisher [25], is used to study wave propagation in large biological and chemical systems. It is also used to study logistic growth-diffusion phenomena, chemical wave propagation, neutron population in a nuclear reactor and chemical kinetics (see [13] and the references therein). The generalized Fisher equation (23) was introduced in order to account for more involved competition, by allowing a higher degree of nonlinearity (see, for instance, [5]). The study of the travelling wave solutions to (23) leads to consider solutions of the form u(x, t) = w(y), where y = x + k4 t, being k4 a non-null constant. The resulting equation is

k4 w − w + k1 w(1 − wk2 )(k3 − wk2 ) = 0. By setting w =

v1/k2 ,

(24)

equation (24) is transformed into the polynomial equation

−k2 vv + (k2 − 1)v2 + k2 k4 vv − k1 k22 v2 (v − 1)(k3 − v) = 0. 

(25)

If the constants satisfy some special relationships between them, we are able to find some reductions of the first type considered in Section 2 for equation (25). A. When k1 (k3 (1 + k2 ) − 1)2 = k24 (1 + k2 ), equation (24) admits the reduction

v =

k2 k4 v(v − 1), k3 k2 + k3 − 1

(26)

whose solutions are

  v = 1 + c1 exp

k2 k4 y k3 (k2 + 1) − 1

 −1 ,

where c1 is a constant. B. When k2 = −2, k1 (k3 + 1)2 + k24 = 0, which is an special case of A, equation (25) does also admit the reduced equation

v =

   −k4 v k4 (k3 − 1) y + c1 − 2v . 1 + k3 + (k3 − 1) tanh 1 + k3 2(k3 + 1)

This is a Bernoulli equation which admits solutions that can be expressed in terms of the hypergeometric function 2 F1 :

 

v=

3   k2k−1

(k3 − 3) cosh η 1 − tanh η

 

(k3 − 3) cosh η





c2 tanh

where c1 is a constant, η = η(y) = c1 +

η +1

 k 2−1 3

3

k3

+ G(y) 4 k3 −1

(k3 −1)k4 y and 2(k3 +1)



k3

− F(y) 4 k3 −1 eη

 1 2 k3 − 3 k3 + 1 , ; (tanh(η) + 1) , ;2 − k3 − 1 1 − k3 k3 − 1 2   2k3 k3 − 3 1 2 ,− ; ; (tanh(η) + 1) . G(y) = 2 F1 − k3 − 1 k3 − 1 k3 − 1 2 F(y) = 2 F1



 

,

J. Ramírez et al. / Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49

43

C. When k2 = − 12 , k3 = −1 we have obtained the reduction

v =

v (4f 2 v2 + 2f  − 2k4 f + k1 ), 4f

where f is a solution to

4ff  = 8f 2 + (6k1 − 4k4 f )f  + k1 (4f 2 − 2k4 f + k1 ), however we are not able to obtain explicit solutions for this reduced equation. As far as we know the obtained solutions for (25) are new. 3.3. Another modified Korteweg-de Vries equation Due to the importance of the Korteweg-de Vries equation (KdV) in many contexts, several modifications of the basic equation have been considered in the literature. In this paper we consider the following equation, proposed by Au and Fung [20,21] as a model of a KdV soliton propagating with varying velocity:

ut + uxxx + (k1 − 6u2 )ux = 0,

(27)

where k1 is a constant. Equation (27) differs from the classical mKdV equation by a first local-derivative; the symmetries of equation (27) has been studied by Huber [22]. However, as far as we know, no study on the reductions derived from the Lie point symmetry

X = (x + 2k1 t)∂x + 3t∂t − u∂u

(28)

is known. The similarity variables that correspond to the symmetry (28) are

y=

x − k1 t , t1/3

u = t−1/3 v(y).

By using these variables, equation (27) can be transformed into the equation

3v − (18v2 + y)v − v = 0.

