Lie symmetries and singularity analysis for generalized shallow-water equations
aa r X i v : . [ n li n . S I] J un Lie symmetries and singularity analysis for generalized shallow-waterequations
Andronikos Paliathanasis ∗ Institute of Systems Science, Durban University of TechnologyPO Box 1334, Durban 4000, Republic of South Africa
June 3, 2020
Abstract
We perform a complete study by using the theory of invariant point transformations and the singularityanalysis for the generalized Camassa-Holm equation and the generalized Benjamin-Bono-Mahoney equation.From the Lie theory we find that the two equations are invariant under the same three-dimensional Liealgebra which is the same Lie algebra admitted by the Camassa-Holm equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of thesimilarity solutions for the two equations of our study. The reduced equations are studied by using the pointsymmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partialdifferential equations from where we infer that the generalized equations of our study pass the singularitytest and are integrable.Keywords: Lie symmetries; invariants; shallow water; Camassa-Holm; Benjamin-Bono-Mahoney
The Lie symmetry analysis plays a significant role in the study of nonlinear differential equations. The existenceof a Lie symmetry for a given differential equation is equivalent with the existence of one-parameter pointtransformation which leaves the differential equation invariant. The later property can be used to reducethe number of independent variables on the case of partial differential equations (PDE), or reduce the orderof an ordinary differential equation (ODE) [1], that is achieved thought the Lie invariants. In addition, Liesymmetries can been used for the determination of conservation laws. One the most well-know applications ofthe latter are the two theorems of E. Noether [1]. However, there are also alternative methods to determinethe conservation laws by using the Lie point symmetries without imposing a Lagrange function, some of thesealternative approaches are described in [1–5] and references therein.The are many applications of the Lie symmetries on the analysis of differential equations, for the determina-tion of exact solutions, to determine conservation laws, study the integrability of dynamical systems or classifyalgebraic equivalent systems [6–13]. Integrability is a very important property of dynamical systems, hence itworth to investigate if a given dynamical system is integrable [14–21]. ∗ Email: [email protected]
1n alternative approach for the study of the integrability of nonlinear differential equations is the singularityanalysis. In contrary with the symmetry analysis, singularity analysis is based on the existence of a pole for thedifferential equation. The first major result of the singularity analysis is the determination of the third integrablecase of Euler’s equations for a spinning by Kowalevskaya [22]. Since then, the have been many contributionsof the singularity analysis, mainly by the French school led by Painlev´e [23–25] and many others [26–30].Nowadays, the application of singularity analysis is summarized in the ARS algorithm [31–33] which has madethe singularity analysis a routine tool for the practising applied mathematicians.Singularity analysis and symmetry analysis have been applied in a wide range of differential equations arisingfrom all areas of applied mathematics, for instance see [34–46] and references therein. The two methods aresupplementary, on the study of integrability of differential equations. Usually, the symmetry method is appliedto reduce the given differential equation into an algebraic equation, or into another well-known integrabledifferential equation. On the other hand, for differential equation which posses the Painlev´e property, i.e. itpass the singularity test, its solution is written in terms of Laurent expansions, a recent comparison of the twomethods is presented in [47].In this work, we study the integrability of generalized Camassa-Holm (CH) and Benjamin-Bono-Mahoney(BBM) equations [48, 49] by using the Lie point symmetries and the singularity analysis. These two equationsdescribe shallow-water phenomena.The Camassa-Holm equation is a well-known integrable equation. It was originally discovered by Fuchssteineret al. in [50], however become popular a decade later by the study of Camassa and Holm where they provedthe existence of peaked solutions, also known as peakons. On the other hand, BBM equation also known asregularized long-wave equation discovered in [52] and it is an extension of the KdV equation. The two equationsare related, in the sense they have a common operator and partial common Hamiltonian structure. The planof the paper is as follows.In Section 2 we present the basic elements on the mathematical tools of our consideration, that is, the Liepoint symmetries and the singularity analysis. Our main analysis is included in Sections 3 and 4 where westudy the existence of similarity solutions for the generalized CH and BBM equations, as also we prove theintegrability of these two equations by using the singularity analysis. Finally, we discuss our results and drawour conclusions in Section 5.
