Lie Symmetries and Similarity transformations for the Generalized Boiti-Leon-Pempinelli equations
aa r X i v : . [ n li n . S I] A ug Lie Symmetries and Similarity transformations for the GeneralizedBoiti-Leon-Pempinelli equations
K. Krishnakumar , A. Durga Devi and A. Paliathanasis , Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University,Kumbakonam 612 001, India. Department of Physics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam 612001, India. Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia 5090000, Chile
Abstract:
We perform a detailed classification of the Lie point symmetries and of the resultingsimilarity transformations for the Generalized Boiti-Leon-Pempinelli equations. The latter equationsfor a system of two nonlinear 1+2 partial differential equations of second- and third-order. Thenonlinear equations depend of two parameters, namely n and m , from where we find that for variousvalues of these two parameters the resulting systems admit different number of Lie point symmetries.For every case, we present the complete analysis for the admitted Lie group as also we determine allthe possible similarity solutions which follow from the one-dimensional optimal system. Finally, wesummarize the results by presenting them in a tabular way. MSC Subject Classification:
Key Words and Phrases:
Symmetries; Similarity solutions; Lie invariants; BLP equations;
Nonlinear ordinary and partial differential equations are playing very prominent role in field of Math-ematics, Physics, Mathematical Physics and Mathematical modeling etc.. Through the study ofsolutions of such differential equations one can understand the behavior or consequence of a natu-ral phenomena very clearly. Though various methods are available in the literature, Lie’ symmetrymethod successfully using by researchers due to its systematic and algorithmic way to derive the so-lution of the differential equations. Actually this method was developed by Sophus Lie, during theperiod 1872 − u ty − (cid:0) u − u x (cid:1) xy − v xxx = 0 ,v t − v xx − uv x = 0 . while in this work we are interesting on the generalization u ty − ( u n − u x ) xy − αv xxx = 0 , (1.1) v t − v xx − βu m v x = 0 , where nm = 0 and αβ = 0 . The BLP equation describes interactions of two waves with different dispersion relations and arethe two-dimensional generalizations of the sine- and sinh-Gordon equations. Some exact and analyticsolutions determined in [39, 40, 41, 42, 43], while recently in [44] a complete analysis of the Lie pointsymmetries and of the B¨acklund transformations performed.In this work we classify the admitted Lie point symmetries of the generalized BLP equationsfor various values of the free parameters n, m , while the latter are constrained by the theory ofLie symmetries. Such analysis is important for the detailed study of nonlinear partial differentialequations and for the determination of new exact solutions, for other examples we refer the reader in[45, 46, 47, 48] and references therein. The plan of the paper is as follows.In Section 2, we present the basic properties and definitions for the theory of Lie point symmetriesof differential equations. We discuss the main mathematical methods which are applied in this work.Section 3 includes the main analysis of this work where we present the complete classification of theLie point symmetries for system (1.1). We found four different possible cases for the set of variables { n, m } . For every set we determine the Lie point symmetries and we present the commutators and theadjoint representation. The latter are used for the derivation of the one-dimensional optimal systemwhich are applied for the derivation of new similarity solutions for the generalized BLP equations(1.1). Finally, our results are summarized in Section 4, where we draw our conclusions.2 Lie’s Theory
