Lie Symmetry Analysis and Similarity Solutions for the Jimbo-Miwa Equation and Generalisations
Amlan K Halder, Andronikos Paliathanasis, Rajeswari Seshadri, Pgl Leach
aa r X i v : . [ n li n . S I] J un LIE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THEJIMBO - MIWA EQUATION AND GENERALISATIONS
AMLAN K HALDER, A PALIATHANASIS, RAJESWARI SESHADRI, AND PGL LEACH
Abstract.
We study the Jimbo - Miwa equation and two of its extended forms, as proposed by Wazwaz et al,using Lie’s group approach. Interestingly, the travelling - wave solutions for all the three equations are similar.Moreover, we obtain certain new reductions which are completely different for each of the three equations. Forexample, for one of the extended forms of the Jimbo - Miwa equation, the subsequent reductions leads to asecond - order equation with Hypergeometric solutions. In certain reductions, we obtain simpler first - orderand linearisable second - order equations, which helps us to construct the analytic solution as a closed - formfunction. The variation in the nonzero Lie brackets for each of the different forms of the Jimbo - Miwa alsopresents a different perspective. Finally, singularity analysis is applied in order to determine the integrabilityof the reduced equations and of the different forms of the Jimbo - Miwa equation. Introduction
The Jimbo - Miwa equation in 1 + 3 space dimensions is a partial differential equation (PDE) and along withits two different extended forms is our subject of study. The equation [19] was proposed in 1983 and has gainedconsiderable attention in the past with respect to its study of integrability using various standard techniques.It is well known that this equation is a member of the KP hierarchy and possesses wider physical applications.Significant works were conducted by Dorizzi et al[15], Wang et al[31], Wazwaz et al [32, 33], Cao et al[9] andmany others. Cao et al[9] also provides a comprehensive list of most of the work done with respect to thealgebraic structure of the equation.The Jimbo - Miwa equation is defined to be(1.1) u xxxy + 3 u y u xx + 3 u x u xy + 2 u ty − u xz = 0 , while the extended forms given by Wazwaz et al [32] are,(1.2) u xxxy + 3 u y u xx + 3 u x u xy + 2 u ty − u xz + u yz + u zz ) = 0 , and(1.3) u xxxy + 3 u y u xx + 3 u x u xy + 2( u tx + u ty + u tz ) − u xz = 0 . We study the determination of solutions for the three equations by using the method of Lie point symmetries.Lie symmetries are a powerful tools for the analysis of nonlinear differential equations. The main idea of Liesymmetries is to determine the transformations which leave the given differential equation invariant. Then,by using normal coordinates, the solution of the differential equation can be written in terms of the invariantfunctions for the Lie symmetry vector and in that way to reduce the number of independent variables in caseof PDEs, or the order of a ordinary differential equation (ODE) [8, 16, 26]. Applications of Lie symmetries canbe found for instance in [18, 20, 22, 24, 25, 27, 34, 35, 43] and references therein.The application of Lie’s theory for equations (1.1), (1.2) and (1.3) reveals that equation (1.1) admits six Liepoint symmetries, equation (1.2) admits ten Lie point symmetries while equation (1.3) is invariant under a six- dimensional group of point transformations.The Lie symmetries are applied in order to determine similarity solutions for the equations under our con-sideration. What is more interesting is that the three equations of our study admit the same travelling-wavereduced equations with slight variation among each other. Reductions with other similarity variables providedifferent results for each equation. For example, with respect to equation (1.1), certain reductions lead to
Date : 28:05:2019.1991
Mathematics Subject Classification.
Key words and phrases.
Symmetry analysis; Similarity solutions; Closed-form solution; Singularity analysis. homogenous solvable PDEs of less order or to second - order equations which are maximally symmetric. Specif-ically, for equation (1.2) certain similarity variable reduces the equation to a second - order equation possessinga Hypergeometric solution. Certain reductions of (1.2) and (1.1) also lead to a third - order or second - orderequation with zero Lie point symmetries. We analyse such equations using singularity analysis. According tothe authors’ knowledge most of the results obtained here are new and cannot be found in the literature. Wetook the aid of a symbolic manipulation code developed by Dimas et al [12, 13, 14]. The paper also discuss theintegrability of the three forms of Jimbo - Miwa using singularity analysis. It is shown that equations (1.1),(1.2) and (1.3) satisfy the test.The paper is arranged as follows: In Section 2 , the preliminaries of Lie point symmetry analysis and singularityanalysis are mentioned. In Section 3 , and its subsequent subsections Lie’s point symmetry analysis of equation(1.1) is discussed elaborately. In sections 4 and 5 , analysis of equations (1.2) and (1.3) is discussed. Section 6 , details the singularity analysis of the second - order equation obtained by the subsequent reductions of equation(1.1). In Section 7 , the singularity analysis for the PDEs is presented. In the end, a brief conclusion and properreferences are mentioned. 2. Preliminaries
In this Section, we briefly discuss the mathematical tools that we apply in this work to study the PDEs of ourconsideration. More specifically, we give the basic definitions for the theory of Lie symmetries and singularityanalysis.2.1.
Lie symmetries.
Let(2.1) F ( x , u α , u β , u δ , u αβ , ... ) = 0 , where x = ( x α , x β , x δ ) are the set of independent variables, u α = ∂u∂x α , u β = ∂u∂x β , and u αβ = ∂u∂x α x β , and thesubsequent terms can be defined henceforth. Under an infinitesimal point transformation,¯ x α = x α + ǫξ α ( x α , x β , x δ , u ) + O ( ǫ ) , ¯ x β = x β + ǫξ β ( x α , x β , x δ , u ) + O ( ǫ ) , ¯ x δ = x δ + ǫξ δ ( x α , x β , x δ , u ) + O ( ǫ ) , ¯ u = u + ǫη ( x α , x β , x δ , u ) + O ( ǫ ) , (2.2)equation (2.1) is said to be invariant if and only if,(2.3) F ( x , u ) = F (¯ x , ¯ u ) . The transformations forms a symmetry group, say, G, the generator of which can be defined as,(2.4) Γ = η ( x α , x β , x δ , u ) ∂ u + ξ α ( x α , x β , x δ , u ) ∂ x α + ξ β ( x α , x β , x δ , u ) ∂ x β + ξ δ ( x α , x β , x δ , u ) ∂ x δ . Therefore Γ, which is the generator of the infinitesimal transformations, can be considered as a Lie pointsymmetry of equation (2.1). Now one can use equation (2.4) to discuss the reduction of the correspondingPDEs using the characteristic functions, obtained by solving the associated Lagrange’s system which is, dx α ξ α = dx β ξ β = dx δ ξ δ = duη . Singularity analysis.