In this section we briefly discuss the application of Lie’s theory on differential equations as also the main stepsof the singularity analysis.
Consider the vector field X = ξ i (cid:0) x k , u (cid:1) ∂ i + η (cid:0) x k , u (cid:1) ∂ u , (1)to be the generator of the local infinitesimal one-parameter point transformation,¯ x k = x k + εξ i (cid:0) x k , u (cid:1) , (2)¯ η = η + εη (cid:0) x k , u (cid:1) . (3)2hen X is called a Lie symmetry for the differential equation, H (cid:0) y i , u, u i , u ij , ..., u i i ...i n (cid:1) , if there exists afunction λ such that the following condition to hold X [ n ] H = λH (4)where X [ n ] is called the second prolongation/extension in the jet-space and is defined as X [ n ] = X + (cid:16) D i η − u ,k D i ξ k (cid:17) ∂ u i + (cid:16) D i η [ i ] j − u jk D i ξ k (cid:17) ∂ u ij + ... + (cid:16) D i η [ i ] i i ...i n − − u i i ...k D i n ξ k (cid:17) ∂ u i i ...in . (5)The novelty of Lie symmetries is that they can be used to determine similarity transformations, i.e. differ-ential transformations where the number of independent variables is reduced [1]. The similarity transformationis calculated with the use of the associated Lagrange’s system, dx i ξ i = duu = du i u [ i ] = ... = du ij..i n u [ ij...i n ] . (6)The similarity transformation in the case of PDEs is used to reduce the number of indepedent variables.The solutions derived by the application of Lie invariants are called similarity solutions. The modern treatment of the singularity analysis is summarized in the ARS algorithm, established by Ablowitz,Ramani and Segur in [31–33]. There are three basic steps which are summarized as follows: (a) determine theleading-order term which describes the behaviour of the solution near the singularity, (b) find the position ofthe resonances which shows the existence and the position of the integration constants and (c) write a Laurentexpansion with leading-order term determined in step (a) and perform the consistency test. More details on theARS algorithm as also on the conditions which should hold at every step we refer the reader in n the review ofRamani et al. [53], where illustrated applications are presented.It is important to mention that when a differential equation passes the conditions and requirement of theARS algorithm we can infer that the given differential equation is algebraically integrable.
We work with the generalized CH equation defined in [48, 49] u t − u xxt + ( k + 2) ( k + 1)2 u k u x = (cid:18) k u k − u x + u k u xx (cid:19) x , (7)where k ≥ k = 1 CH equation is recovered. The Lie symmetryanalysis for the CH equation presented before in [36]. It was found that the CH is invariant under a threedimensional Lie algebra.For the generalized CH equation (7) the application of Lie’s theory provides us that the admitted Lie pointsymmetries are three, and more specifically they are X = ∂ t , X = ∂ x and X = t∂ t − ku∂ u . The commutators and the adjoint representation of the Lie point symmetries are presented in Tables 1 and 2respectively. 3able 1: Commutators of the admitted Lie point symmetries by the differential equation (7)[ , ] X X X X kX X X − kX Ad (exp ( εX i )) X j X X X X X X X − kεX X X X X X e εk X X X The results presented in Tables 1 and 2 can be used to classify the admitted Lie algebra as also to determinethe one-dimensional optimal systems [54]. A necessary analysis to perform a complete classification of thesimilarity solutions.As far as the admitted Lie algebra is concerned, from Table 1 it is find to be the { A ⊗ s A } in theMorozov-Mubarakzyanov Classification Scheme [55–58].In order to find the one-dimensional optimal systems we consider the generic symmetry vector X = a X + a X + a X , (8)from where we find the equivalent symmetry by considering the adjoint representation. We remark that theadjoint action admits two invariant functions the φ ( a i ) = a and φ ( a i ) = a which are necessary to simplifythe calculations on the derivation of the one-dimensional systems. More specifically