Consider a system of equation as follows △ ( t, x, y, u, v, u t , v t , u x , v x , u y , v y , u tt , v tt , u tx , v tx , u ty , v ty , u xx , u xy , ... ) = 0 , (2.1) △ ( t, x, y, u, v, u t , v t , u x , v x , u y , v y , u tt , v tt , u tx , v tx , u ty , v ty , u xx , u xy , ... ) = 0 , (2.2)where t, x, y are taken to represent the independent variables and u, v are taken to represent dependentvariables, that is u = u ( t, x, y ) and v = v ( t, x, y ), while u t = ∂u∂t .Infinitesimal point transformation for each variables is defined as in the following manner,˜ t ( t, x, y, ǫ ) = t + ǫξ ( t, x, y ) + ◦ ( ǫ ) = t + ǫξ ( t, x, y ) + ◦ ( ǫ ) , ˜ x ( t, x, y, ǫ ) = x + ǫξ ( t, x, y ) + ◦ ( ǫ ) = x + ǫξ ( t, x, y ) + ◦ ( ǫ ) , ˜ y ( t, x, y, ǫ ) = y + ǫξ ( t, x, y ) + ◦ ( ǫ ) = y + ǫξ ( t, x, y ) + ◦ ( ǫ ) , ˜ u ( t, x, y, ǫ ) = u + ǫη ( t, x, y ) + ◦ ( ǫ ) = u + ǫη ( t, x, y ) + ◦ ( ǫ ) , ˜ v ( t, x, y, ǫ ) = v + ǫη ( t, x, y ) + ◦ ( ǫ ) = v + ǫη ( t, x, y ) + ◦ ( ǫ ) , where X is called infinitesimal generator defined as X = ∂ ˜ t∂ǫ ∂ t + ∂ ˜ x∂ǫ ∂ x + ∂ ˜ y∂ǫ ∂ y + ∂ ˜ u∂ǫ ∂ u + ∂ ˜ v∂ǫ ∂ v , or equivalently X = ξ ( t, x, y ) ∂ t + ξ ( t, x, y ) ∂ x + ξ ( t, x, y ) ∂ y + η ( t, x, y ) ∂ u + η ( t, x, y ) ∂ v . The invariant condition for the system (2.1) and (2.2), based on the Lie’s theory is given by △ ( t, x, y, u, v ) = △ (˜ t, ˜ x, ˜ y, ˜ u, ˜ v ) △ ( t, x, y, u, v ) = △ (˜ t, ˜ x, ˜ y, ˜ u, ˜ v )By using infinitesimal transformation and invariant condition one can find the infinitesimal generatorwhich is known as symmetries of the system (2.1) and (2.2). Therefore, if X is a Lie point symmetryfor equation △ ≡ △ ≡ X [ k ] △ = λ △ , mod △ = 0 X [ k ] △ = φ △ , mod △ = 0where λ and φ are arbitrary functions, and X [ k ] is the k-th extension of X in the jet-space. Thesesymmetries play a crucial role to perform the reduction of the order of the differential equations aswell as number of independent variables. The latter system known as determining system providesthe Lie point symmetries for a given system of differential equations.3 .1 Lie invariants By applying the Lie’s theory one can find the symmetries of a given differential equations. By using theLie symmetry with some additional regularity assumptions we obtain transformations which is calledLie’s invariant. The invariant reduces the given equation into another which involve fewer independentvariables than the original equation. The solution of the reduced equation is called invariant solutionto the given equation.
Optimal system forms by a list of n − parameter subalgebras if every n − parameter subalgebras isequivalent to a unique member of the list under some element of the adjoint representation. P.J.Olver discussed the procedure to find adjoint representation and optimal system which are givingan unique combination of symmetries to perform the reductions of a given differential equation [26].Later researcher follows slightly different method from Olver which proposed by Ibragimov [49]. Themethod is given a simple way to find the optimal system [50, 51, 52, 44]. We apply the theory of Lie point symmetries and we have obtained four sets for the set of variables { n, m } where we find the symmetries of the system (1.1). In particular we found that of { n, m } = (2 , { n, m } = { , } and for { n, m } = ( n, n − n = 1) theadmitted Lie point symmetries are three plus infinity plus infinity but with different admitted Liealgebra. On the other hand for arbitrary { n, m } the admitted Lie point symmetries are the commonelements, of the common subalgebra for the previous cases.For each case we present the commutator table and the adjoint representation. From the latterresults we determine the