Our second method of investigation is known as singularity analysis which evolved at about the same time assymmetry analysis, i.e. towards the end of the nineteenth century. In essence, it is the determination of theexistence of singularities in the dependent variable(s) in the complex plane of the independent variable. Indeed,it is interesting to note how analysis developed following the pioneering work of Cauchy in complex analysis.Equations which satisfy the analysis are said to possess the Painlev´e Property. This is an interesting descriptoras the first known application of singularity analysis was due to Sophie Kowalevski who used the analysis todetermine the third integrable case of the spinning top [21] . The treatment of the Painlev´e Property in theclassic text of E L Ince [17] is very clear. Later works by Ramani, Grammaticos and Bountis [29] and MichaelTabor [30] have kept the idea of proving integrability through singularity analysis before the public eye. The first two cases were due to Euler and Lagrange in the seventeenth century.
IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS3
A major development is to be found in the works of Ablowitz, Ramani and Segur [1, 2, 3] who introduced astraightforward approach to determining whether a given differential equation possesses the Painlev´e Property.The approach has become known as the ARS algorithm and is very simple in its concept although there aretimes when its application faces some serious challenges. There are three steps:(1) Determine the existence or otherwise of a singularity by making a substitution of y = a ( x − x ) p intothe differential equation (for simplicity we consider a single dependent variable).This is called the leading - order term. The nature of the singularity is indicated by the value of p and its location is x . Originally, it was considered to be a negative integer, hence, the method ofpolelike expansions, but over the years it was accepted that p could equally be a fraction which couldbe positive or negative as differentiation of a positive fractional power eventually leads to a negativepower and so a singularity. As a practical point, it must be borne in mind that a fraction requires thecomplex plane to be divided into segments by branch - line cuts. A multitude of these is not good ifnumerical work has eventually to be undertaken. The terms which contribute to the evaluation of p arecalled the dominant terms.(2) A differential equation of order greater than one needs additional constants of integration and the ideais to construct a Laurent expansion about the singularity.The additional constants of integration enter the expansion at powers called resonances and theseare identified by making the substitution y = a ( x − x ) p + m ( x − x ) p + s into the dominant terms .The result is a polynomial in m . A given coefficient enters the expansion when its coefficient is m . Thecoefficient is arbitrary, i.e. to be determined by the initial conditions, if m is zero. The coefficient of m is a polynomial in s and the solution of polynomial equals zero gives the requisite values of s . Oneof the resonances must take the value −
1. This value is associated with the location of the moveablesingularity.(3) The Laurent expansion is substituted into the complete equation with the coefficients of the resonantterms bring arbitrary and the other coefficients determined.This can be a tedious process, but, if all goes well, the solution is inferred to be analytic albeitpossibly restricted in applicability by branch cuts.Parallel procedures for partial differential equations can be found in [36, 37, 38, 39, 40, 41, 42].3.
The Symmetry Analysis for equation (1.1)
For (1.1), we compute the Lie - Point symmetries,Γ a = ∂ t , Γ a = t∂ t − u ∂ u + x ∂ x + 2 z ∂ z , Γ a = ∂ z , Γ a = t∂ x + 2 x ∂ u , Γ a = c ( z ) ∂ y + 3 tc ′ ( z )4 ∂ x + (cid:18) xc ′ ( z )4 − tyc ′′ ( z )4 (cid:19) ∂ u , Γ a = E ( t, z ) ∂ u , where c ( z ) and E ( t, z ) are arbitrary functions.The nonzero Lie brackets are, [Γ a , Γ a ] = Γ a , [Γ a , Γ a ] = ∂ x , [Γ a , Γ a ] = − a , [Γ a , Γ a ] = 2Γ a . Case 1:
To study the travelling - wave reductions, we follow a simple procedure, by which we consider m Γ a − c Γ a , to Usually the letter r is used, but we use s in deference to the pioneering work of Kowalevskaya. AMLAN K HALDER, A PALIATHANASIS, RAJESWARI SESHADRI, AND PGL LEACH reduce the equation (1.1) to a new PDE of dimension 1 + 2, where m is the wave number and c denotes thefrequency. The similarity variables for this reduction are, w = mz − ct,u ( t, x, y, z ) = v ( x, y, w ) . (3.1)The possible reduced PDE is(3.2) v xxxy + 3 v x v xy + 3 v y v xx − cv yw − mv xw = 0 . The Lie - Point symmetries are, Γ b = ∂ w , Γ b = v∂ v − w∂ w − x∂ x − y∂ y , Γ b = ∂ x , Γ b = w∂ x − (cid:18) cx my (cid:19) ∂ v Γ b = ∂ y . Case 1a:
The translation, with respect to the variables x, w and y is obtained for the equation (3.2). Therefore, we usethe operator k Γ b + l Γ b + Γ b , where k and l are the wave numbers, to obtain the reduced ode,(3.3) k lp ′′′′ ( q ) + 6 k lp ′ ( q ) p ′′ ( q ) − (2 cl + 3 km ) p ′′ ( q ) = 0 , where q = kx + ly + w and v ( x, y, w ) = p ( q ). This can be ascertained by the reader that the equation, (3.3),is the corresponding travelling - wave reduction for the PDE, equation (1.1). We compute the symmetries forequation (3.3). They are, Γ c = ∂ q , Γ c = ∂ p , Γ c = q∂ q + (cid:18) mqkl + 2 cq k − p (cid:19) ∂ p . Case 1b:
We first look at the reduction using Γ c , namely,(3.4) g ′′′ ( h ) = 10 g ′ ( h ) g ′′ ( h ) g ( h ) − g ′ g ( h ) + (cid:18) (2 cl + 3 km ) g ( h ) k l − g ( h ) k (cid:19) g ′ ( h ) , where h = p ( q ) and g ( h ) = p ′ ( q ) . Equation (3.4) possesses a lone symmetry, ∂ h . We use this to reduce (3.4) toa second - order equation.(3.5) b ′′ ( a ) = 3 b ′ b ( a ) + 10 b ′ ( a ) a + (cid:18) ak − a ck − a mk l (cid:19) b ( a ) + 15 b ( a ) a , where a = g ( h ) and b ( a ) = g ′ ( h ) . Equation (3.5) is maximally symmetric. Therefore, the solution of (1.1), canbe obtained in accordance to equation (3.5). Case 1c:
Next, we use Γ c to study the reduction of equation (3.3).The reduced third - order equation is,(3.6) g ′′′ ( h ) = (cid:18) cl + 3 kmk l − g ( h ) k (cid:19) g ′ ( h ) , where h = q and g ( h ) = p ′ ( q ). The reduced third - order equation has two symmetries. They are ∂ h and h∂ h + (2 c − gk ) l +3 km k l ∂ g . The reduction with respect to ∂ h leads to the second - order equation(3.7) b ′′ ( a ) = 3 b ′ b ( a ) + (6 ak l − cl − km ) b ( a ) k l , IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS5 where a = g ( h ) and b ( a ) = g ′ ( h ) . Equation (3.7) possesses 8 symmetries and hence is linearisable.The reduction with respect to h∂ h + (2 c − gk ) l +3 km k l ∂ g leads to the second - order equation,(3.8) b ′′ ( a ) = 3 b ′ b ( a ) + 9 b ′ ( a ) b ( a ) + (6 a + 26 k ) b ( a ) k − (12 a + 24 ak ) b ( a ) k , where a = (6 g ( h ) k l − cl − km ) h k l and b ( a ) = 3 k lh (3 hg ′ ( h ) k l + 6 g ( h ) k l − cl − km ) . Equation (3.8) has zero Lie - Point symmetries. Singularity analysis of this equation is treated in Section 6 . The reduction with respect to Γ c leads to a third - order equation with zero Lie point symmetries. We omitmentioning the equation here considering the high nonlinearity of the third - order equation. The equation isunder study and we shall be discussing it in our subsequent work.3.1. Further reductions for the Jimbo - Miwa equation.