there are four possiblecases, { φ φ = 0 } , { φ = 0 , φ = 0 } , { φ = 0 , φ = 0 } and { φ = 0 , φ = 0 } .Consequently, with the use of the invariant functions φ and φ and Table 2 we find that the possibleone-dimensional optimal systems are X , X , X , cX + X and X + αC . (9)We proceed with the application of the latter one-dimensional system in order to reduce the PDE (7) into anODE. In this section we proceed with the application of the Lie point symmetries to the nonlinear generalized CHequation. In order to solve the reduced equation we apply the Lie point symmetries and when it is no possibleto proceed the reduction process, we consider the singularity analysis by applying the ARS algorithm.4 .1.1 Reduction with X : Static solution
The application of the Lie point symmetry vector X indicates that the solution u is static, i.e. u = U ( x ) wherenow function U ( x ) satisfies the third-order ODE( k + 2) ( k + 1)2 U k U x − (cid:18) k U k − U x + U k U xx (cid:19) x = 0 . (10)The latter equation can be easily integrated, and be written in the equivalent form U xx + k U U x − U U k + ( k + 2)2 U = 0 , (11)where U is a constant of integration. Equation (11) admits the following conservation law12 U k ( U x ) − U U − U K +2 U , (12)in which U is a second constant of integration. Equation (12) can be integrated by quadratures.Conservation law (12) is nothing else than the Hamiltonian function of the second-order ODE (11). Beforewe proceed with another reduction let us now apply the singularity analysis to determine the analytic solutionof equation (10). Singularity analysis
We substitute U ( x ) = U χ p , χ = x − x , in (10) and we find the polynomial expression( k + 1) ( k + 2) χ p ( k +1) − + ( p ( k + 1) −
2) ( p ( k + 2) − χ p ( k +1) − = 0 , (13)Hence we can infer that the only possible leading-order behaviour to the terms with x p ( k +1) − , where from therequirement ( p ( k + 1) −
2) ( p ( k + 2) −
2) = 0 , (14)provides p = 2 k + 2 or p = 2 k + 1 , (15)while constant U is undetermined.Consider now the leading order term p . In order to find the resonances we replace U ( x ) = U χ p + µχ p + s in (15) and we linearize around the µ = 0. From the linear terms of µ , the coefficient of the leading order terms χ − k + s are s ( s + 1) ( s ( k + 2) − s = − , s = 0 and s = 2 k + 2 . (16)We remark that because k is always a positive integer number, p and s are always rational numbers. Resonance s indicates that the singularity is movable, the position of the singularity is one of the three integrationconstants. Resonances s shows that the coefficient constant of the leading-order term should be arbitrary sinceit is also one of the integration constants of the problem. The third constant it is given at the position of theresonance s and depends on the value k . Moreover, because all the resonances are positive the solution will begiven by a Right Painlev´e Series. In order to complete the ARS algorithm we should perform the consistencytest. For that we select a special value of k .We select k = 2, and we consider the right Laurent expansion U ( x ) = U χ + U χ + U χ + ∞ X i =3 U i χ + i , (17)5e find that U is the third integration constant while the first coefficient constants are U ( U , U ) = −
78 ( U ) U , U = 54 ( U ) ( U ) , U = − U ) ( U ) + U , ... ; (18)Hence, the consistency test is satisfied and expression (17) is one solution of the third-order ODE (10).We work similar and for the second leading-order behaviour p = k +1 , the resonances are derived to be s = − , s = 0 , s = − k + 1 , (19)from where we infer that the solution is given by a Mixed Painlev´e Series. However, we perform the consistencytest for various values of the integer number k , and we conclude that for this leading-order behaviour thedifferential equation does not pass the singularity test.In order to understand better why only the leading-order behaviour p passes the singularity test, let usperform the same analysis for the second-order ODE (11). By replacing in U ( x ) = U χ p in (11) we find that