one-dimensional optimal system which is applied for the derivation of all thepossible similarity transformations and reduction, and when it is feasible of exact solutions. n = 2 , m = 1 If m = 1 and n = 2 then the system (1.1) yields the following equation u ty − (cid:0) u − u x (cid:1) xy − αv xxx = 0 ,v t − v xx − βuv x = 0 . (3.1)The admitted Lie symmetries are X = ∂ t , X = ∂ x , X = ψ ( y ) ∂ y − ψ ′ v∂ v , X ∞ = φ ( y ) ∂ v ,X = 2 t∂ t + x∂ x − u∂ u . X m , X n ] = X m X n − X n X m and Ad (cid:2) e ǫ X I (cid:3) X J = X J − ǫ [ X I , X J ] + ǫ [ X I , [ X I , X J ]] − ... respectively are given by the followingtables (1) and (2). Table 1: Commutator Table for case 1.1 with ψ = 1[ X I , X J ] X X X X X X X X X X ψ ′ φ + φ ′ ψ ) ∂ v X − ( ψ ′ φ + φ ′ ψ ) ∂ v X − X − X ψ = 1 (cid:2) Ad ( e ǫX i ) X j (cid:3) X X X X X X X X X − ǫX X X X X X − ǫX X X X X X X e ǫ X e ǫ X X X m = 1 , n = 2Optimal System Similarity variable Reductions X + c X X = x √ t , Y = y − c log t, ( I ) αV XXX − U XXY + 2
U U XY + 2 U X U Y + u = 1 √ t U [ X, Y ] , v = V [ X, Y ] 12 XU XY + c U Y Y + U Y = 0 V XX + βU V X + XV X + c V Y = 0 X + c X + c X r = t + c x + c y, u = U ( r ) , v = V ( r ) ( II ) c U ′′ − c c (cid:16) U ′ + 2 U U ′′ (cid:17) + c c U ′′′ + αc V ′′′ = 0(1 − c βU ) V ′ − c V ′′ = 0 X u = U [ x, y ] , v = V [ x, y ] ( III ) 2( U x U y + U U xy ) − U xxy + αV xxx = 0 βU V x + V xx = 0 X u ( t, x, y ) = U ( t, y ) , v ( t, x, y ) = V ( t, y ) ( IV ) U ty = 0 ,V t = 0 X u ( t, x, y ) = U ( t, x ) , v ( t, x, y ) = V ( t, x ) . ( V ) βU V x + V xx − V t = 0 ,V xxx = 0 X X = x √ t, Y = y, ( V I ) 2 αV XXX + (1 + 4 U X ) U Y + ( X + 4 U ) U XY − U Y XX = 0 u ( t, x, y ) = √ t U ( X, Y ) , v ( t, x, y ) = V ( X, Y ) ( X + 2 βU ) V X + 2 V XX = 0 s discussed in [51] the optimal system are X + c X , X + c X + c X , X , X , X , X . Nowwe have investigated invariant solution for (3.1) corresponding to each optimal system and tabulatedthem in table (3). For example the characteristic equation of X + c X is written as dt t = dxx = dyc = du − u = dv . (3.2)The solution of the characteristic equations yields four invariant of X + c X : X = x √ t , Y = y − c t, δ = u √ t, δ = v. (3.3)Thus, the solution of (3.1) is given by the invariant form u √ t = U (cid:20) x √ t , y − c t (cid:21) , v = V (cid:20) x √ t , y − c t (cid:21) . (3.4)Then u = 1 √ t U [ X, Y ] , v = V [ X, Y ] . (3.5)Based on this transformation (3.1) reduced to( I ) αV XXX − U XXY + 2
U U XY + 2 U X U Y +12 XU XY + c U Y Y + 12 U Y = 0 , (3.6) V XX + βU V X + 12 XV X + c V Y = 0 . (3.7)For further reductions and invariant solutions, again we find the symmetries of the system ( I ) thentabulated our results in table (4). The symmetries of ( I ) are ∂ Y and ∂ V .Table 4: Solution for equation ( I )Symmetry Similarity variable Reduction Solution ∂ Y U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = H ( X ) (cid:0) X + βG (cid:1) H ′ + H ′′ = 0 G = − I X +4 I +2 I X β ( I +2 I X ) ∂ Y + c ∂ V U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = c Y + H ( X ) c c + (cid:0) X + βG (cid:1) H ′ + H ′′ = 0 G = − I X +4 I +2 I X + c c β ( I +2 I X ) Next take X + c X + c X , then (3.1) is reduced in to the following equation( II ) c U ′′ − c c (cid:16) U ′ + 2 U U ′′ (cid:17) + c c U ′′′ + αc V ′′′ = 0 , (3.8)(1 − c βU ) V ′ − c V ′′ = 0 , (3.9)7here u = U ( r ) , v = V ( r ) and r = t + c x + c y are invariants obtained from X + c X + c X . Here ′ represents the differentiation with respect to r . Integrating (3.8) twice we obtain I + I r + c U − c c U + c c U ′ − c αV ′ = 0 . (3.10)From (3.9), we get V = I Z exp (cid:20)Z (cid:18) − c βUc (cid:19) r. (cid:21) r. . (3.11)Substituting (3.11) in (3.10) then we have U ′ + 1 c U = 1 c U − I + I rc c + c I αc exp (cid:20)Z (cid:18) − c βUc (cid:19) r. (cid:21) , (3.12)where I , I and I are constants of integration.Similarly we have performed the reduction for all other optimal system then the resultant equationsare tabulated in table (3). The follows give the further reductions and invariant solutions for theresultant equation which are tabulated in table (3). Reductions and Solutions for ( III )The equation (