In this subsection, we study the reductions of equation (1.1) with respect to Γ a . The similarity variables are, w = xt ,w = zt ,u ( t, x, y, z ) = v ( y, w , w ) t . The reduced PDE of 1 + 2 form is,(3.9) 9 v w w + (2 − v w w ) v y + 4 w v yw + 2 w v yw − v w v yw − v yw w w = 0 . The symmetries of equation (3.9) are, Γ d = y∂ y + w ∂ w , Γ d = ∂ y , Γ d = ∂ w + 2 w ∂ v , Γ d = 2 √ w ∂ w − y √ w ∂ v , Γ d = c ( w ) ∂ v , where c ( w ) is an arbitrary function with respect to w . Case 2:
We use Γ d for the reduction. The similarity variables are w y = w , v ( y, w , w ) = v ( w , w ) . The reducedPDE in 1 + 1 dimensions is,0 = − w v w w + 9 v w w − w w v w w + 9 w v w v w w +3 w v w ( − v w w ) + 3 w v w w w w . The Lie point symmetries are Γ e = ∂ v and Γ e = w ∂ v + ∂ w . We use Γ e for reduction. The similarityvariables are v ( w , w ) = w + v ( w ) . The reduced ode is v ′ + w v ′′ = 0 which is maximally symmetric.Finally, the closed - form similarity solution of equation (1.1) is, u ( t, x, y, z ) = x t + K t + K t log (cid:18) zyt (cid:19) + K t + K t , where K , K , K , and K are arbitrary constants. Case 3: Reduction with respect to Γ d : The similarity variable is v ( y, w , w ) = v ( w , w ). The reduced PDE is v w w = 0 . The solution of equation(1.1) can be given as,(3.10) u ( t, x, y, z ) = f (cid:16) xt (cid:17) + g (cid:16) zt (cid:17) t . AMLAN K HALDER, A PALIATHANASIS, RAJESWARI SESHADRI, AND PGL LEACH
Case 4: Reduction with respect to Γ d : The similarity variable is, v ( y, w , w ) = w v ( y, w ) . The reduced PDE is v yw = 0 . Similarly, to the above case, the solution of the PDE can be easily determined.Therefore, the solution of the equation (1.1) can be given in terms of the solution of the reduced PDE.
Case 5: Reduction with respect to Γ d : The similarity variables are v ( y, w , w ) = − yw w + v ( y, w ) . Hence, the reduced PDE is,(3.11) 9 y w + 2 v y + 4 w v yw = 0 . The Lie - point symmetries for (3.11) are derived to be,Γ f = c ( y ) ∂ y , Γ f = c ( w ) ∂ w − v c ( w )2 w ∂ v , Γ f = v ∂ v , Γ f = E ( y, w ) ∂ v , where c ( y ), c ( w ) and E ( y, w ) are arbitrary functions. None of the above mentioned symmetries or linearcombination of any of them leads to a satisfactory reduction. The Painlev´e analysis of the equation (3.11) isunder study to ascertain its integrability. Case 6: Reduction with respect to Γ a : Next, we study the reduction with respect to Γ a for equation (1.1). The similarity variable is u ( t, x, y, z ) = x t + v ( t, y, z ) . The reduced PDE of dimension 1 + 2 is,(3.12) v y + tv ty = 0 . The reduced PDE possesses the following Lie point symmetries,Γ g = E ( t, z ) ∂ t , Γ g = E ( y, z ) ∂ y , Γ g = E ( z ) ∂ z + vE ( z ) ∂ v , Γ g = E ( t, y, z ) ∂ v , where E i ’s are arbitrary functions of the variables mentioned against them. Equation (3.12) can easily beintegrated v + tv t + f ( t ) = 0, from which we find v ( t ) = v t − R f ( t ) dtt . Reductions for the Jimbo - Miwa equation.
In this subsection, we study subsequent reductions for equations (3.2) and (1.1). We start with Γ b , for whichwe consider different possibilities for the similarity variables and study the subsequent reductions. We considerfirstly the similarity variables, x w = w ,xy = w ,v ( x, y, w ) = v ( w , w ) x . The reduced PDE is,0 = − w v ( w , w ) (cid:0) w v w w − v w + w v w w (cid:1) + 2 cw w v w w − mw v w w − mw w v w w − mw v w + 27 w w v w w w w + 27 w w v w w w w +27 w w v w v w w + 54 w w v w w w + 9 w w v w w w w + 9 w w v w v w w +18 w w v w w w + 6 w w v w w + 3 w v w (cid:0) w (cid:0) w v w w + w v w w (cid:1) + 2 w v w w (cid:1) + w v w w w w − w v w . IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS7
The Lie-Point symmetries are,Γ h = (cid:18) mw cw − w (cid:19) ∂ w + (cid:18) m c + w (cid:19) ∂ w − (cid:18) m w cw + 3 mv cw + cw mw w − v (cid:19) ∂ v , Γ h = w ∂ v . Case 7:
We use Γ h , for reduction and the corresponding similarity variables can be represented as v ( w , w ) = p ( q , q ), where w = q and w = q . We assume that p ( q , q ) = p a ( q ) p a ( q ), where p a and p a arearbitrary functions of q and q respectively. The computation becomes less tedious for two particular cases,firstly when p a ( q ) = F , where F is an arbitrary constant. The reduced second - order ode which is maximallysymmetric is,(3.13) 2 p ′ a + 3 q ( p ′′ a ) = 0 . Therefore, the solution of equation (1.1) can be given in terms of equation (3.13).Similarly, when p a ( q ) = F , where F is arbitrary, we obtain a fourth - order equation, namely,(3.14) 6 p a ( q ) p ′ a ( q ) − q p a ′ ( q ) − q p a ( q ) p ′′ a ( q ) + 6 q p ′ a ( q ) p ′′ a ( q ) + q p ′′′′ a ( q ) = 0 . This equation has a single symmetry, ∂ q , which reduces equation (3.14) to a third - order equation with zeropoint symmetries. The equation is, p ′′′ a ( q ) = (cid:18) p ′ a ( q ) p a ( q ) + 6 p a ( q ) (cid:19) p ′′ a ( q ) − p ′ a ( q ) p a ( q ) − p ′ a ( q ) + (cid:0) (3 q − p a ( q ) − p a ( q ) (cid:1) p ′ a ( q ) − p a ( q ) + (9 q − p a ( q ) , (3.15)where q = p a ( q ) and p a ( q ) = q p ′ a ( q ) . Singularity analysis of the equation (3.15) is under study.