theunique leading-order behaviour is that with p = k +2 with arbitrary U . The two resonances now are calculatedto be s = − s = 0, from where we can infer that the solution is given by a Right Painlev´e series andthe two integration constants is the U and the position of the singularity. In that case, since we know the twointegration constants it is not necessary to perform the consistency test. X : Stationary solution Reduction with the vector field X provides the stationary solution u ( t, x ) = U ( t ), where U t = 0, that is u ( t, x ) = u . This is the trivial solution. X : Scaling solution I From the symmetry vector X we derive the Lie invariants u = U ( x ) t − k , x (20)hence, by replacing in (7) we end up with the following third-order ODE2 kU k U xxx − (cid:0) − k U k − U x (cid:1) U xx + k (cid:16) (1 − k ) kU k − ( U x ) − ( k + 2) ( k + 1) U k (cid:17) U x + 2 U = 0 , (21)The latter equation admits only one Lie point symmetry, the vector field X . The latter vector field can be usedto reduce equation (21) into a second-order nonautonomous ODE, with no symmetries. Hence, we proceed withthe application of the singularity analysis for equation (21).We replace U ( x ) = U χ p in (21) where we find the following expression − U χ p +2 U p ( p − χ p − + U k +10 k ( k + 2) ( k + 1) χ p ( k +1) − − U k +10 pk ( p ( k + 2) −
2) ( p ( k + 1) − χ ( p ( k +1) − = 0 . (22)From the latter term we find that the only possible leading terms with k positive integer number are p − p ( k + 1) −
3) from where we find that p = k while U is given by the following expression U − k = 2 (2 − k ) , (23)from where we infer that there is a leading-order behaviour only for k = 2.6he resonances are calculated to be s = − , s = k − k , s = k − k , (24)from where we can infer that for k >
2, the solution is given by a Right Painlev´e Series. We perform theconsistency test by choosing k = 3. Hence the Laurent expansion is written as U ( χ ) = U x + U x + ∞ X i =2 U i χ i , (25)where U and U are two integration constants of the solution while the rest coefficient constants are U i = U i ( U , U ). cX + X : Travel-wave solution The travel-wave similarity solution is determined by the application of the Lie invariants of the symmetry vector cX + X where c − is the travel-wave speed. The invariant functions for that vector field are determined to be u ( t, x ) = U ( ξ ) , ξ = x − c − t, (26)where U ( ξ ) satisfies the following third-order ODE2 (cid:0) − cU k (cid:1) U ξξξ − ckU k − U ξ U ξξ − (cid:0) c ( k − U k − U ξξ + 2 − c ( k + 2) ( k + 1) U k (cid:1) U ξ = 0 . (27)Equation (27) is autonomous and admit only one symmetry vector the ∂ ξ . It can easily integrated as follows2 (cid:0) − cU k (cid:1) U ξξ − c (cid:0) U k − − U k − (cid:1) ( U ξ ) + c ( k + 2) U k − − U + U = 0 , (28)which can be solved by quadratures.Let us now apply the singularity analysis to write the analytic solution of equation in (28) by using Laurentexpansions. We apply the ARS algorithm and we find the leading order term U ( ξ ) = U ( ξ − ξ ) p with p = k +2 and U arbitrary. The resonances are calculated to be s = 0 and s = 0, which means that the solution isgiven by a Right Painlev´e Series with integration constants the position of the singularity ξ and the coefficientconstant of the leading order term U . The step of the Painlev´e Series depends on the value of k , for instancefor k = 2, p = and the step is , while for k = 3 , p = and the step is . X + αX : Scaling solution II We complete our analysis by determine the similarity solution given by the symmetry vector X + αX . Thethat specific symmetry the Lie invariants are calculated u ( t, x ) = U ( ξ ) t − k , ξ = x + 1 αk ln t. (29)Therefore, by selecting ξ to be the new independent variable and U ( ξ ) the new dependent variable we endup with the third-order ODE2 (1 + αk ) U ξξξ − α (cid:0) − k U k − U ξ (cid:1) U ξξ + (cid:16) a ( k − U k − ( U ξ ) − − αkU k k ( k + 2) ( k + 1) (cid:17) U ξ + 2 αU = 0 . (30)7he latter equation is autonomous and admit only one