III ) admits the following symmetries X = ∂ x , X = f ( y ) ∂ y − V f ′ ( y ) ∂ V , X = x∂ x − U ∂ U , X = f ( y ) ∂ V Table 5: Adjoint representation Table with f = 1 and f = 0 (cid:2) Ad ( e ǫX i ) X j (cid:3) X X X X X X X − ǫX X X X X X e ǫ X X X The table 5 shows the adjoint representation of symmetries of the resultant equation (
III ) withconditions f = 1 and f = 0. The corresponding optimal system, similarity variables with respect tothe optimal system, reductions and solutions are tabulated in table 6:8able 6: Solution for equation ( III )Optimal Similarity Reduction Solutionsystem variable X U ( x, y ) = G ( y ) , U ( x, y ) = G ( y ) V ( x, y ) = H ( y ) V ( x, y ) = H ( y ) X U ( x, y ) = G ( x ) , H ′′′ = 0 , H = I + I x + I x V ( x, y ) = H ( x ) βGH ′ + H ′′ = 0 G = − I β ( I +2 I x ) X U ( x, y ) = G ( y ) x , (1 + 2 G ) G ′ = 0 , G ( y ) = I V ( x, y ) = H ( y ) H(y)=H(y) X + c X r = y − c x , 2( G ′ + GG ′′ )+ H = I R exp h βc R Gdr i dr + I U ( x, y ) = G ( r ) , c G ′′′ + c αH ′′′ = 0 , c G ′ = I + I r − G − V ( x, y ) = H ( r ) βGH ′ − c H ′′ = 0 c αI exp h βc R Gdr i X + c X r = y − c x , c G ′′′ + 3 c G ′′ + 2 G ′ + H = I R exp h − c R (1 − βG ) dr i dr + I U ( x, y ) = G ( r ) x GG ′ + 2 c ( G ′ + GG ′′ )+ ( ∗∗ ) c G ′′ + 3 c G ′ + 2 G = V ( x, y ) = H ( r ) 2 c αH + 3 c αH ′′ + c αH ′′′ = 0 I − G − GG ′ − (1 − βG ) H ′ − c H ′′ = 0 ( G + βG + c G ′ ) I c αβ exp h − c R (1 − βG ) dr i The trivial symmetry of equation ( ∗∗ ) is ∂ r and the corresponding canonical variable are G ′ = W ( G ) . Based on these transformation theequation ( ∗∗ ) reduced in to the following c W W ′ + 3 c W + 2 G = I − G − GW − I c αβ ( G + βG + c W ) exp (cid:20) − c Z (1 − βG ) dr (cid:21) , (3.13)where ′ represents the differentiation with respect to G . eductions and Solutions for ( IV )The solution of system ( IV ) is given by U ( t, y ) = Z f ( t ) dt + g ( y ) , (3.14) V ( t, y ) = h ( y ) . (3.15) Reductions and Solutions for ( V )The solution of system ( V ) is given by U ( t, x ) = f ′ ( t ) + g ′ ( t ) x + h ′ ( t ) x − h ( t ) β ( g ( t ) + 2 xh ( t )) , (3.16) V ( t, y ) = f ( t ) + g ( t ) x + h ( t ) x . (3.17) Reductions and Solutions for ( V I )The trivial symmetry of the system (
V I ) is ∂ Y . Based on the symmetry the similarity variables are U ( X, Y ) = G ( X ) and V ( X, Y ) = H ( X ) and then the reduced systen is given by H XXX = 0 , (3.18)( X + 2 βG ) H X + 2 H XX = 0 . (3.19)The solution of above system are given by H = I + I X + I X , (3.20) G = − I + I X + 2 I X β ( I + 2 I X ) , (3.21)where I , I and I are constants of integration. m = 1 and n = 1 If m = 1 and n = 1 then the system (1.1) becomes u ty − ( u − u x ) xy − αv xxx = 0 ,v t − v xx − βuv x = 0 . (3.22)The above system admitted the following Lie symmetries Y = ∂ t , Y = ∂ x , Y = ψ ( y ) ∂ y − ψ ′ v∂ v , Y ∞ = φ ( y ) ∂ v ,Y = 2 t∂ t + ( x − t ) ∂ x + (cid:18) − βuβ (cid:19) ∂ u . Table 7 and table 8 give the commutator relation and adjoint representation, respectively, for thesymmetries of system (3.22). Indeed, the adjoint representation tabulated with condition ψ ( y ) = 1and φ ( y ) = 0 . Y I , Y J ] Y Y Y Y Y Y Y − Y ) Y Y Y ψ ′ φ + φ ′ ψ ) ∂ v Y − ( ψ ′ φ + φ ′ ψ ) ∂ v Y − Y + Y − Y (cid:2) Ad ( e ǫY i ) Y j (cid:3) Y Y Y Y Y Y Y Y Y − ǫY + ǫY Y Y Y Y Y − ǫY Y Y Y Y Y Y e ǫ Y − e ǫ ( e ǫ − Y e ǫ Y Y Y m = 1 , n = 1Optimal System Similarity variable Reductions