Case 8:
The similarity variable considered using Γ b is v ( x, y, w ) = − cx +3 mxy w + v ( y, w ), where y and w are the newindependent variables. The reduced PDE is,(3.16) v y + wv yw = 0 . The reduced PDE (3.16) is similar to equation (3.12) which is also a reduced PDE of 1 + 1 dimensions. Also, itcan be easily observed that equation(3.16) is a variant of equation (3.11). The Lie point symmetries are almostin similar nature, hence we omit mentioning them here.
AMLAN K HALDER, A PALIATHANASIS, RAJESWARI SESHADRI, AND PGL LEACH The Symmetry Analysis for equation (1.2)
The Lie-point symmetries are,Γ i = ∂ x , Γ i = ∂ y , Γ i = t ∂ x + (cid:18) t z (cid:19) ∂ y + t∂ z , Γ i = 12 ∂ y + ∂ z , Γ i = t∂ t + (cid:18) t x z (cid:19) ∂ x + (cid:18) t z (cid:19) ∂ y + z ∂ z − (cid:16) u y (cid:17) ∂ u , Γ i = ∂ t + 38 ∂ y , Γ i = z ∂ x + (cid:18) y − z − t (cid:19) ∂ y + z∂ z − y ∂ u , Γ i = 3 c ( t )2 ∂ x + xc ′ ( t ) ∂ u , Γ i = − c ( t )2 ∂ x + ( zc ′ ( t ) − xc ′ ( t )) ∂ u , Γ i = c ( t ) ∂ u , where c , c , and c are arbitrary functions of t. The nonzero Lie brackets are,[Γ i , Γ i ] = Γ i , [Γ i , Γ i ] = − ∂ u , [Γ i , Γ i ] = − ∂ u ∂ y , [Γ i , Γ i ] = − Γ i , [Γ i , Γ i ] = − (cid:18) t + 2 z (cid:19) ∂ u − t ∂ x − (cid:18) t z (cid:19) ∂ y − t ∂ z , [Γ i , Γ i ] = − Γ i − Γ i , [Γ i , Γ i ] = − Γ i − Γ i , [Γ i , Γ i ] = − (cid:18) t z (cid:19) ∂ u + t ∂ x + 3 t + 2 z ∂ y + t∂ z , [Γ i , Γ i ] = − ∂ u
24 + ∂ x
12 + ∂ y ∂ z , [Γ i , Γ i ] = − ∂ u ∂ x ∂ y ∂ z , [Γ i , Γ i ] = − ∂ t + ∂ u − ∂ x − ∂ y , [Γ i , Γ i ] = − (cid:18) t z (cid:19) ∂ u , [Γ i , Γ i ] = − ∂ u . Case 9:
We study the reduction firstly with the travelling - wave simplification. It can be easily verified that Γ i − Γ i and Γ i − i are symmetries. Therefore, we use linear combination of Γ i , Γ i , Γ i − Γ i and Γ i − i , i.e. k Γ i + l Γ i + m (Γ i − Γ i ) − c (Γ i − i ), where k, l, m are wave numbers and c is the frequency, to reduce theequation to a fourth-order ode,(4.1) k lp ′′′′ ( q ) + 6 k lp ′ ( q ) p ′′ ( q ) − (cid:0) cl + 3 km + 3 lm + 3 m (cid:1) p ′′ ( q ) = 0 , where q = kx + ly + mz − ct and p ( q ) = u ( t, x, y, z ). It is to be noted here that equation (4.1) is similar to theequation (3.3). The only difference can be found in the coefficient of second derivative of p with respect to q .Also, the subsequent reductions are similar to the previous section. IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS9
Further Reductions of Equation (1.2).Case 10:
We reduce equation (1.2) using Γ i . It is to be mentioned here that other symmetries such as, Γ i and Γ i donot provide favourable reductions.The similarity variables for Γ i are, − t + 8 y − z w ,z √ t = w , x − z − t t = w ,u ( t, x, y, z ) = 45 t − t y + 48 t z + 640 v ( w , w , w )640 t . The reduced PDE of dimension 1 + 2 is9 v w w + (2 − v w w ) v w − w v w w − v w v w w − v w w w w + 3 w v w w = 0 . The Lie point symmetries are, Γ j = 2 w ∂ w + w ∂ w , Γ j = ∂ w , Γ j = ∂ v , Γ j = − ∂ w + w ∂ v , Γ j = w ∂ v . Case 11:
We study the reduction with respect to Γ j . The similarity variables are w = w √ w , v ( w , w , w ) = v ( w , w ) . The reduced PDE of 1 + 1 dimensions is,3( − w ) v w w + w v w (5 − v w w ) − w (cid:0) (2 w + 9 v w ) v w w + 3 v w w w w (cid:1) = 0 . The Lie point symmetries are Γ k = ∂ w − w ∂ v , Γ k = ∂ v . Case 12:
We study the reduction with Γ k . The similarity variable is, v ( w , w ) = − w v ( w ) . The reduced ode is,(4.2) 7 w v ′ ( w ) − v ′′ ( w ) + 3 w v ′′ ( w ) = 0 . Equation (4.2) is maximally symmetric. Also, one interesting observation regarding (4.2) is that the solutionof the equation is in terms of Hypergeometric function. Therefore, solution of (1.2) can be given in terms ofequation (4.2).