(29)

By integration we get the following 2nd-order nonlinear ODE:

3v − (yv + 6v3 ) + k2 = 0,

(30)

where k2 is an arbitrary constant. This equation is in fact equation (12) with b = 2 and k1 = −k2 . By using the results we have obtained for the Gardner equation, we obtain the following solutions for (27): 



1 c1 Ai (ξ (t, x)) + Bi (ξ (t, x)) u(x, t) = − √ , 3 6t c1 Ai(ξ (t, x)) + 6 Bi(ξ (t, x))



ξ (t, x) =

−x + k1 t √ 3 6t



,

where c1 is an arbitrary constant. 3.4. The Schwarzian Korteweg-de Vries equation The Schwarzian Korteweg-de Vries equation in (2 + 1)-dimensions is the equation

Wt +

Wxx Wz W 2 Wz 1 Wx Wxz Wx Wxxz − − + x 2 − 4 2W 4W 8 2W



∂x−1



Wx2 W2



= 0, z

 where we denote ∂x−1 f = f dx. The Schwarzian Korteweg-de Vries (SKdV) was introduced by Toda [26] as an extension of the classical KdV equation to (2 + 1) dimensions which is invariant under Möbius transformations. It is related to the equations of Calogero–Bogoyalevlenskii– Schiff and Ablowitz–Kaup–Newell–Segur. The complete Lie group classification of this equation has been studied in [27] and the non-classical symmetries appear in [28]. The Exp-function method and the extended (G /G) − expansion method have been successfully applied to obtain exact solutions to the (2 + 1)-dimensional SKdV equation (see, for instance, [29]). However, some of the reductions shown in these papers lead to ODEs that, as far we know, cannot be solved by means of standard methods. This is the case of equation (19) in [28]:

k1 3x2 (2v2 v − 3vv2 − v5 ) − 4x2 v3 + 2k2 v3 + 4k3 (xvv − xv2 + vv − xv4 − 4k3 xv2 ) = 0,

(31)

where k1 = ε , k3 = c1 , k2 = c2 (being ε , c1 , c2 the constants that appear in [28]). Although equation (31) is a polynomial equation of the form (1), the first method discussed in Section 2 does not provide solutions for k3  0; the reductions we have found correspond to k3 = 0. In this case (31) becomes

k1 3x2 (2v2 v − 3vv2 − v5 ) − 4x2 v3 + 2k2 v3 = 0. For some special values of the constants k1 , k2 , we have been able to find new reductions to (32).

(32)

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It can be checked that, for k1  0, (32) admits a reduction of the form

v = ±v2 + a1 (y)v, where a1 (y) =

(33)

B

1 y

− 2 B , being

     y y B = k1 y Yd  + c1 Jd (  ) , k1 k1



  2k2 + k1  d= 2 k1

where Jd and Yd are the Bessel functions of the first and second kinds, respectively. The corresponding solution to (32) is given by

v=

y  B2 (c2 ±

y B2

dy)

,

where c1 , c2 are constant. In order to consider different reductions, we have used a change of variable of the form v(y) = c0 (y) + w1(y) . We have found 1 that if c0 (y) is a constant, namely c0 (y) = k4  0, then v = k4 + w is a solution for (32) when w = w(y) is a solution to the Riccati equation

w = a2 w2 +



 a2 ± k4 w ± 1, k4

(34)

where a2 can be expressed in terms of the Bessel functions J, Y as:

     2k4 c1 JM+1 √y + YM+1 √y k1 k1 k4 (1 + 2M)     , −   a2 = ±k24 + y k1 c1 JM √y + YM √y k1

being c1 an arbitrary constant and M =

√

2k2 +k1

2

√

k1

k1

.

The corresponding solution to equation (34) can be used to obtain the following solutions for equation (32):

 −1  

a2 (y) v = k24 k4 + R(y) c2 + dy , R(y)  where c2 is an arbitrary constant and R(y) = exp(

(35) a2 (y)∓k24 dy). k4

In the special case that k4 = 0, the corresponding equation (34) becomes

w = a1 w ± 1, where a1 = − 12 a21 +

k2 k1 y2

(36) −



2 . k1

The solutions to this last equation can be expressed in terms of the Bessel equation as

    2 c1 JM+1 √y + YM+1 √y k1 k1 1 + 2M     , −   a1 = y k1 c1 JM √y + YM √y k1

where M =

√

2k2 +k1

2

√

k1

k1

. The corresponding solution to (36) is

 

w = exp(R) c2 ± exp(−R) , where R(y) = a1 (y)dy. Therefore, a solution to (32) is

   −1

v = exp(R) c2 ± exp(−R) . Although the reductions we have been able to find correspond to particular cases of the constants, these reductions cannot be obtained by using the previous methods we have cited in the introduction. 4. Equations that are reducible to generalized elliptic equations In this section we consider several families of equations that can be reduced by means of the second method we have described in Section 2. Most of the obtained solutions, as far as we know, are new in the literature.

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45

4.1. The Korteweg-de Vries equation The classical Korteweg-de Vries (KdV) equation is

ut + uux − β uxxx = 0,

(37)

where β  0 is a constant. This equation was introduced to study the time evolution of some dispersive wave phenomena, specifically to describe the propagation of unidirectional shallow water waves, etc. [30]. Equation (37) is a very well-known equation that has been studied from many points of view (see [31] and [17]). The Lie point symmetries of this equation appear, for instance, in [1] and [32]. One of these symmetries, X = xx + 3tt − 2uu , leads to the similarity variables

y=

x , t1/3

u = t−2/3 v(y).

(38)

The corresponding reduced equation is

−3β v + 3vv − yv − 2v = 0,

(39)

where the independent variable is y. For (39) we search solutions of the form (7), where w satisfies an equation of the form (1). We have found that a family of 1 solutions for (39) is given by v = c1 − y + , where c1 is a constant and w is a solution to w

w2 ((−3c1 + 4y)w + 3β w ) − 3ww (1 + 6β w ) + 18β w3 + (6y − 5c1 )w4 − 5w3 = 0. This equation admits the reduction

 

w =

w(1 + c1 w) ((c1 − 2y)w + 1). 3β

(40)

For c1  0 we are not able to find explicit solutions to (40), but for the special case c1 = 0 the corresponding equation (40) admits solutions that can be expressed in terms of the Airy functions Ai, Bi; they lead to the following family of solutions to the KdV equation:

  2   c2 Bi (ξ ) + c3 Ai (ξ ) x 3 6β u=− +2 , t c2 Bi(ξ ) + c3 Ai(ξ ) t2



ξ = 3

x

 .

6β t

It can be checked that our family of solutions to equation KdV strictly contains the family of solutions given in [32] for the case c1 = 0. On the other hand, the reductions we have found in this subsection correspond to equations of the form (3) whose coefficients are not constant. Therefore, these solutions cannot be found by using any of the previous methods we have cited in the introduction.

4.2. The Boussinesq equation Let us consider the Boussinesq equation in the form that appears in [12]:

k1 utt + uxxxx + k2 (u ux )x = 0,

(41)

where k1 , k2 are constants. The Boussinesq equation was introduced in 1872 [33] to describe weakly nonlinear dispersive water waves. It has also found applications to ion acoustics solitons and one-dimensional lattice waves. The classical Boussinesq equation corresponds to the case k1 = 1, k2 = 1. Several studies on equation (41) appear in [12]. Equation (41) admits the classical symmetry [34]

X = x∂x + 2t∂t − 2u∂u . The corresponding similarity variables are y = xt−1/2 , u = t−1 v(xt−1/2 ) and the reduced ODE is

4v + (k1 y2 + 4k2 v)v + 4k2 (v )2 + 7k1 yv + 8k1 v = 0.

(42)

This equation lacks Lie point symmetries and, as far as we know, no explicit solutions for (42) are known. Nevertheless, we have 5k 3k 1 , where k3 is an arbitrary constant and w is a solution found that equation (42) has solutions of the form v = 4k1 y2 + 2k1 k3 + w of the reduced equation

2

2

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J. Ramírez et al. / Commun Nonlinear Sci Numer Simulat 29 (2015) 37–49

w =

1 4



w (9k1 (y2 + k3 )w + 4k2 ), −3k2

(43)

which is of the type (3). By using the change of variable w = −V2 , equation (43) is transformed into the equation

1 V =  (9k1 (y2 + k3 )V 2 − 4k2 ), 8 3k2

(44)

which is a Riccati equation and admits solutions of the form

 √ √ 4 3/4 3k1 k4 F1 + 2 z F2 −3 k2   V=  , √  9k1 k4 z F1 + 4(m − 1)F3 + 2 33/4 4 k1 3(z + 1) F2 + 2(2m − 3)z F4 √ z , 3k1

2 where y = √ 4

4m−3) k3 = 4(√ , k4 is an arbitrary integration constant and the functions Fi , for 1  i  4, can be expressed in 3k1

terms of the hypergeometrical function F = 1 F1 :

  1 1 F1 = F m − ; ; z , 2 2   1 3 F3 = F m − ; ; z , 2 2

  3 F2 = F m; ; z , 2   5 F4 = F m; ; z . 2

The corresponding solutions to the Boussinesq equation could be given through the function v(y) =

