Abstract
Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the “conditional symmetry” generator and imposing the extra condition given by the characteristic of the symmetry. Several of examples starting from the Boussinesq and including nonautonomous KortewegDe Vries like equations are given to showcase the methodology introduced.
Differential equations invariant under conditional symmetries
Decio Levi, Miguel A. Rodríguez,Zora Thomova
INFN, Sezione Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
Departamento de Física Teórica II, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
SUNY Polytechnic Institute,
100 Seymour Road,
Utica, NY 13502, USA
July 6, 2021
1 Introduction
La filosofia è scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l’Universo), ma non si può intendere, se prima non il sapere a intender la lingua, e conoscer i caratteri ne quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile intenderne umanamente parola; senza questi un aggirarsi vanamente per un oscuro labirinto.^{1}^{1}1Philosophy is written in this grand book, which stands continually open before our eyes (I say the ’Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth. As Galileo Galilei said in Il Saggiatore (1623) [20], our world is described in mathematical formulas and is up to us to comprehend it. This was the starting point of the scientific revolution which goes on up to nowadays and gave us the present world technology, i.e. cellular phones, lasers, computers, nuclear resonance imaging, etc.
Our capability of solving complicated physical problems described by mathematical formulas (say equations) is based on the existence of symmetries, i.e. transformations which leave the equations invariant. In correspondence with a symmetry we find an exact solution and this throws a light upon the phenomena we are dealing with.
Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of differential equations. He showed the following main property: if an equation is invariant under a oneparameter Lie group of point transformations then we can construct an invariant solution. This observation unified and extended the available integration techniques. Roughly speaking, Lie point symmetries are a local group of transformations that map every solution of the system to another solution of the same system. In other words, it maps the solution set of the equation into itself.
A partial differential equation (PDE) is invariant under a symmetry group if the corresponding infinitesimal symmetry generator is such that
(1) 
where by the symbol pr we mean the prolongation of the infinitesimal generator to all derivatives appearing in the equation . In particular, if we consider a second order differential equation in of independent variables and and dependent variable ,
(2) 
(where the subscripts on denote partial derivatives) the infinitesimal generator will be given by
(3) 
where , and are functions of their arguments to be determined by solving (1). Its prolongation is given by
where the functions and are algorithmically derived in terms of , and for example in[31, 8, 5, 6, 24, 32, 39].
A function is an invariant of a symmetry if it is such that
(5) 
Eq. (5) is a first order PDE which can be solved on the characteristic and provide in general the set of invariants , , , , . Then a PDE invariant with respect to the infinitesimal generator (3) is given by
(6) 
Lie methods provide a well established technique to search for exact solutions of differential or difference equations of any type, integrable or non–integrable, linear or nonlinear. However, many equations may have no symmetries and there is no simple algorithm to prove the existence of symmetries other than looking for them. Moreover, the obtained solutions do not always fulfill the conditions imposed by the physical requests (boundary conditions, asymptotic behavior, etc.). So, one starts to look for extension or modification of the construction which could overcome some of these problems. One looks for more symmetries,

not always expressed in local form in terms of the dependent variable of the differential equations,

not satisfying all the properties of a Lie group but just providing solutions.
In the first class are the potential symmetries introduced by Bluman et. al. [4, 9], the nonlocal symmetries by Vinogradov et. al. [25, 43, 23, 35, 36, 28, 13, 30] and in the second one are the conditional symmetries [7, 26, 17, 19, 29, 18].
In this paper we will be interested in showing that one can construct equations having given conditional symmetries, i.e. by starting from the symmetries, their invariants and imposing the condition.
In Section 2 we will provide the theory behind the construction of the conditional symmetries clarifying in this way the difference between symmetries and conditional symmetries. Then in Section 3 we will construct equations having conditional symmetries starting from a series of symmetries. Section 4 is devoted to the summary of the result, some concluding remarks and prospects of future works.
2 What is a conditional symmetry?
The Bluman and Cole nonclassical method [7] consists of adding an auxiliary firstorder equation to (2) build up in terms of , namely
(7) 
the infinitesimal symmetry generator (3) written in characteristic form [31] set equal to zero. Equation (7) is as yet unspecified and it will be determined together with the vector field , as it involves the same functions and .
We now look for the simultaneous symmetry group of the overdetermined system of equations (2) and (7), using the classical method. It is easy to prove that (7) is invariant under the first prolongation of (3)
(8) 
without imposing any conditions on the functions and . Consequently, we are left with need to solve the following invariance condition
(9) 
The equation (9) gives nonlinear determining equations for , and which provide at the same time the classical and non–classical symmetries. In fact, as noted in [14], since all solutions of the classical determining equations necessarily satisfy the nonclassical determining equations, the solution set may be larger in the nonclassical case. As appears in (9) as a condition imposed on the determining equations we can call the resulting symmetries conditional symmetries.
There are several works devoted to using nonclassical symmetries to construct solutions of PDEs that are different from the ones obtained by “classical” method using the Lie point symmetries. Among them, let us cite as an example, [16, 10, 33, 37, 34, 1, 21, 2, 22].
In this paper we want to look at the conditional symmetries from a different perspective. Given an infinitesimal group generator characterized by a vector field for specific values of the functions and , we want to construct equations which have this symmetry as a conditional symmetry and not as a Lie point symmetry. Taking in account that an equation invariant under a given symmetry is written in terms of its invariants (6), a second order PDE invariant under a conditional symmetry will be given formally by
(10) 
The constraint in (10) is to be interpreted as the differential equation (7) and all of its differential or multiplicative consequences (see Section 3 for details contained in the explicit examples).