Case 13:
We study the reduction using Γ j + Γ j , the similarity variables w , w , are the newly defined independentvariable and v ( w , w , w ) = w + v ( w , w ) . The reduced PDE is,(4.3) 2 − v w w + 9 v w w = 0 . The symmetries of equation (4.3) are,Γ l = v ∂ v , Γ l = ∂ w , Γ l = ( − c ( w − w ) + c ( w + w )) ∂ w + ( − c ( w − w ) + c ( w + w )) ∂ w , Γ l = E ( w , w ) ∂ v , where c ( w − w ) , c ( w + w ) and E ( w , w ) are arbitrary functions. Case 14:
We use Γ l for reduction. The similarity variable is v ( w , w ) = v ( w ) . The reduced ode is,(4.4) 2 + 9 v ′′ ( w ) = 0 . Equation (4.4) is linearisable. Therefore, the solution of (1.2) can be given in terms of equation (4.4).
Case 15:
We study the reduction with respect to Γ j . The similarity variables are v ( w , w , w ) = − w + v ( w , w ) . The new independent variables are w and w . The reduced PDE is,(4.5) 9 v w w + 4 v w + 3 w v w w = 0 . The Lie - point symmetries are, Γ m = ∂ w , Γ m = v ∂ v , Γ m = E ( w , w ) ∂ v . Case 16:
The reduction using Γ m , leads to the well - known linearisable second - order equation, which is v ′′ ( w ) = 0 , where the similarity variable is v ( w , w ) = v ( w ) . Next, using Γ m + Γ m , which is a symmetry,the corresponding reduced ode is,(4.6) 4 v ( w ) + 3 w v ′ ( w ) + 9 v ′′ ( w ) = 0 , where the similarity variables are v ( w , w ) = e w v ( w ) . Equation (4.6) is a second - order linear equationand it is maximally symmetric.5.
The Symmetry Analysis for equation (1.3)
Similarly, for equation (1.3), we only mention the Lie - point symmetries here. The results with respect tothe travelling - wave reduction are similar and hence we omit it here.Γ o = ∂ t , Γ o = ∂ x , Γ o = ∂ y , Γ o = ∂ z , Γ o = f a ( t ) ∂ u , Γ o = g a ( z ) ∂ u , where f a ( t ) and g a ( z ) are the arbitrary functions. It is clear that the Lie brackets of the point symmetries, fromΓ o to Γ o , of equation (1.3) do not possess any nonzero output and so the algebra is abelian .5.1. Further reduction for equation (1.3).Case 17:
We study the reduction with respect to Γ o + Γ o . The similarity variable is u ( t, x, y, z ) = f ( t ) + v ( x, y, z ) , where f ( t ) = R f a ( t ) dt. The reduced PDE in 1 + 2 dimensions is,(5.1) − v xz + 3 v x v xy + 3 v y v xx + v xxxy = 0 . The Lie - point symmetries are, Γ p = x∂ v − z∂ y , Γ p = ∂ y , Γ p = ∂ z , Γ p = h ( z ) ∂ x − yh ′ ( z ) ∂ v , Γ p = h ( z ) ∂ v , where h ( z ) and h ( z ) are arbitrary functions with respect to z . IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS11
Case 18:
The similarity variable with respect to Γ p is v ( x, y, z ) = − xyz + v ( x, z ) . The reduced PDE in 1 + 1 dimensionsis,(5.2) v x + zv xz + xv xx = 0 . The Lie - point symmetries are, Γ q = E ( x, z ) ∂ x + E ( x, z ) ∂ v , Γ q = E ( z ) ∂ z , Γ q = v E ( z ) ∂ z , where E ( x, z ) , E ( z ) and E ( z ) are the arbitrary functions.The similarity variable for Γ q is v ( x, z ) = xv ( z ) and the reduced ode is of Euler type, namely,(5.3) v ( z ) + zv ′ ( z ) = 0 . Next, the similarity variable using Γ q + Γ q , is of the form v ( x, z ) = e E ( z ) + v ( x ) , where E ( z ) = R E ( z ) E ( z ) dz. The reduced ode is,(5.4) v ′ ( x ) + xv ′′ ( x ) = 0 . Therefore, the solution of equation (1.3) can be given in terms of equation (5.3) and (5.4).
Case 19:
We use Γ p + Γ p for the reduction. The similarity variables are w = z − y, v ( x, y, z ) = v ( x, z − y ) . Thereduced PDE is,(5.5) 3 v xw + 3 v x v xw + 3 v w v xx + v xxxw = 0 . The Lie - point symmetries of equation (5.5) are,Γ r = x∂ x − (2 x + v ) ∂ v , Γ r = ∂ x , Γ r = ∂ v , Γ r = h ( w ) ∂ w , where h ( w ) is an arbitrary function of w . We use Γ r for reduction. The similarity variable, v ( x, w ) = − x + v ( w ) x , leads to the reduced first - orderode,(5.6) ( − v ( w )) v ′ ( w ) = 0 , which can be solved easily. Case 20:
We now study the reduction with respect to Γ o + Γ o , for which the similarity variables are u ( t, x, y, z ) = g ( z ) + v ( t, x, y ) , and g ( z ) = R dzg a ( z ) . The reduced PDE of dimension 1 + 2 is,(5.7) 3 v x v xy + 3 v y v xx + v xxxy + 2 v ty + 2 v tx = 0 . The Lie - point symmetries are, Γ s = ∂ t , Γ s = t∂ t + x ∂ x + y ∂ y − v ∂ v , Γ s = ∂ y , Γ s = t∂ x + (cid:18) x y (cid:19) ∂ v , Γ s = ∂ x , Γ s = h ( t ) ∂ v . where h ( t ) is an arbitrary function of t. We use Γ s + Γ s + Γ s to study the further reduction. The similarity variables are, x − t = w ,y − t = w ,v ( t, x, y ) = v ( w , w ) . The reduced PDE of dimension 1 + 1 is, − v w w − v w w + 3 v w v w w − v w w + 3 v w v w w + v w w w w = 0 . The Lie - point symmetries are, Γ t = ∂ w , Γ t = ∂ w , Γ t = ∂ v . The similarity variables with respect to Γ t + Γ t + Γ t are, w − w = w ,v ( w , w ) = w + v ( w ) . The reduced ode is,(5.8) v ′′′′ ( w ) − v ′ ( w ) v ′′ ( w ) + 3 v ′′ ( w ) = 0 . Equation (5.8) has three symmetries, which are,Γ u = ∂ w , Γ u = ∂ v , Γ u = w ∂ w + ( w − v ) ∂ v . Case 21:
The reduction with respect to Γ u , leads to a third - order ode,(5.9) v ′′′ ( w
4) = 10 v ′ ( w v ′′ ( w v ( w − v ′ v ( w + ( − v ( w + 6 v ( w v ′ ( w , where w = v ( w ) and v ( w ) = v ′ ( w ) . The equation (5.9) has a sole symmetry, ∂ w . The reduced third -order ode is further reduced to the second - order equation, which is,(5.10) v ′′ ( w
5) = 3 v ′ ( w ) v ( w ) + 10 v ′ ( w ) w + (3 w − w ) v ( w ) w + 15 v ( w w , where w = v ( w ) and v ( w ) = v ′ ( w ) . The equation (5.10) is maximally symmetric.