point symmetry, the vector field ∂ ξ , which can beused to reduce by one the order of the ODE. The resulting second-order ODE has no symmetries. Hence, thesingularity analysis is applied to study the integrability of (30).In order to perform the singularity analysis we do the change of variable V = U − . Hence by replacing V ( ξ ) = V ( ξ − ξ ) p in (30) we find the leading-order terms p = − p = − k > , (31)while V is arbitrary.The resonances are calculated to be p : s = − , s = 0 and s = 1; (32) p : s = − , s = 0 and s = − . (33)We apply the consistency test where we find that only only the leading-order term p provides a solution, whichis given by the following Right Painlev´e Series V ( ξ ) = V ( ξ − ξ ) − + ∞ X i =1 V i ( ξ − ξ ) − i . Until now we applied the singularity analysis to study the integrability of the ODEs which follow by the similarityreduction for the generalized CH equation. However, it is possible to apply the singularity analysis directly inthe PDE. We follow the steps presented in [30].Before we proceed with the application of the ARS algorithm we make the change of transformation u ( t, x ) = v ( t, x ) − in (7). For the new variable we search for a singular behaviour of the form v ( t, x ) = v ( t, x ) φ ( t, x ) p ,where v ( t, x ) is the coefficient function and φ ( t, x ) p is the leading-order term which describe the singularity.The first step of the ARS algorithm provides two values of p, p = − p = − , where v ( t, x ) isarbitrary. A necessary and sufficient condition in order these two leading-order terms to exists is φ ,t φ ,x = 0.Otherwise other leading-order terms follow, however these possible cases studied before. The resonances forthese two leading-order terms are those given in (32) and (34).Consequently, the following two Painlev´e Series should be studied for the consistency test v ( t, x ) = v ( t, x ) φ ( t, x ) − + ∞ X i =1 v i ( t, x ) φ ( t, x ) − i , (34) v ( t, x ) = v ( t, x ) φ ( t, x ) − + ∞ X i =1 v i ( t, x ) φ ( t, x ) − i . (35)By replacing (34) we find that the second integration constant is v ( t, x ). On the other hand, the series (35)does not pass the consistency test. We conclude that that the generalized CH equation passes the singularitytest and it is an integrable equation.We proceed our analysis with the BBM equation. 8 Generalized Benjamin-Bono-Mahoney equation
The generalized BBM equation is u t − u xxt + βu k u x = 0 , (36)where k is a positive integer number. Equation (36) can been seen as the lhs of (7) when β = ( k +2)( k +1)2 andreduce to the BBM equation when k = 1. For the case of k = 1 the Lie symmetry analysis for the BBM equationpresented recently in [59, 60].We apply the Lie theory in order to determine the point transformations which leave equation (36) invariant.We found that the equation (36) admits three point symmetries which are the vector fields X , X , X presentedin Section 3. Hence, the admitted Lie algebra is the 2 A ⊗ s A and there are five one-dimensional optimalsystems as presented in (9). We proceed with the application of the Lie point symmetries for the determinationof similarity solutions. For equation (36) the application of the Lie symmetries X and X provide the trivial solution u ( t, x ) = u forboth cases. X : Scaling solution I The application of the Lie invariants which given by the symmetry vector X gives u ( t, x ) = U ( x ) t k where U ( x ) satisfies the second-order ODE U xx + βkU k U x − U = 0 . (37)The latter equation is autonomous and admit the point symmetry ∂ x which can be used to reduce equation(37) into the following first-order ODE y z = βkz k y − zy , (38)where y ( z ) = ( U x ) − and z = U ( x ).However, equation (37) can be easily solved analytical by using the singularity analysis. Indeed from theARS algorithm we find the leading-order behaviour U ( χ ) = (cid:18) k + 1 βk (cid:19) k x − k , (39)with resonances s = − s = 1 + kk . (40)In order to perform the consistency test we have to select specific value for the parameter k . Indeed for k = 2 we write