Y + c Y X = x − c t, Y = y, ( V II ) U Y XX − (1 + c ) U Y X − αv XXX = 0 u = U [ X, Y ] , v = V [ X, Y ] V XX + ( c + βU ) V X = 0 Y + c Y X = √ t ( x + t ) , Y = y − c log t, ( V III ) U Y + c U Y Y + XU Y X − U Y XX + 2 αv XXX = 0u=U[X,Y], v=V[X,Y] c V Y + ( X + 2 βU ) V X + 2 V XX = 0 Y + c Y + c Y r = t + c x + c y ( IX ) c U ′′ − c c U ′′ + c c U ′′′ − αc V ′′′ = 0u=U(r), v=V(r) (1 − c βU ) V ′ − c V ′′ = 0 Y u ( t, x, y ) = U ( x, y ) , v ( t, x, y ) = V ( x, y ) ( X ) U xy − U xxy + αV xxx = 0 ,βU V x + V xx = 0 Y u ( t, x, y ) = U ( t, y ) , v ( t, x, y ) = V ( t, y ) ( XI ) U ty = 0 ,V t = 0 Y u ( t, x, y ) = U ( t, x ) , v ( t, x, y ) = V ( t, x ) , ( XII ) βU V x + V xx − V t = 0 V xxx = 0 Y X = t + x √ t , Y = y, ( XIII ) 2 αV XXX + U Y + XU XY − U Y XX = 0 u ( t, x, y ) = β + U ( X,Y ) √ t , v ( t, x, y ) = V ( X, Y ) ( X + 2 βU ) V X + 2 V XX = 0 ow we have examined the optimal system, similarity variables for the optimal system, the resultantequation based on the similarity variables are tabulated in table 9. The follows, giving the reductionsand solutions for the resultant equations which are tabulated in table 9. Reductions and solutions for ( V II )The resultant system (
V II ) have the symmetries ∂ X , f ( Y ) ∂ Y − V f ′ ( Y ) ∂ V , f ( Y ) ∂ V . If f ( Y ) = f ( Y ) = 1 then the optimal system, corresponding similarity variables, reductions and solutions of( V II ) tabulated in table 10. Table 10: Solution for equation (
V II )Optimal Similarity Reduction Solutionsystem variable ∂ X + c ∂ Y r = X − c Y, c (1 + c ) G ′′ − c G ′′′ − αH ′′′ = 0 H = I R exp (cid:2) − R ( c + βG ) dr (cid:3) dr + I U ( X, Y ) = G ( r ) , ( c + βG ) H ′ + H ′′ = 0 G ′ − (1 + c ) G = I r + I − V ( X, Y ) = H ( r ) αc I exp (cid:2) − R ( c + βG ) dr (cid:3) ∂ Y + c ∂ V U ( X, Y ) = G ( X ) , H ′′′ = 0 , H = I + I X + I X V ( X, Y ) = Yc + H ( X ) ( c + βG ) H ′ + H ′′ = 0 G = − c I +2 I +2 c I Xβ ( I +2 I X ) ∂ Y U ( X, Y ) = G ( X ) , H ′′′ = 0 , H = I + I X + I X V ( X, Y ) = H ( X ) ( c + βG ) H ′ + H ′′ = 0 G = − c I +2 I +2 c I Xβ ( I +2 I X ) Reductions and solutions for ( V III )The resultant system (
V III ) have the symmetries ∂ Y , ∂ V . The optimal system, corresponding simi-larity variables, reductions and solutions of ( V III ) tabulated in table 11.Table 11: Solution for equation (
V III )Symmetry Similarity variable Reduction Solution ∂ Y U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = H ( X ) (cid:0) X + βG (cid:1) H ′ + H ′′ = 0 G = − I X +4 I +2 I X β ( I +2 I X ) ∂ Y + c ∂ V U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = Yc + H ( X ) c c + (cid:0) X + βG (cid:1) H ′ + H ′′ = 0 G = − c + c ( I X +4 I +2 I X )2 c β ( I +2 I X ) Reductions and solutions for ( IX ) 13ntegrating the resultant equation ( IX ) we obtain the following reduced equation V = I Z exp (cid:20)Z (cid:18) − c βUc (cid:19) r. (cid:21) r. , (3.23) U ′ + 1 c U = 1 c U − I + I rc c + c I αc exp (cid:20)Z (cid:18) − c βUc (cid:19) r. (cid:21) , (3.24)where I , I and I are constants of integration. Reductions and solutions for ( X )The system ( X ) admits the symmetries Y = ∂ x , Y = f ( y ) ∂ y − V f ′ ( y ) ∂ V , Y = f ( y ) ∂ V . Theadjoint representation given by the table 12 and corresponding optimal system, similarity variables,reductions and solutions are given in table 13.Table 12: Adjoint representation Table with f = 1 and f = 0 (cid:2) Ad ( e ǫY i ) Y j (cid:3) Y Y Y Y Y Y Y Y Table 13: Solution for equation ( X )Optimal Similarity Reduction Solutionsystem variable Y U ( x, y ) = G ( x ) , U ( x, y ) = G ( x ) V ( x, y ) = H ( x ) V ( x, y ) = H ( x ) Y U ( x, y ) = G ( x ) , H ′′′ = 0 , H = I + I x + I x V ( x, y ) = H ( x ) βGH ′ + H ′′ = 0 G = − I β ( I +2 I x ) Y + c Y r = y − c x , c G ′′′ + G ′′ + c αH ′′′ = 0 , H = I R exp h βc R Gdr i dr + I U ( x, y ) = G ( r ) x , ( βG ) H ′ − c H ′′ = 0 c G ′ + G = I r + I − V ( x, y ) = H ( r ) c I α exp h βc R Gdr i Reductions and solutions for ( XI ) 14he solution of the system ( XI ) is given by U ( t, y ) = Z f ( t ) dt + g ( y ) , (3.25) V ( t, y ) = h ( y ) . (3.26) Reductions and solutions for ( XII )The solution of the system (