As is one of the invariants which depends on both the and derivatives of , the argument of in equation (10) may not be expressed solely in terms of the invariants of due to the conditions , provided (9) is satisfied. As we will see in the following section this arbitrariness is a necessary condition to build PDE’s which have the given conditional symmetry. The condition and its differential consequences must not be used everywhere on the invariant equation to get (9) as, if we would do so, the global substitution of the condition and its consequences would turn the invariant PDE into an ODE in with parametric dependence on .
3 A series of examples including the Boussinesq equation, but not only.
The Boussinesq equation
(11) 
was introduced in 1871 by Boussinesq to describe the propagation of long waves in shallow water [11, 12] and is of considerable physical and mathematical interest. It also arises in several other physical applications including onedimensional nonlinear lattice waves[41, 40], vibrations in a nonlinear string[42], and ion sound waves in a plasma[38].
If is different from zero the resulting determining equations do not fix it and we can always put it equal to one. The same phenomena happens when and we have . In this case we can put .
The conditional symmetries of the Boussinesq equation for were obtained in [26], and in [15] by non group techniques. The case has been considered later and can be found in [14].
In [26, 14] we find the following generators of conditional symmetries for (11):
(12)  
(13)  
(14)  
(15)  
(17)  
(18) 
where is a special case of the Weierstrass elliptic function [3] with satisfying the differential equations , and , and are arbitrary constants.
The generators were obtained assuming , thus are defined in (123) up to an arbitrary function while and were obtained assuming and , thus are defined in (17) and (18) up to an arbitrary function .
3.1 Conditional invariant equations associated to
For the infinitesimal generator and its prolongation up to fourth order we obtain the following invariants:
(19)  
The condition is given by i.e. and we can construct a nonlinear evolution PDE in term of the invariants (19). It is:
(20) 
As we have:
So, the Boussinesq equation (11) has the conditional symmetry given by .
Additionally, we want to construct a KdV like nonautonomous equation which may have the conditional symmetry given by . Let us consider a different subset of the invariants (19)
(21) 
So we we have:
(22) 
To verify if effectively (22) has a conditional symmetry we need to compute its Lie point symmetries. The Lie point symmetries of (22) are
(23)  
We observe that . The invariants of are:
(24)  
and (22) is given by . Effectively (22) does not have as a conditional symmetry and instead is invariant with respect to .
3.2 Conditional invariant equations associated to
For the infinitesimal generator and its prolongation up to fourth order we obtain the following invariants:
(25)  
The condition is given by i.e. and we can construct a nonlinear evolution PDE in term of the invariants (25). It is:
(26) 
As
So the Boussinesq equation (11) has the conditional symmetry given by .
A KdV like nonautonomous equation which may have the conditional symmetry given by can be found by considering a different subset the invariants (25), i.e.
(27)  
We get:
(28) 
Lie point symmetries of (28) are
(29)  
We observe that . The invariants of are:
(30)  
and (28) is given by . Thus the equation (28) is invariant under the the of (29) and not conditionally invariant under the .
3.3 Conditional invariant equations associated to
For the infinitesimal generator and its prolongation up to fourth order we obtain the following invariants:
(31)  
The condition is given by i.e. and we can construct a nonlinear evolution PDE in term of the invariants (31), provided the condition is satisfied. It is:
(32) 
As and
So, the Boussinesq equation (11) has the conditional symmetry given by .
3.4 Conditional invariant equations associated to
For the infinitesimal generator with and its prolongation up to fourth order we obtain the following invariants:
(35)  
The condition is given by i.e. 35). It is: and we can construct a nonlinear evolution PDE in term of the invariants (
(36) 
As
and
So, the Boussinesq equation (11) has the conditional symmetry given by .
3.5 Conditional invariant equation associated to
We now consider a symmetry generator unrelated to the symmetries of the Boussinesq equation given by [27]. For the infinitesimal generator and its prolongation up to third order we obtain the following invariants:
(41) 
The condition is given by i.e. and we can construct a nonlinear evolution PDE in term of the invariants (41). It is:
(42) 
i.e. a nonlinear dispersive nonautonomous KdV like equation. Lie point symmetries of (42) are
(43)  
We observe that . The invariants of are:
(44) 
and (42) is given by . Thus the equation (42) is invariant under and not conditionally invariant under .
4 Conclusions
In this article we presented a construction of nonlinear PDE’s having given conditional symmetries. They are obtained starting from the invariants of the symmetry and imposing the extra condition given by equating to zero the characteristic of the symmetry. Starting from the conditional symmetries of the Boussinesq equation we reconstructed the Boussinesq equation itself as well as other KdV type nonlinear equations (22, 28, 34, 39, 42). Equations (22, 28, 34, 39, 42) are nonautonomous as it is well known that the KdV equation has no conditional symmetries [15]. However, not all obtained equations are conditionally invariant even if we constructed them in such a way. The obtained equations can still have the generator as a point symmetry due to the arbitrary multiplicative factor or under which the condition is defined. This is what happens in cases of the KdVlike equations (22), (28), (42).
Work is in progress on solving by symmetry reduction the obtained KdV like equations and on the construction of conditional symmetry preserving discretizations of the Boussinesq equation.
Acknowledgments.
DL has been supported by INFN ISCSN4 Mathematical Methods of Nonlinear Physics. DL thanks ZT and the SUNY Polytechnic Institute for their warm hospitality at Utica when this work was started. DL thanks the Departamento de Física Téorica II of the Complutense University in Madrid for its hospitality. MAR was supported by the Spanish MINECO under project FIS 201563966P. ZT thanks the summer student Douglas Nedza for the verification of some of the computations.
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