Differential Equations Invariant Under Conditional Symmetries
 DOI
 10.1080/14029251.2019.1591731How to use a DOI?
 Keywords
 Lie symmetries; partial differential equations; conditional symmetries
 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. Series of examples starting from the Boussinesq and including nonautonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.
 Copyright
 © 2019 The Authors. Published by Atlantis and Taylor & Francis
 Open Access
 This is an open access article distributed under the CC BYNC 4.0 license (http://creativecommons.org/licenses/bync/4.0/).
1. Introduction
As Galileo Galilei said in Il Saggiatore (1623) [21], our world is described in mathematical formulas and it 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. Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group of symmetries 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 reduce the equation and possibly construct an invariant solution. This observation unified and extended the available integration techniques such as separation of variables or integrating factors. Roughly speaking, Lie point symmetries are a local group of transformations which map every solution of the system into another solution of the same system. In other words, it maps the solution set of the equation into itself.
A partial differential equation (PDE) ℰ = 0 is invariant under a continuous group of Lie point transformations if the corresponding infinitesimal symmetry generator
Given an infinitesimal generator of a symmetry
Eq. (1.5) is a first order PDE which can be solved on the characteristic and provide the set of invariants and differential invariants ℐ_{j}, j = 0,1,... depending on x, y, u and its partial derivatives up to the second order. Then a PDE invariant with respect to the infinitesimal generator (1.3) can be written as
Lie method is a well established technique to search for exact solutions of differential or difference equations of any type, integrable or nonintegrable, 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 looks 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, 8], the nonlocal symmetries by Vinogradov et al. [13,24,26, 31,34, 39,40,46] while in the second one are the conditional symmetries [5,19,20, 32,35].
The inverse problem, that is, to construct a differential equation with a prescribed set of Lie point symmetries has been discussed and used in many references (see for instance [6]), looking for models enjoying properties related to these symmetries. The basic idea is that by computing the invariants of the symmetries and then constructing the quantities written in terms of these invariants we can obtain equations that have prescribed symmetries and whose solutions have the given transformation properties, for example the Lorentz transformation.
A very important application in this field is the symmetry preserving discretization where we construct discretized equations which possess the same symmetries or a subgroup of the symmetry group of a given ODE or PDE [9,18,28–30]. The construction of the Ordinary Difference Equation (OΔE) or the Partial Difference Equation (PΔE) is obtained through the computation of discrete invariants of the symmetry group of the ODE or PDE or one of its subgroup looking for a function of these invariants which yields the ODE or the PDE in the continuous limit.
This paper is part of the work of extending the project of symmetry preserving discretization to the case of conditional symmetries. To do so we have to understand at first how to construct a differential equation possessing a given conditional symmetry starting from the infinitesimal generator. This is the content of this paper. The symmetry preserving discretization is left to future work.
As in other approaches to inverse problems, it is not evident that the equations we can find might have an immediate interest. This is the case, for example, of integrable equations associated to given spectral problems [12]. Here, as we will see in the examples, we have presented KdVlike equations which can be of interest in the description of shallow water waves in different physical contexts [37].
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 verify the proposed construction in the case of the Boussinesq equation (3.1) and, in correspondence with its conditional symmetries, construct new conditionally invariant equations. Section 4 is devoted to the summary of the result, some concluding remarks and prospects of future works.
2. What is a conditional symmetry?
Conditional symmetries were introduced by Bluman and Cole with the name nonclassical method [5] by adding an auxiliary firstorder equation to (1.2), build up in terms of the coefficients of the infinitesimal generator
Eq. (2.3) gives nonlinear determining equations for ξ, η and ϕ which provide at the same time the classical and nonclassical symmetries. In fact, as noted in [15], since all solutions of the classical determining equations necessarily satisfy the nonclassical determining equations (2.3), the solution set may be larger in the nonclassical case. As 𝒞 = 0 appears in (2.3) as a condition imposed on the determining equations one has called the resulting symmetries conditional symmetries.
There are several works devoted to using the nonclassical method 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, [2, 17, 22, 23, 25, 38, 41]. Moreover we can find programs to compute them [3] and algorithms [14] for showing the existence of nontrivial nonclassical symmetries for given PDEs. In the case of integrable equations let us mention the works of Sergyeyev [43, 44] where he considered the classification of all (1 + 1)dimensional evolution systems that admit a generalized (Lie–Bäcklund) vector field as a generalized conditional symmetry.
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
The constraint {𝒞 = 0} in (2.4) is to be interpreted as the differential equation (2.1) and all of its differential consequences (see Section 3 for the details presented in the explicit examples).
The condition 𝒞 = 0 and its differential consequences must not be used everywhere on the invariant equation to get (2.3) as, if we would do so, the global substitution of the condition and its consequences would turn the invariant PDE into an ODE in one of the independent variables with parametric dependence on the other.