XII ) is given by U ( t, x ) = f ′ ( t ) + g ′ ( t ) x + h ′ ( t ) x − h ( t ) β ( g ′ ( t ) + 2 xh ( t )) , (3.27) V ( t, y ) = f ( t ) + g ( t ) x + h ( t ) x . (3.28) Reductions and solutions for ( XIII )The trivial symmetry of the system (
XIII ) is ∂ Y . Based on the symmetry the similarity variable is U ( X, Y ) = G ( X ) and V ( X, Y ) = H ( X ) and the reduced equation is given by H XXX = 0 , (3.29)( X + 2 βG ) H X + 2 H XX = 0 . (3.30)The solution of above system are given by H = I + I X + I X , (3.31) G = − I + I X + 2 I X β ( I + 2 I X ) , (3.32)where I , I and I are constants of integration. n = 1 , m = n − If m = n − n = 1 then the system (1.1) yields the following equation u ty − ( u n − u x ) xy − αv xxx = 0 ,v t − v xx − βu n − v x = 0 . (3.33)The above system admitted Lie symmetries as we are given below: Z = ∂ t , Z = ∂ x , Z = ψ ( y ) ∂ y − ψ ′ v∂ v , Z ∞ = φ ( y ) ∂ v ,Z = 2 t∂ t + x∂ x + (cid:18) − n (cid:19) u∂ u + (cid:18) − n − n (cid:19) v∂ v . The commutator table for the symmetries are displayed in table 14. Table 15 gives adjoint represen-tation of the symmetries with condition ψ ( y ) = 1 and φ ( y ) = 0 . Optimal system, similarity variablesand corresponding resultant equations are given in table 16.15able 14: Commutator Table for case 1.3[ Z I , Z J ] Z Z Z Z Z Z Z Z Z Z ψ ′ φ + φ ′ ψ ) ∂ v Z − ( ψ ′ φ + φ ′ ψ ) ∂ v (cid:16) n − n − (cid:17) Z Z − Z − Z − (cid:16) n − n − (cid:17) Z (cid:2) Ad ( e ǫZ i ) Z j (cid:3) Z Z Z Z Z Z Z Z Z − ǫZ Z Z Z Z Z − ǫZ Z Z Z Z Z Z e ǫ Z e ǫ Z Z Z m = n − , n = 1Optimal System Similarity variable Reductions Z + c Z X = x √ t , Y = y − c log t, ( XIV ) αV XXX − U XXY + nU n − U XY + 12 XU XY + u = t − n ) U [ X, Y ] , v = t − n − n ) V [ X, Y ] n ( n − U n − U X U Y + c U Y Y + n − U Y = 0 V XX + βU n − V X + 12 XV X + c V Y − (cid:16) n − n − (cid:17) V = 0 Z + c Z + c Z r = t + c x + c y, ( XV ) c U ′′ − c c (cid:16) n ( n − U n − U ′ + nU n − U ′′ (cid:17) + c c U ′′′ + αc V ′′′ = 0 u = U ( r ) , v = V ( r ) (1 − c βU n − ) V ′ − c V ′′ = 0 Z u = U [ x, y ] , v = V [ x, y ] ( XV I ) n ( n − U n − U x U y + nU n − U xy − U xxy + αV xxx = 0 βU n − V x + V xx = 0 Z u ( t, x, y ) = U ( t, y ) , v ( t, x, y ) = V ( t, y ) ( XV II ) U ty = 0 ,V t = 0 Z u ( t, x, y ) = U ( t, x ) , v ( t, x, y ) = V ( t, x ) ( XV III ) βU n − V x + V xx − V t = 0 V xxx = 0 Z x = X √ t, y = Y, ( XIX ) A B U y + nA n B n (( n + 1) A − nA n ) U n − U X U Y + u ( t, x, y ) = (2(1 − n ) t ) − n ) U ( X, Y ) , ( n − A B XU XY + n ( n − A n +22 B n U n − U XY + v ( t, x, y ) = (2( n − t ) n − n − V ( X, Y ) A n B − n αV XXX + A n B − n U XXY = 0( n − V − ( n − XV X + (2(1 − n )) βU n − V X − n − V XX = 0where A = 2 n − and B = (1 − n ) − n ) . able 17: Solution for equation ( XIV )Symmetry Similarity variable Reduction Solution ∂ Y U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = H ( X ) ( n − n − H − ( X + 2 βG n − ) H ′ − H ′′ = 0 G = h − n − β ( I +2 I X )(2 − n ) I + I X +4( n − I + nI X i n − ∂ Y + c e ( n − Y ( n − c ∂ V U = G ( X ) , H ′′′ = 0 , H = I + I X + I X V = ( n − c c ( n − e ( n − Y ( n − c + H ( X ) ( n − n − H − ( X + 2 βG n − ) H ′ − H ′′ = 0 G = h − n − β ( I +2 I X )(2 − n ) I + I X +4( n − I + nI X i n − eductions and solutions for ( XIV )The system (
XIV ) has two symmetries which are given as ∂ Y and e ( n − Y ( n − c ∂ V . Correspondingreductions and solutions are tabulated in table 17.
