Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to find the symmetries and solutions of the differential equations (DEs) by following an algorithm. This analysis leads to reduce the order of partial differential equations (PDEs). Many physical problems are converted into non-linear DEs and these DEs or system of DEs are then solved with several methods such as similarity methods, Lie Bäcklund transformation, and Lie group of transformations. LSA is suitable for providing the conservation laws corresponding to Lie point symmetries or Lie Bäcklund symmetries. Short pulse equation (SPE) is a non-linear PDE, used in optical fibers, computer graphics, and physical systems and has been generalized in many directions. We will find the symmetries and a class of solutions depending on one-parameter (ε) obtained from Lie symmetry groups. Then we will construct the optimal system for the Lie algebra and invariant solutions (called similarity solutions) from Lie subalgebras of generalized SPE.
Non-linear phenomenon can be translated in terms of non-linear differential equations (DEs). The solutions to these DEs may not be possible to find by using routine methods. There are some numerical as well as analytical methods to solve some particular kinds of non-linear DEs. However, Sophus Lie in 1875 gave a general method to solve these DEs. This method is general, algorithmic, and free of estimations. Lie’s method helped to find exact solutions of DEs with the help of Lie group transformations, which is suitable for system of non-linear ordinary differential equations (ODEs) and partial differential equations (PDEs) [1,2]. Inspired by the work of Galois and Abel on the classification of discrete groups, Lie used their work for continuous groups in the nineteenth century. Lie succeeded to reduce order of an ODE by using methodology that the ODE is invariant under one-parameter Lie group of transformations. Lie’s fundamental theorems clearly showed that continuous groups of DEs are specified in respect of their symmetry generators. Furthermore, if system of PDEs is invariant under Lie group of transformations, then one can determine solution’s classes called invariant (similarity) solutions [1,3,4]. The non-linear PDEs arise in many applied fields such as biology, fluid, engineering, mechanics, plasma physics, and optics. Some physical models can be represented in mathematical form through short pulse equation (SPE). SPE is a non-linear PDE that was derived by Schäfer and Wayne  as to model non-linear DE, which describes the propagation of ultra short optical pulses. In optics, pulses are the light flashes. Figure 1 gives an overview of a positively chirped ultra-short pulse of light in the time domain. Thus, physically the short pulses possess the duration of picoseconds, which allows high data transmission in optics . Rauch and Alterman used the approach of short pulses in Fourier domain where pulse is broad [5,7,8]. Their studies motivated other quality researchers to focus on other PDEs rather than cubic non-linear Schrodinger equation (NLSE). This idea inspired Schafer and Wayne to concentrate on ultra short pulses in 2004. They demonstrated that SPE gives strongly preferable approximation to the solution of Maxwell’s equation of electric field in fiber as compared with the cubic NLSE . Chung et al. proved it numerically that as we shorten the pulse length, the NLSE approximation becomes less precise as compared with SPE for the solution of Maxwell’s equation . Liu and Li evaluated exact travelling wave solution and analytic solution corresponding to SPE . Matsuno derived a class of one- and two-phase non-singular periodic solutions of SP model equation and discussed their properties in ref. . Liu et al. determined conditions for wave breaking in SPE . Feng et al. worked on integrable discretization of SPE . They have generated determinant formulas of N-soliton solutions of SPE. It is well-known that solitons are the solutions of a particular non-linear PDE. The two-component short pulse, complex short pulse, and coupled complex SPEs were proposed almost 20 years ago in ref. . These equations were derived from the negative-order flows of the WadatiŰKonnoŰIchikawa (WKI) hierarchy. Qiao et al. discussed the category of non-linear evolution equations associated with the spectral problem and provided an approach for constructing their algebraic structure and -matrix. First authors introduced the category of NLEEs, which is composed of various positive-order and negative-order hierarchies of NLEEs both integrable and non-integrable. Authors also proved that the whole category of NLEEs possessed a generalized Lax representation . Feng’s paper on complex short pulse and coupled complex SPEs computed the soliton solutions by the typical Hirota methods in ref. . Feng worked on a complex SPE and a coupled complex SPE, which describe propagation of ultra-short pulse in optical fibers . Sakovich concluded that vector form of SPE must be integrable as a result of Painlevé analysis . Liu et al. gave a complete Lie group classification for the generalized SPE. They determined the reductions and exact solutions along with power series solutions for SPE . For further details of SPE and its solutions, see refs [19,20, 21,22].