3. A series of examples including the Boussinesq equation
The Boussinesq equation
If η in (1.3) is different from zero the resulting determining equations for conditional symmetries do not fix it and we can always put it equal to one. If η = 0 and ξ ≠ 0, we can put ξ = 1 for the same reason.
The conditional symmetries of the Boussinesq equation for η ≠ 0 were obtained in [32], and in [16] by non group techniques. The case η = 0 has been considered later and can be found in [15]. Moreover it is worthwhile to notice that the condition is the same if we consider
In [15,32] we find the following generators of the conditional symmetries for (3.1):
The generators
3.1. Conditional invariant equations associated to X ^ 1
For the infinitesimal generator
The condition is given by ℐ_{3} = 0 i.e. 𝒞 = 2y + yu_{x} + u_{y} = 0 and we can construct the Boussinesq equation in terms of the invariants (3.9). It is:
As 𝒞_{x} = u_{xy} + yu_{xx} we have:
So, the Boussinesq equation (3.1) has the conditional symmetry given by
Now, we want to construct an autonomous equation which have the conditional symmetry given by
So we we have:
To verify if effectively (3.12) has
As it does not exist a function η(x, y, u) such that
A KdV like nonautonomous equation which has the conditional symmetry given by
We get:
To verify if effectively (3.15) has
The invariants of Ẑ_{1} are:
3.2. Conditional invariant equations associated to X ^ 2
For the infinitesimal generator
The condition is given by ℐ_{3} = 0 i.e.
A KdV like nonautonomous equation which may have the conditional symmetry given by
Lie point symmetries of (3.21) are
We observe that
3.3. Conditional invariant equations associated to X ^ 3
For the infinitesimal generator
The condition is given by ℐ_{3} = 0 i.e.
Then, the Boussinesq equation (3.1) has the conditional symmetry given by the vector field
A KdV like nonautonomous equation which may have the conditional symmetry given by
We get:
The equation (3.27) has no point symmetries. Then (3.27) is a conditionally invariant KdVlike equation.
3.4. Conditional invariant equations associated to X ^ 6
For the infinitesimal generator
The condition is given by ℐ_{3} = 0 i.e.
Then, the Boussinesq equation (3.1) has the conditional symmetry given by
A KdV like nonautonomous equation which may have the conditional symmetry given by
Eq. (3.30) reads:
Equation (3.31) has only Lie point symmetry
So, (3.31) is a conditionally invariant equation.
3.5. Conditional invariant equation associated to Y ^ = ∂ y + x 2 y ∂ x − 1 y ∂ u
The generator
For the infinitesimal generator Ŷ and its prolongation up to third order we obtain the following invariants:
The condition is given by ℐ_{3} = 0 i.e. 𝒞 = 2 + xu_{x} + 2yu_{y} = 0 and we can construct, apart from the Frank–Kamenetskii partial differential equation (3.33), a nonlinear KdV like evolution PDE in terms of the invariants (3.34). It is:
We observe that Ẑ_{2} = yŶ. The invariants of Ẑ_{2} are:
For this infinitesimal generator we can construct a conditionally invariant wave equation too,
In correspondance with Ŵ_{3}, after some nontrivial calculations, we get the following invariants
Then (3.38) is written in terms of the invariants (3.40) as
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 and its differential consequences. We showed in the case of the Boussinesq equation that effectively the construction proposed works and allows to produce other conditionally invariant nonlinear equations (3.12, 3.15, 3.21, 3.27, 3.31, 3.35, 3.38). Equations (3.15, 3.21, 3.27, 3.31, 3.35) are nonautonomous KdV–like equations and it is well known that the KdV equation has no nonsingular conditional symmetries [16]. 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
An important point not touched in this work but on which we are presently working is understanding a priori if this technique can be used to characterize the class of nonlinear PDEs that allows conditional symmetries (see the paper by Kunzinger and Popovych [27] for results on this problem). Why most of the KdV like equation we have constructed are not conditionally invariant?
Moreover work is also in progress on solving by symmetry reduction the obtained conditionally invariant 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 of the Universidad Complutense de Madrid for its hospitality. MAR was supported by the Spanish MINECO under project FIS 201563966P. D. Nedza, summer student of ZT, contributed to the verification of some of the computations.
References
Cite this article
TY  JOUR AU  Decio Levi AU  Miguel A. Rodríguez AU  Zora Thomova PY  2021 DA  2021/01/06 TI  Differential Equations Invariant Under Conditional Symmetries JO  Journal of Nonlinear Mathematical Physics SP  281 EP  293 VL  26 IS  2 SN  17760852 UR  https://doi.org/10.1080/14029251.2019.1591731 DO  10.1080/14029251.2019.1591731 ID  Levi2021 ER 