the Laurent expansion U ( χ ) = (cid:18) β (cid:19) x − + ∞ X i =1 U i x − i , (41)and by replacing in (37) we find that U = 0 , U = 0 , U = − r β , U = 0 , U = − √ β U ) , .... (42)where U is the second integration constant. We conclude that that the equation (37) passes the Painlev´e test.9 .1.2 Reduction with cX + X : Travel-wave solution The travel-wave solution of the generalized BBM equation is u = U ( ξ ), where ξ = x − c − t and U ( ξ ) satisfiesthe differential equation U ξξξ + (cid:0) cβU k − (cid:1) U ξ = 0 . (43)The latter equation can be integrated easily U ξξ + (cid:18) cβk + 1 U k +1 − U (cid:19) + U = 0 , (44)that is 12 ( U ξ ) + (cid:18) cβ ( k + 1) ( k + 2) U k +2 − U (cid:19) + U U − U = 0 , (45)where U , U are two integration constants. The latter differential equation can be solved easily by quadratures.As far as the singularity analysis is concerned for equation (44), the ARS algorithm provides the leading-orderbehaviour U ( ξ ) = U ( ξ − ξ ) − k , U k = − k + 1) ( k + 2) βck , (46)with resonances s = − s = 2 ( k + 2) k . The consistency test has been applied for various values of the positive integer k , and we can infer that equation(44) is integrable according to the singularity analysis. X + αX : Scaling solution II From the Lie symmetry X + αX we find the similarity reduction u ( t, x ) = U ( ξ ) t − k , ξ = x + αk ln t where U ( ξ ) is a solution of the following differential equation U ξξξ − αU ξξ − (cid:0) αβU k + 1 (cid:1) U ξ + αU = 0 . (47)Equation (47) can be reduced to the following second-order ODE by use of the point symmetry vector ∂ ξ , z y zz + z ( y z ) − αzy z − (cid:0) αβz k + 1 (cid:1) y + αz = 0 , (48)where z = U ( x ) and y ( z ) = U x .We apply the ARS algorithm for equation (47) and we find that it passes the singularity test for the leadingorder behaviour U ( ξ ) = U ( ξ − ξ ) − k , U k = 2 ( k + 2) ( k + 1) αβk , (49)with resonances s = − , s = 2 ( k + 1) k , s = 2 ( k + 2) k . (50) We complete our analysis by applying the singularity test in the generalized BBM equation in a similar way aswe did in Section 3.2 for the generalized CH equation. Indeed we find the leading order term u ( t, x ) = v ( t, x ) φ ( t, x ) − k , ( v ( t, x )) k = 2 ( k + 2) ( k + 1) αβk φ t φ x , (51)and resonances those given in (50). We performed the consistency test and we infer that the generalized BBMequation passes the singularity test for any value of the positive integer parameter k .10 Conclusion
In this work we studied the existence of similarity solutions of the generalized CH and generalized BBM equation.The approached that we used is that of the Lie point symmetries. We determined the admitted invariantpoint transformations for the differential equations of our consideration and we determined the one-dimensionaloptimal systems by using the adjoint representation of the admitted Lie algebra. The two differential equationsof our consideration are invariant under the same Lie symmetry vectors which form the same Lie algebra withthe CH and the BBM equations.For each of the equations we perform five different similarity reductions where the PDEs are reduced to third-order ODEs. The integrability of the resulting equations is studied by using symmetries and/or the singularityanalysis. In the case of the generalized CH equation most of the reduced ODEs can not be solved by using Liesymmetries, hence the application of the singularity analysis was necessary to determine the analytic solutionsof the reduced equations.Finally, we study the integrability of the PDEs of our consideration by applying the singularity analysisdirectly on the PDEs and not on the reduced equations. From the latter analysis we found that the generalizedCH and BBM equations pass the singularity analysis and their solutions are given by Right Painlev´e Series.This work contribute to the subject of the integrability of generalized equations describe shallow-waterphenomena. The physical implication of the new analytic solutions will be presented in a future communication.