Lie paved the way to solve non-linear system of PDEs by using algorithm of Lie group symmetries. Sun discussed exact and explicit solutions of non-linear Burgers’ equation by using the Lie symmetry method . Liu et al. established Lie symmetry analysis (LSA) for the non-linear general Burgers’ equation. They evaluated its exact solutions and similarity reductions . LSA for the Heisenberg equation is performed by Zhao and Han , which allowed them to obtain the corresponding Lie symmetries, optimal system of sub-algebra, invariant solutions, and conservation laws for this equation. Khalique and Adem used LSA to generate exact solutions of generalized (3 + 1)-dimensional Kadomtsev–Petviashivili equation . We actually give some generalizations of the work of Liu and Li  on cases of SPE and add more facts such as optimal system of one-dimensional Lie symmetry sub-algebras. The SPE is defined by
where is the unknown real function and it shows the magnitude of the electric field, and are dependent variables, and and are non-zero real parameters. In subscripts, and denote the differentiation. Some cases have also been done for special SPE for and in Eq. (1) which is
For and in Eq. (1), Liu and Li  obtained exact solutions by using LSA for
Matsuno worked on modified SPE . The modified short pulse equation is a direct reduction of a coupled SPE given as
Authors studied multi-soliton solutions and the Cauchy problem for a two-component short pulse system, . Zhijun Qiao discussed a finite-dimensional involutive system, and the WKI hierarchy of non-linear evolution equations and their commutator representations in ref. . Recently, the researchers are working on the more general and modified forms of PDEs by taking specific functions to generalize the results corresponding to these PDEs. In the present article, we have considered the generalized form of SPE from Eq. (1), which contains the arbitrary (general) functions and by replacing in first term and in second term at right-hand side of Eq. (1), respectively, given as
Our results will be more general in nature covering a broad class of already available results on SPE. The generalized short pulse equation (GSPE) (5) gives many non-linear wave equations depending upon the functions and . For the generalized SPE, the following cases have been discussed for the first time:
For , and ,
For and .
2 Main results
We focus on LSA on the GSPE given above in two separate cases.
Case 1: We will deal with the situation for , , and
Case 2: We will deal with the situation for and .
2.1 LSA for generalized SPE for , , and
The generalized SPE corresponding to Case 1 for ( ) and , we have
First, we consider the Lie group of transformations with independent variables and dependent variable for the equation
where is the group parameter.
Now, consider the corresponding one-parameter Lie group of transformations given as:
where is represented as the group parameter.
The infinitesimal generator of (7) is defined in the vector form as
where , and are infinitesimal functions of group variables. The generalized SPE involves second-order derivatives, therefore, the second prolongation of the infinitesimal generator has the following form:
The derivatives w.r.t. and are involved in (6) when prolongation (10) is applied onto (6). The general formula of th prolongation of with coefficients ( ; ) is determined in ref. . We only need the values of and which are given in ref.  as
where and are the total derivatives.
In order to calculate the symmetry algebra of (6), apply the second prolongation (10) of the infinitesimal generator (9) along with invariance condition onto Eq. (6):
then simplification gives the following equation:
Two terms and that are involved in the aforementioned equation are derived as follows:
Substituting their values in (14), we have an equation
The coefficients of monomials are then compared and this comparison will give us the following overdetermined system of equations:
These are the determining equations in terms of infinitesimals and derivatives of infinitesimals with respect to independent and dependent variables. One can obtain the most general Lie groups for (6) by finding the values of infinitesimal functions from the above determined equations. There is no hard and fast method to solve the above system of equations. Therefore, to solve the above system, we are considering such a value of , which satisfies the above system of equations involving in (18)
where , , and are arbitrary constants. Using this above value of in the last two equations of system (18), we will achieve the values of , and that are given as
, and are arbitrary constants. Consequently, the infinitesimal generators of SPE (6) for the one-parameter Lie groups of transformations by (6) are obtained by substituting the values of infinitesimal functions (20) into (9) and simplifying this equation. Thus, these infinitesimal generators are given as follows:
The set of generators is closed under the one-parameter Lie groups that are generated by infinitesimal generators ( ) where the entries will give the transformed points .
is a group parameter and .
For each symmetry group , if satisfies (6), then for are solutions of (6).
where and is any positive number and .
2.1.1 Optimal system of one-dimensional subalgebra for generalized SPE (6)
Optimal system provides us the classes of equivalent Lie subalgebras and each similar class of Lie subalgebra in a set forms an optimal system. Optimal system leads to reduce the number of independent variables in DE, which makes it easier to obtain the solution of this DE. As we know that symmetry group of system of PDEs maps one solution of the system to another one of that system of PDEs. In this subsection, we will find a one-dimensional optimal system of Lie subalgebras admitted by (6) with the help of Lie brackets and values in adjoint table.