Case 22:
The reduction with respect to Γ u , leads to the third - order equation,(5.11) v ′′′ ( w
4) = (6 v ( w − v ′ ( w , where w = w and v ( w ) = v ′ ( w ) . Equation (5.11) has two symmetries, which are ∂ w and w ∂ w +(1 − v ) ∂ v . The reduction with respect to ∂ w leads to the second - order equation,(5.12) v ′′ ( w
5) = 3 v ′ v ( w
5) + ( − w + 3) v ( w ) , where w = v ( w ) and v ( w ) = v ′ ( w ) . The equation (5.12) possess 8 symmetries and hence is maximallysymmetric. Next, we use w ∂ w + (1 − v ) ∂ v , for reduction. The reduced second - order ode is,(5.13) v ′′ ( w
5) = 3 v ′ ( w ) v ( w ) + 9 v ( w ) v ′ ( w ) + (26 − w ) v ( w ) + (12 w − w ) v ( w ) , where w = (2 v ( w ) − w and v ( w ) = w ( w v ′ ( w )+2 v ( w ) − . Equation (5.13) possesses zero Lie point sym-metries and it is similar to equation (3.8). The singularity analysis fails to infer any specific conclusion forequation (5.13). Therefore, the solution of equation (1.3) can be given in terms of equation (5.10) and (5.12).
IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS13
Case 23:
The reduction with respect to Γ u leads to a third - order equation with zero Lie point symmetries. The equationis, v ′′′ ( w
4) = (cid:18) v ′ ( w ) v ( w ) + 10 v ( w ) (cid:19) v ′′ ( w ) − v ′ ( w ) v ( w ) − v ′ ( w ) + (cid:0) ( − w − v ( w ) + 6 v ( w ) (cid:1) v ′ ( w ) + 18 v ( w ) − (30 w + 50) v ( w ) +(12 w + 24 w ) v ( w ) , where w = (2 v ( w ) − w ) w and v ( w ) = w ( w v ′ ( w )+ v ( w ) − w ) . Singularity analysis of this equation is understudy.
Case 24:
The reduction with respect to Γ s , leads to a PDE of dimension 1 + 1, with similarity variable, v ( t, x, y ) = v ( xt , yt ) t , w v w w + 4 v w + 2 w v w w + 2 w v w w − v w v w w + v w (4 − v w w )+2 w v w w − v w w w w . (5.14)Equation (5.14) has a sole point symmetry, ∂ v . The subsequent reductions do not yield any favourable reduc-tions.
Case 25:
The reductions with respect to Γ s , leads to a PDE of 1 + 1 dimension. Similar equations are obtained in section3 , equation (3.12), where we have discussed it in detail. The equation is,(5.15) v y + tv ty = 0 , where the similarity variable is v ( t, x, y ) = x + xy )3 t + v ( t, y ) . Singularity analysis
We study the singularity analysis of equation (3.8). We suggest that readers refer [4, 5, 28, 31, 33] tounderstand the preliminaries.Firstly, we write it in a convenient format,(6.1) − k + 3 x ) y ( x ) + 12 x (2 k + x ) y ( x ) − ky ( x ) y ′ ( x ) − ky ′ ( x ) + ky ( x ) y ′′ ( x ) = 0 , where we mention b ( a ) = y ( x ). We substitute y → αx p in equation (6.1) and look for the possible values of theexponent p . The substitution leads to,(6.2) − a kx p − a kpx − p − a kp x − p − a kpx − p − a x p + 24 a kx p + 12 a x p = 0One of the possible values obtained from the dominant terms is − . For p = −
1, the possible values of theleading-order coefficient α are 0 , , , . Next we substitute y → αx − + mx − s into (6.1) to compute theresonances ( s denotes the resonance). The substitution leads to,0 = − a kx + 9 a kx − a kx + 24 a kx − a x + 12 a x − akmx − s + 27 a kmx − s − a kmx − s +120 a kmx − s + 3 akmsx − s − a kmsx − s + akms x − s − a mx − s + 60 a mx − s − km x − s + 27 akm x − s − a km x − s + 240 a km x − s + 3 km sx − s − akm sx − s − km s x − s − a m x − s + 120 a m x − s +9 km x − s − akm x − s + 240 a km x − s − km sx − s − am x − s +120 a m x − s − km x − s + 120 akm x − s − m x − s + 60 am x − s + 24 km x − s + 12 m x − s . (6.3) We consider the linear terms with respect to m from equation (6.3), − akx − s + 27 a kx − s − a kx − s + 120 a kx − s + 3 aksx − s − a ksx − s + aks x − s − a x − s + 60 a x − s . (6.4)We list the corresponding values of resonances for various nonzero values of the leading - order coefficient, α . (cid:0) α = → s = ( , (cid:1) , (cid:0) α = → s = ( − , ) (cid:1) , (cid:0) α = → s = ( − , − ) (cid:1) . It is to be observed that the genericvalue of s , which is −
1, is not obtained for any of the possible values of the leading - order coefficient[4, 5] andhence we cannot infer about the integrability of equation (3.8).7.