5k1 2 y 4k2

+

3k1 k 2k2 3

−

1 V2

but

the analytical expression of the solution is too involved to be included here. Nevertheless, let us observe that when k3 = 0 the solutions of equation (44) can be expressed in terms of the Bessel function Iν :

 k2 k5 I− 1 (ξ ) − k4 I 1 (ξ ) 4 4

, V= √   2 33/4 4 k1 ξ k4 I− 3 (ξ ) − k5 I 3 (ξ ) 4



ξ=

 1 3k1 y2 , 8

(45)

4

where k4 , k5 are constant. It can be checked, by using the properties of the Hermite polynomial and of the confluent hypergeometric functions (see [19]), that when k4 = 0 and m is a negative integer then the function V is a rational functions in the variable y, which provides rational solutions for the Boussinesq equation. For m = 0 and m = −1, the corresponding solutions u(x, t) to (41) appear in the following table: m

Solution to the Boussinesq equation (41)

0

u(x, t) = −

−1

u(x, t) =

k1 x4 + 12t2 . k2 x2 t2

  −3(k21 x8 − 8k1 3k1 tx6 + 96k1 t2 x4 − 48 3k1 t3 x2 + 144t4 )  2 4 2 2 k2 t x ( 3k1 x − 6t)

Since the reductions we have found do not have constant coefficients, the solutions we have found cannot be obtained by using the previous methods we have described in the introduction. 4.3. The Kawahara equation The generalized Kawahara equation is

ut + (1 + u2 )ux + a uxxx + b uxxxxx = 0,

(46)

where a and b are constants. This equation is a model for plasma waves, capillary-gravity water waves and others dispersive phenomena when the cubic KdV-type dispersion is weak. It was introduced by Kawahara [35] and several generalizations have been addressed in a number of research papers (see [6] and [36] and the references therein). Equation (46) has been considered by Assas [6] to find some solutions by mean of the modified extended direct algebraic method. The search of travelling wave solutions for (46) leads to the 5th-order equation

bv5) + av3) + (1 − λ + v2 )v = 0,

(47)

whose independent variable is y = x − λt. By using the second of the methods described in Section 2, for equation (47) we have found the reduction of the type (3) given by



(v ) = ±  2

  4a2 a 2 −2 v v2 − + 5(1 − λ) − v + c0 , 5b 3 5b 5b

where c0 is an arbitrary constant.

(48)

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47

Equation (48) admits the following solution, which can be expressed in terms of the Jacobi elliptic function sn:



−5b 2 sn v = 3Am 2

 √ 5A(m + 1)b − a A , (y + y0 ), m + √ 2 −10b

(49)

where y0 is a constant and the new constants A and m are related to the remaining constants a, b, λ and c0 by the algebraic equations