Lie brackets for the commutator table are defined as
The commutator table for is easy to be seen
for . Here, we will generate the Adjoint table for infinitesimal operators for by using the formula given as
Thus, the adjoint table is as follows:
Taking a non-zero generator of the form
we suppose that the coefficients for .
Case 1: If we take , we have the symmetry generator as follows:
Applying the on and representing it with . Therefore,
Now we apply on . Therefore, we have the symmetry generator as
According to this algorithm, any subalgebra obtained from the above is equivalent to .
Case 2: If we take and , in this case, we have the symmetry generator of the form:
Applying the onto :
Subcase 2.1: If , then
Subcase 2.2: If , then
Case 3: If we take and let . Then, the generator becomes .
Case 4: Let us consider and . In this case, the generator is .
In this way, we will have the comprising form of optimal system for the symmetries of generalized SPE (6):
2.1.2 Reductions and invariant solutions
Now, we reduce non-linear PDE (6) into a non-linear ODE against each symmetry by constructing new invariant solutions from the characteristic equation.
Reduction through .
The corresponding characteristic equation is
Eq. (40) gives invariants and . So, the invariant solution will be
where and are any constants. The aforementioned equation will reduce Eq. (7) in
This implies that is a solution of trivial ODE and it gives .
which implies that
where and are any constants. This will lead to similarity solution of the form:
Thence, substituting this equation into (6)
Hence, the solution set will be
The corresponding characteristic equation of will be written as:
Therefore, we have the invariants
Here, in the aforementioned equations and are arbitrary constants.
The invariant solution in the form of is
We need the derivatives , and of the above value of for substitution into (6). Therefore, we have
inserting these values in (6) and simplifying the calculations lead to
Thus, it reduced our PDE into an ODE, which is non-linear.
The corresponding Lagrange equation is
and the invariants are
are arbitrary constants and the corresponding solution to this in the form of function will be written as:
substituting this value into (6), we will have
For this generator, the invariants are
where and are any constants. The invariant solution corresponding to generator is found to be
substituting this solution into Eq. (6) gives an ODE of the following form:
2.2 LSA for and
We deal some part of the second case and leave the remaining as an open case to discuss. The non-linear SPE for the functions and is
For this, we have discussed the one-parameter Lie group of transformations, infinitesimal generator, and second prolongation with coefficients in (8), (9), (10), (11), and (12), respectively. Applying the second-order prolongation onto (61):
simplification allows the equation of the form:
solving this equation in the aforementioned manner of Section 2.1 for values of and . Also, comparing the values of coefficients will generate a system of equations which is:
For the same value of in (19), infinitesimals are
The infinitesimal generators for the one-parameter Lie group of transformations are as follows:
The commutator table for ( ) is
The adjoint table has been constructed by using the above definition in (25).
The symmetry generators and form a two-dimensional abelian Lie symmetry algebra.
In this article, LSA is performed for the generalized non-linear SPE by taking the general functions for , in place of and in place of , respectively. Lie point symmetries are evaluated for these cases. Exact solutions are found from one-parameter Lie groups . In first case, optimal system of one-dimensional Lie sub-algebra is constructed for GSPE, which leads to their invariant solutions. The graphs are obtained for the functions ( ), which are solutions of our GSPE. By using the symmetries, the reductions are made which basically converts non-linear SPE into non-linear ODE. LSA is a powerful tool to deal with PDEs especially non-linear and fractional PDEs that represent many physical models. LSA enables us to find the symmetries and solutions for the travelling waves. Many of the non-linear PDEs have been solved by this algorithm of LSA. Physicists and mathematicians are working on different forms of non-linear PDEs from nineteenth century to the present such as coupled PDEs, complex PDEs, modified forms of non-linear PDEs, vector forms of PDEs, solitary wave solutions, multi-soliton solutions, N-soliton solutions, their relations with other PDEs, etc. Their work is helping to overcome many difficulties in mathematics and applied sciences. Others can do work on generalized forms of different non-linear PDEs to evaluate their symmetries and exact solutions. Also, they can find their solitary solutions and construct conservation laws corresponding to the symmetries of non-linear PDEs and enable us to attain interesting results related to these PDEs.
The authors are thankful to reviewers’ comments and suggestions, which improved the quality of this article.
Funding information: Science & Technology Bureau of ChengDu 2020-YF09-00005-SN supported by Sichuan Science and Technology program 2021YFH0107 Erasmus+ SHYFTE Project 598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
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