Reductions and solutions for ( XV )The resultant equation further reduced to the following equations V = I Z exp (cid:20)Z (cid:18) − c βU n − c (cid:19) r. (cid:21) r. , (3.34) U ′ + 1 c U = 1 c U n − I + I rc c + c I αc exp (cid:20)Z (cid:18) − c βU n − c (cid:19) r. (cid:21) , (3.35)where I , I and I are constants of integration. Reductions and solutions for ( XV I )The equation (
XV I ) admits the following symmetries Z = ∂ x , Z = f ( y ) ∂ y − V f ′ ( y ) ∂ V , Z = x∂ x − (cid:18) n − (cid:19) U ∂ U + (cid:18) n − n − (cid:19) V ∂ V , Z = f ( y ) ∂ V . The adjoint representation with conditions f ( y ) = 1 and f ( y ) = 0 is tabulated in table 18 andcorresponding optimal system, similarity variables, reduced system and solutions are given by table19. Table 18: Adjoint representation Table with f = 1 and f = 0 (cid:2) Ad ( e ǫZ i ) Z j (cid:3) Z Z Z Z Z Z Z − ǫZ Z Z Z Z Z e ǫ Z Z Z XV I )Optimal Similarity Reduction Solutionsystem variable Z U ( x, y ) = G ( y ) , U ( x, y ) = G ( y ) V ( x, y ) = H ( y ) V ( x, y ) = H ( y ) Z U ( x, y ) = G ( x ) , H ′′′ = 0 , H = I + I x + I x V ( x, y ) = x n − n − H ( y ) βG n − H ′ + H ′′ = 0 G = h − I β ( I +2 I x ) i n − Z U ( x, y ) = ( x − nx ) − n G ( y ) , (2 − n ) αGH − (1 − n ) − n − n GG ′ + G ( y ) = ( − β ) − n n (1 − n ) n − n − G n G ′ = 0 ,V ( x, y ) = H ( y ) G + βG n = 0 H ( y ) = 0 Z + c Z r = y − c x , n ( n − G n − G ′ ) + nG n − G ′′ + H = I R exp h βc R G n − dr i dr + I U ( x, y ) = G ( r ) , c G ′′′ + c αH ′′′ = 0 , c G ′ = I + I r − G n − V ( x, y ) = H ( r ) βG n − H ′ − c H ′′ = 0 c αI exp h βc R G n − dr i Z + c Z r = y − c x , n (cid:16) n − n − (cid:17) αH + n (1 − n ) − n G n − G ′ + U ( x, y ) = x − n (1 − n ) − n G ( r ) c n (1 − n ) − n − n G n − G ′ + c ( n − n − αH ′ + V ( x, y ) = x n − n − H ( r ) c (1 + n )(1 − n ) − n − n G ′′ − c n (1 − n ) − n − n G n − G ′′ − c ( n − αH ′′ − c (1 − n ) − n − n G ′′′ − n (1 − n ) − n G ′ − c ( n − αH ′′′ = 0 (cid:16) n − n − (cid:17) (1 + βG n − ) H + c ( n − H ′ − c βG n − H ′ − c ( n − H ′′ = 0 eductions and solutions for ( XV II )The solution of the system (
XV II ) is given by U ( t, y ) = Z f ( t ) dt + g ( y ) , (3.36) V ( t, y ) = h ( y ) . (3.37) Reductions and solutions for ( XV III )The solution of the system (
XV III ) is given by U ( t, x ) = (cid:20) f ′ ( t ) + g ′ ( t ) x + h ′ ( t ) x − h ( t ) β ( g ( t ) + 2 xh ( t )) (cid:21) n − , (3.38) V ( t, y ) = f ( t ) + g ( t ) x + h ( t ) x . (3.39) Reductions and solutions for ( XIX )The trivial symmetry of the system (
XIX ) is ∂ Y . Based on this symmetry the system is reducedin to H ′′′ = 0 , (3.40)( n − H − ( n − XH ′ +(2(1 − n )) βG n − H ′ − n − H ′′ = 0 . (3.41)The solution of above system are given by H = I + I X + I X , (3.42) G = " (2(1 − n )) β ( I + 2 I X )(2 − n ) I + I X + 4( n − I + nI X − n . (3.43) n, m For arbitrary m and n :The system (1.1) admitted Lie symmetries as followsΓ = ∂ t , Γ = ∂ x , Γ = ∂ y , Γ ∞ = ψ ( y ) ∂ v The commutator table for the symmetries are displayed in table 20. Table 21 gives adjoint rep-resentation of the symmetries with condition ψ ( y ) = 0. Optimal system, similarity variables andcorresponding resultant equations are given in table 22.21able 20: Commutator Table for case 1.4[Γ I , Γ J ] Γ Γ Γ Γ Γ Γ − Γ (cid:2) Ad ( e ǫ Γ I )Γ J (cid:3) Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ m and n Optimal System Similarity variable ReductionsΓ + c Γ + c Γ r = t + c x + c y, ( XX ) c U ′′ − c c (cid:16) n ( n − U n − U ′ + nU n − U ′′ (cid:17) + c c U ′′′ + αc V ′′′ = 0 u = U ( r ) , v = V ( r ) (1 − c βU m ) V ′ − c V ′′ = 0Γ u ( t, x, y ) = U ( x, y ) , v ( t, x, y ) = V ( x, y ) ( XXI ) n ( n − U n − U x U y + nU n − U xy − U xxy + αV xxx = 0 βU m V x + V xx = 0Γ u ( t, x, y ) = U ( t, y ) , v ( t, x, y ) = V ( t, y ) ( XXII ) U ty = 0 ,V t = 0Γ u ( t, x, y ) = U ( t, x ) , v ( t, x, y ) = V ( t, x ) ( XXIII ) βU m V x + V xx − V t = 0 V xxx = 0 eductions and solutions for ( XX ) V = I Z exp (cid:20)Z (cid:18) − c βU m c (cid:19) r. (cid:21) r. , (3.44) U ′ + 1 c U = 1 c U n − I + I rc c + c I αc exp (cid:20)Z (cid:18) − c βU m c (cid:19) r. (cid:21) , (3.45)where I , I and I are constants of integration. Reductions and solutions for ( XXI )The system (
XXI ) admits the following symmetries Γ = ∂ x , Γ = ∂ y . The correspondingoptimal system, similarity variables, reductions and solutionsnare given in table 23.Table 23: Solution for equation ( XXI )Optimal Similarity Reduction Solutionsystem variableΓ U ( x, y ) = G ( y ) , U ( x, y ) = G ( y ) V ( x, y ) = H ( y ) V ( x, y ) = H ( y )Γ U ( x, y ) = G ( x ) , H ′′′ = 0 , H = I + I x + I x V ( x, y ) = H ( x ) βG m H ′ + H ′′ = 0 G = h − I β ( I +2 I x ) i m Γ + c Γ r = y − c x , n ( n − G n − G ′ + nG n − G ′′ + H = I R exp h βc R ( G m ) dr i dr + I + c G ′′′ + c αH ′′′ = 0 c G ′ + G n = I r + I − U ( x, y ) = G ( r ) x , ( βG ) H ′ − c H ′′ = 0 c I α exp h βc R ( G m ) dr i V ( x, y ) = H ( r ) Reductions and solutions for ( XXII )The solution of the system (
XXII ) is given by U ( t, y ) = Z f ( t ) dt + g ( y ) , (3.46) V ( t, y ) = h ( y ) . (3.47) Reductions and solutions for ( XXIII ) 24he solution of the system (
XXIII ) is given by U ( t, x ) = (cid:20) f ′ ( t ) + g ′ ( t ) x + h ′ ( t ) x − h ( t ) β ( g ( t ) + 2 xh ( t )) (cid:21) m , (3.48) V ( t, y ) = f ( t ) + g ( t ) x + h ( t ) x . (3.49) In this work, we studied the generalized BLP system by using the theory of Lie symmetries. Specifi-cally, we performed a detailed classification of the admitted Lie point symmetries for the generalizedBLP system by constraint the free parameters of the system with the Lie conditions.We found four different sets for the unknown parameters in which the resulting systems admitsdifferent Lie point symmetries. For each system we determined the commutators and the Adjointrepresentation for the admitted Lie point symmetries. The latter applied to determine the one-dimensional optimal system, an important information in order to determine all the possible andindependent similarity transformations which lead to invariant solutions. Finally invariant solutionshave been presented for all the cases of study.This work contributes on the application of Lie’s theory on nonlinear differential equations. In afuture work we want to investigate the existence of conservation laws which follow from the admittedgroup properties for the generalized BLP system.
Acknowledgements
KK thank Prof. Dr. Stylianos Dimas, S´ao Jos´e dos Campos/SP, Brasil for providing new version ofSYM-Package.
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