5b(3A2 b(m2 − m + 1) + 2(λ − 1)) + a2 = 0, 6a3 + 50ab(λ − 1) − 25a2 bA(m + 1) − 25b2 (6c0 + 10A(λ − 1)(m + 1) + 5A3 b(m + 1)3 ) = 0. Since the reductions we have found correspond to an ODE with constant coefficients, the solutions we have found for (46) could also be obtained by using some of the methods described in the introduction. Nevertheless, as far as we know, these solutions are new in the literature. Conclusions Two new methods to obtain first-order reductions of polynomial ODEs are presented in this paper. These methods generalize previous methods by several authors, that usually consider reductions with constant coefficients, whereas the reductions introduced in this paper contain non-constant coefficients. The procedures can be easily implemented in symbolic computation programs. The methods are applied to obtain new reductions of several well-known equations of Mathematical Physics (KdV, Gardner, Boussinesq, Burgers-Fisher, etc.). Acknowledgments The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper. The authors acknowledge the financial support of the Junta de Andalucía research group FQM-377. Appendix A.- Algorithm and program to obtain reductions by using the first method For the corresponding program, we assume that the value of k in (2) is already known. We first load several packages, in order to use the following specific functions: dsubs and casesplit from PDEtools; CoefficientList from PolynomialTools and Flatten from ListTools. We also load the package DEtools and we use several auxiliary variables: aux1, aux2, . . . , answ. 1. We assign the left hand side of equation (1) to the variable ec. The independent and dependent variables of the equation are y and v, respectively. 2. We substitute v = ak (y)vk +  + a0 (y) and its derivatives in the variable ec. We substitute v(y) by U and write the result in the variable aux1. 3. The list of the coefficients of the powers of U in aux1 is assigned to the variable aux2. Each member in the list corresponds to an algebraic or differential equation in the unknown functions ak (y), . . . , a0 (y). 4. We search the solutions to the overdetermined system defined by the list saved in aux2. We must indicate to the program the constants that appear in the equations and we can constrain the execution time with the order ctl. The resulting systems of ODEs in the unknowns ak (y), . . . , a0 (y) are saved in the variable answ. 5. Each system of equations is solved, if possible, and the results are set into the reduced equation (2). This equation must be solved to obtain solutions of (1). The following program contains the MAPLE code to find reductions of the Gardner equation by using the first of the two reduction methods discussed in Section 2. In this case, according with our discussion in the Subsection 2.1, we take k = 2. with(DEtools); with(PDEtools); with(PolynomialTools); with(ListTools); ec := -diff(v(y),y$2)+b*v(y)^3+(1/3)*y*v(y)+k1; aux1 := simplify(radnormal(numer(normal(eval(dsubs(diff(v(y), y) = a2(y)*v(y)^2+a1(y)*v(y)+a0(y), ec), [v(y) = U])))), size); aux2 := CoefficientList(aux1, U); (* we can put also aux2 := [op(aux2), b != 0] to avoid the case b=0 *) answ := casesplit(aux2, parameters = [k1, b], ctl = 100); eval(diff(v(y), y) =a2(y)*v(y)^2+a1(y)*v(y)+a0(y), op(1,answ[1])) B.- Algorithm and program to obtain reductions by using the second method We now describe the program we have used to obtain reductions of the second type considered in Section 2. In this case we use a bilineal transformation of the form (7). The program is very similar to the one we have just described. We explain the main differences with respect to the previous program.

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1 1. After loading the packages, we first substitute v = c0 (y) + w and its derivatives into the given equation. In the resulting    k equation we substitute w and its derivatives by w = ak (y)w + · · · + a0 (y) and its derivatives. 2. We set w(y) = U and, as before, we write  the result with a common-denominator. After the pertinent simplification and

rationalization of the result we set S = ak (y)U k + · · · + a0 (y). The corresponding numerator is saved in the variable aux1. 3. The list of the coefficients of the powers of U in aux1 is assigned to the variable aux2. 4. The list of the coefficients of the powers of S in aux2 is assigned to the variable aux3. Each member in the list corresponds to an algebraic or differential equation in the unknown functions ak (y), . . . , a0 (y), c0 (y). 5. We search the solutions to the overdetermined system defined by the list saved in aux3. We must indicate to the program the constants that appear in the equations and we can constrain the execution time with the order ctl. The results for ak (y), . . . , a0 (y), c0 (y) and their derivatives are saved in the variable answ. The remaining steps are as for the first program. The corresponding code for the Korteweg-de Vries equation, by setting k = 4, is as follows: with(DEtools); with(PDEtools); with(PolynomialTools); with(ListTools); ec := -3*b*diff(v(y),y$3)+3*v(y)*(diff(v(y), y))-y*(diff(v(y), y))-2*v(y); aux1 := eval(simplify(eval(radnormal(numer(normal(eval(dsubs(diff(w(y), y) = sqrt(a4(y)*w(y)^4+a3(y)*w(y)^3+a2(y)*w(y)^2+a1(y)*w(y)+a0(y)), dsubs(v(y) = 1/w(y)+c0(y), ec)), w(y) = U))), expanded), sqrt(a4(y)*U^4+a3(y)*U^3+a2(y)*U^2+a1(y)*U+a0(y)) = S), power, symbolic), csgn(U) = 1): aux2 := CoefficientList(aux1, U); aux3 := Flatten(map(proc (y) CoefficientList(y, S) end proc, aux2)); aux3 := [op(aux3), b != 0]; answ := casesplit(aux3, parameters = [b], ctl = 100); sol := pdsolve(answ[1], build); simplify(eval(diff(w(y), y) = sqrt(a4(y)*w(y)^4+a3(y)*w(y)^3+a2(y)*w(y)^2+a1(y)*w(y)+a0(y)), sol), power, symbolic); simplify(eval(1/w(y)+c0(y), sol), size); It must be said that the execution of these codes could use a considerable amount of time and memory, depending on the type or complexity of the initial equations. 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