Singularity analysis for the Jimbo-Miwa PDE
We apply the singularity analysis for the PDE (1.1). We do that by replacing u → U φ ( t, x, y, z ) p [10, 11] , in equation (1.1). 0 = 3 pU Φ − p Φ z Φ x − p U Φ − p Φ z Φ x − p U Φ − p Φ y Φ x +6 p U Φ − p Φ y Φ x − pU Φ − p Φ y Φ x ++11 p U Φ − p Φ y Φ x − p U Φ − p Φ y Φ x + p U Φ − p Φ y Φ x − pU Φ − p Φ xz + 3 p U Φ − p Φ x Φ xy ++6 pU Φ − p Φ x Φ xy − p U Φ − p Φ x Φ xy ++3 p U Φ − p Φ x Φ xy + 3 p U Φ − p Φ y Φ xx +6 pU Φ − p Φ y Φ x Φ xx − p U Φ − p Φ y Φ x Φ xx ++3 p U Φ − p Φ y Φ x Φ xx − pU Φ − p Φ xy Φ xx +3 p U Φ − p Φ xy Φ xx − pU Φ − p Φ x Φ xxy + 3 p U Φ − p Φ x Φ xxy − pU Φ − p Φ y Φ xxx + p U Φ − p Φ y Φ xxx + pU Φ − p Φ xxxy + − pU Φ − p Φ y Φ t + 2 p U Φ − p Φ y Φ t + 2 pU Φ − p Φ ty . (7.1)Solving the dominant terms, we obtain the value of p is − . To obtain the coefficient of the leading - orderexponent we substitute the value of p and collect the dominant terms, which gives,(7.2) − U Φ y Φ x + 24 U Φ y Φ x . We solve equation (7.2) to obtain the values of U , which are, U → , U → φ x . For the nonzero U , we compute the resonances, for which we substitute, U ( t, x, y, z ) → U Φ( t, x, y, z ) − + m Φ( t, x, y, z ) − S into equation (1.1). The substitution leads to,0 = 6 m ( − S )( − S ) Φ S Φ y Φ x − U Φ y ( U − x )Φ x + m ( − S )( − S )Φ S Φ y Φ x ( − U + ( S − − S )Φ x ) ++3 m ( − S ) Φ S (Φ x Φ xy + Φ y Φ xx ) ++3 U Φ( U − x )(Φ x Φ xy + Φ y Φ xx ) ++3 m ( − S )Φ ( S )( − U + ( S − − S )Φ x )(Φ x Φ xy + Φ y Φ xx ) − m ( − S )( − S )Φ S ( − xy Φ xx + 3Φ x (Φ z − Φ xxy ) − Φ y (Φ xxx + 2Φ t ))+2 U Φ (3Φ xy Φ xx + 3Φ x ( − Φ z + Φ xxy ) + Φ y (Φ xxx + 2Φ t )) ++ U Φ (3Φ xz − Φ xxxy − ty ) − m ( S − S (3Φ x − Φ xxxy − ty ) . Next, we collect all the linear terms with respect to m, which gives,( − S )( − S )Φ S Φ y Φ x ( − U + ( S − − S )Φ x ) +3( − S )Φ S ( − U + ( S − − S )Φ x )(Φ x Φ xy + Φ y Φ xx ) − ( S − − S )Φ S ( − xy Φ xx + 3Φ x (Φ z − Φ xxy ) − Φ y (Φ xxx + 2Φ t )) − ( S − S (3Φ xz − Φ xxxy − ty ) . IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS15
After substitution of the nonzero value of U, we obtain,(7.3) − y Φ x + 10 S Φ y Φ x + 23 S Φ y Φ x − S Φ y Φ x + S Φ y Φ x . When we factor equation (7.3), we get the values of resonances as s → − , , , . As the generic values of − S is obtained, we next verify the consistency, for which we substitute, U = (2Φ x )Φ( t, x, y, z ) − + U ( t, x, y, z )Φ − ++ U ( t, x, y, z )Φ( t, x, y, z ) − + U ( t, x, y, z )Φ( t, x, y, z ) − + U ( t, x, y, z )Φ( t, x, y, z ) − ++ U ( t, x, y, z )Φ( t, x, y, z ) − + U ( t, x, y, z )Φ( t, x, y, z ) − , into equation (1.1). From our observation, the values for U ( t, x, y, z ) and U ( t, x, y, z ) are,6Φ y Φ x U ( t, x, y, z ) = 3Φ z Φ x − y U x Φ x − U y Φ x ++3Φ xy Φ xx − x Φ xxy − Φ y Φ xxx − y Φ x U ( t, x, y, z ) = 112Φ y Φ x y U x Φ x − U y Φ x − x Φ xz + 3Φ x U xy ++6 U x Φ x Φ xy + 9 U Φ x Φ xy + 3Φ y Φ x U xx − z Φ xx ++3Φ y U x Φ xx + 9 U y Φ x Φ xx + 15 U Φ y Φ x Φ xx − xy Φ xxx + 4Φ x Φ xxxy + Φ y Φ xxxx + 2Φ xy Φ t + 2Φ x Φ ty + 2Φ y Φ tx . As, observed from the values of U ( t, x, y, z ) and U ( t, x, y, z ) , arbitrary term is U ( t, x, y, z ) . Similar observationscan be made from U ( t, x, y, z ) to U ( t, x, y, z ). The calculations are bit tedious and we omit mentioning themhere. We summarize our analysis in the following theorem. Theorem:
The Jimbo-Miwa equation (1.1) and its generalizations (1.2) and (1.3) pass the singularity test,that is, have the Painlev´e property and are integrable. Their solution is expressed by a Right Painlev´e Series.8.
Conclusion
An elaborate study of the reductions of Jimbo - Miwa and its extended forms is discussed. As mentionedabove, new reductions of the equations were obtained. The singularity analysis of certain reduced odes and allthe three forms of Jimbo - Miwa equation are discussed. This paper forms the first part of our subsequent paperson Jimbo - Miwa equations. The conservation laws and the solutions from them are under study. Moreover,we have discussed reductions elaborately, but not exhaustively. Therefore, the remaining reductions will alsobe discussed in our future work.Finally, by applying the singularity analysis for the PDEs of our consideration, without applying any similaritytransformation, we found that the equations of our consideration have the Painlev´e property and the analyticsolution can be expressed in terms of a Right Painlev´e Series.9.
Acknowledgements
AKH expresses grateful thanks to UGC (India), NFSC, Award No. F1-17.1/201718/RGNF-2017-18-SC-ORI-39488 for financial support and Late Prof. K.M.Tamizhmani for the discussions which AKH had with him whichformed the basis of this work. PGLL acknowledges the support of the National Research Foundation of SouthAfrica, the University of KwaZulu-Natal and the Durban University of Technology.