References [1] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York, (1989)[2] N.H. Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)[3] N.H. Ibragimov, J. Phys. A: Math. Theor. 44, 432002 (2011)[4] S.C. Anco, Symmetry 9, 33 (2017)[5] S.A. Hojman, J. Phys. A: Math. Gen. 25, L291 (1992)[6] M.C Nucci and G. Sanchini, Symmetry 8, 155 (2016)[7] P.G.L. Leach, K.S. Govinder and K. Andriopoulos, J. Applied Mathematics 2012, 890171 (2012)[8] S. Jamal and N. Mnguni, Appl. Math. Comp. 335, 65 (2018)[9] S. Jamal, Quaestiones Mathematicae 41, 409 (2018)[10] S. Jamal, A.G. Johnpillai, Indian Journal of Physics (2019) [doi.org/10.1007/s12648-019-01449-z][11] S. Jamal, Int. J. Geom. Meth. Mod. Phys. 16, 1950160 (2019)[12] A. Paliathanasis, Symmetry 11, 1115 (2019)[13] A. Paliathanasis, Class. Quantum Grav. 33, 075012 (2016)[14] T. Sen, Phys. Lett. A 122, 327 (1987) 1115] M. Tsamparlis and A. Paliathanasis, J. Phys. A. Math. Theor, 44, 175205 (2011)[16] M. Tsamparlis and A. Paliathanasis, J. Phys. A. Math. Theor. 45, 275202 (2012)[17] M. Lakshmanan and R. Sahadevan, Phys. Reports 224, 1 (1993)[18] R.Z. Zhdanov, J. Math. Phys. 39, 6745 (1998)[19] P. Nattermann and H.-D. Doebner, J. Nonlinear Math. Phys. 3, 302 (1996)[20] M.S. Velan and M. Lakshamanan, J. Nonlinear Math. Phys. 5, 190 (1998)[21] P.-L. Ma, S.-F. Tian, T.-T. Zhang and X.-Y. Zhang, Eur. Phys. J. Plus 131, 98 (2016)[22] S. Kowalevski, Acta Math
177 (1889)[23] P. Painlev´e, Le¸cons sur la th´eorie analytique des ´equations diff´erentielles (Le¸cons de Stockholm, 1895)(Hermann, Paris, 1897). Reprinted, Oeuvres de Paul Painlev´e, vol. I, ´Editions du CNRS, Paris, (1973)[24] P. Painlev´e, Bulletin of the Mathematical Society of France
201 (1900)[25] P. Painlev´e, Acta Math LXIV
LXVI
806 (1962)[30] R. Conte, The Painlev´e Property: One Century Later, Conte Robert ed (CRM Series in MathematicalPhysics, Springer-Verlag, New York) (1999)[31] M.J. Ablowitz, A. Ramani and H. Segur, Lettere al Nuovo Cimento
333 (1978)[32] M.J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys.
715 (1980)[33] M.J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys.
159 (1989)[54] P.J. Olver, Applications of Lie Groups to Differential Equations, second edition, Springer-Verlag, New York(1993)[55] Morozov VV (1958),
Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika, Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika, Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika, Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika,35