References [1] Ablowitz M J, Ramani A & Segur H (1978), Nonlinear Evolution Equations and Ordinary Differential Equations of Painlev´eType,
Lett Nuovo Cimento , , 333-337.[2] Ablowitz M J, Ramani A & Segur H (1980), A connection between nonlinear evolution equations and ordinary differentialequations of P type I, Journal of Mathematical Physics , , 715-721.[3] Ablowitz M J, Ramani A & Segur H (1980), A connection between nonlinear evolution equations and ordinary differentialequations of P type II, Journal of Mathematical Physics , , 1006-1015.[4] Andriopoulos K & Leach PGL (2009), The occurrence of a triple-1 resonance in the standard singularity, Nuovo Cimento dellaSocieta Italiana di Fisica. B, General Physics, Relativity, Astronomy and Mathematical Physics and Methods , , 1-11. [5] Andriopoulos K & Leach PGL (2011), Singularity analysis for autonomous and nonautonomous differential equa-tions, Applicable Analysis and Discrete Mathematics , , 230-239, (available online at http://pefmath.etf.rs)(doi:10.2298/AADM110715016A) (ISSN 1452-8630).[6] Andriopoulos K & Leach PGL (2006), An interpretation of the presence of both positive and negative nongeneric resonancesin the singularity analysis, Physics Letters A , , 199-203.[7] Andriopoulos K & Leach PGL (2007), Symmetry and singularity properties of second-order ordinary differential equation ofLie’s Type III, Journal of Mathematical Analysis and Applications , , 860-875.[8] Bluman GW & Kumei S(1989), Symmetries and Differential Equations, Springer-Verlag, New York.[9] Cao B (2010), Solutions of Jimbo-Miwa Equation and Konopelchenko-Dubrovsky Equations, Acta applicandae mathematicae , , 181-203.[10] Conte R (1988), Universal invariance properties of Painlev´e analysis and B¨ a cklund transformation in nonlinear partial differ-ential equations, Physics Letters A , , 100-104.[11] Conte R (1989), Invariant Painlev´e analysis of partial differential equations, Physics Letters A , , 383-390.[12] Dimas S & Tsoubelis D (2004), SYM: A new symmetry - finding package for Mathematica. In Proceedings of the 10thInternational Conference in Modern Group Analysis (pp. 64-70).[13] Dimas S & Tsoubelis D (2006), A new Mathematica - based program for solving overdetermined systems of PDEs,
In 8thInternational Mathematica Symposium, Avignon .[14] Dimas S (2008), Partial differential equations, algebraic computing and nonlinear systems, Ph. D. thesis,
University of Patras ,Greece.[15] Dorizzi B, Grammaticos B, Ramani A & Winternitz P (1986), Are all the equations of the Kadomtsev - Petviashvili hierarchyintegrable?
Journal of mathematical physics , , 2848-2852.[16] Ibragimov NH(2000), CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions,and Conservation Laws, CRS Press LLC, Florida.[17] Ince EL (1927), Ordinary Differential Equations, (Longmans, Green & Co, London).[18] Jamal S & Paliathanasis A (2017), Group invariant transformations for the Klein - Gordon equation in three dimensional flatspaces, Journal of Geometry and Physics , , 50-59.[19] Jimbo M & Miwa T (1983), Solitons and infinite dimensional Lie algebras, Publications of the Research Institute for Mathe-matical Sciences , , 943-1001.[20] Kallinikos N & Meletlidou E (2013), Symmetries of charged particle motion under time-independent electromagnetic fields, Journal of Physics A: Mathematical and Theoretical , , 305202.[21] Kowalevski Sophie (1889), Sur la probl`eme de la rotation d’un corps solide autour d’un point fixe, Acta Mathematica , Journalof Mathematical Analysis and Applications , , 487-493.[23] Meleshko SV(2005), Methods for Constructing Exact Solutions of Partial Differential Equations, Springer Science, New York.[24] Meleshko SV & Shapeev VP (2011), Nonisentropic solutions of simple wave type of the gas dynamics equations, Journal ofNonlinear Mathematical Physics , , 195-212.[25] Nucci MC & Sanchini G (2015), Symmetries, Lagrangians and conservation laws of an Easter Island population model, Symmetry , , 1613-1632.[26] Olver PJ(1993), Applications of Lie Groups to Differential Equations, Springer-Verlag, New York.[27] Paliathanasis A, Krishnakumar K, Tamizhmani KM & Leach PGL (2016), Lie symmetry analysis of the Black-Scholes-MertonModel for European options with stochastic volatility, Mathematics , , 28.[28] Paliathanasis A & Leach PGL (2016), Nonlinear ordinary differential equations: A discussion on symmetries and singularities, International Journal of Geometric Methods in Modern Physics , , 1630009.[29] Ramani A, Grammaticos B & Bountis T (1989), The Painlev´e property and singularity analysis of integrable and nonintegrablesystems Physics Reports , , 159-245.[30] Tabor M (1989), Chaos and Integrability in Nonlinear Dynamics: An Introduction, John Wiley, New York .[31] Wang D, Sun W, Kong C & Zhang H (2007), New extended rational expansion method and exact solutions of Boussinesqequation and Jimbo-Miwa equations,
Applied mathematics and computation , , 878-886.[32] Wazwaz AM (2017), Multiple-soliton solutions for extended (3 + 1)-dimensional Jimbo-Miwa equations, Applied MathematicsLetters , , 21-26.[33] Wazwaz AM (2008), Multiple-soliton solutions for the Calogero -Bogoyavlenskii- Schiff, Jimbo- Miwa and YTSF equations, Applied Mathematics and Computation , , 592-597.[34] Webb GM (1990), Lie symmetries of a coupled nonlinear Burgers-heat equation system, Journal of Physics A: Mathematicaland General , , 3885.[35] Webb GM & Zank GP (2006), Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydro-dynamics and gas dynamics. Journal of Physics A: Mathematical and Theoretical , , 545.[36] Weiss John, Tabor M & Carnevale George (1983), The Painlev´e property for partial differential equations, Journal of Mathe-matical Physics , , 522-526.[37] Weiss John (1983), The Painlev´e property for partial differential equations. II: B¨acklund transformation, Lax pairs, and theSchwarzian derivative Journal of Mathematical Physics , , 1405-1413.[38] Weiss John (1984), On classes of integrable systems and the Painlev´e property Journal of Mathematical Physics , , 13-24.[39] Weiss John (1984), B¨acklund transformation and linearizations of the H´enon-Heilles system Physics Letters A , ,329-331. IE SYMMETRY ANALYSIS AND SIMILARITY SOLUTIONS FOR THE JIMBO - MIWA EQUATION AND GENERALISATIONS17 [40] Weiss John (1984), The sine-Gordon equations: Complete and partial integrability,
Journal of Mathematical Physics , ,2226-2235.[41] Weiss John (1985), The Painlev´e property and B¨acklund transformations for the sequence of Boussinesq equations, Journal ofMathematical Physics , , 258-269.[42] Weiss John (1985), Modified equations, rational solutions, and the Painlev´e property for the Kadomtsev-Petviashvili andHirota-Satsuma equations, Journal of Mathematical Physics , , 2174-2180.[43] Xin X, Liu Y, & Liu X (2016), Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations, Applied Mathematics Letters , , 63-71. Department of Mathematics, Pondicherry University, Puducherry - 605014, India
E-mail address : [email protected] Institute for Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of SouthAfrica
E-mail address : [email protected] Department of Mathematics, Pondicherry University, Puducherry - 605014, India
E-mail address : [email protected] School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South AfricaandInstitute for Systems Science, Durban University of Technology, Durban, South Africa
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