Algebraic Properties for Certain Form of the Members of Sequence on Generalized Modified Camassa-Holm Equation
aa r X i v : . [ n li n . S I] D ec Algebraic Properties for Certain Form of the Membersof Sequence on Generalized Modified Camassa-HolmEquation
K Krishnakumar , A Durga Devi ∗ , V Srinivasan and PGL Leach Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemedto be University, Kumbakonam 612 001, India. Department of Physics, Srinivasa Ramanujan Centre, SASTRA Deemed to beUniversity, Kumbakonam 612 001, India. Department of Mathematics, Durban University of Technology, PO Box 1334,Durban 4000, Republic of South Africa.
Abstract
We study the symmetry and integrability of a Generalized ModifiedCamassa-Holm Equation (GMCH) of the form u t − u xxt + 2 nu x ( u − u x ) n − ( u − u xx ) + ( u − u x ) n ( u x − u xxx ) = 0 . We observe that for increasing values of n ∈ N , N denotes naturalnumber, the above equation gives a family of equations in which non-linearity is rapidly increasing as n increases. However, this family hassimilar form of symmetries except the values of n . Interestingly theresultant second-order nonlinear ODE which is to be obtained fromGMCH equation has eight dimensional symmetries. Hence the second-order nonlinear ODE is linearizable. Finally we conclude that theresultant second-order nonlinear ordinary differential equation whichis obtained from the family of GMCH passes the Painlev´e Test alsoit posses the similar form of leading order, resonances and truncatedseries solution too. MSC Numbers:
PACS Numbers:
Keywords:
Generalized Modified Camassa-Holm Equation (GMCH), Sym-metry, Painlev´e Test, Integrability.
Researchers who are working in such a field of Mathematical Physics, Engi-neering, Mathematical Biology etc. are developing mathematical models tounderstand the behavior of the physical phenomena which are arising in theirrespective fields. Especially they are converting their physical phenomena1nto system of differential equations. The deterministic mathematical de-scriptions of such a differential equations can be cast into appropriate math-ematical models through nonlinear ordinary differential equations, partialdifferential equations and so on. Among them partial differential equationshave been mostly used to study the nature of the systems. For example in1834 Kortewegde Vries (KdV) equation which is nonlinear partial differentialequation studied experimentally to analyze waves on shallow water surfacesby John Scott Russell then Lord Rayleigh and Joseph Boussinesq studiedtheoretical investigation about this equation by around 1870 and followed byKorteweg and De Vries in 1895. Similarly, there are many wave equationshave been investigated in various fields which is quietly related to mathemat-ics. Few of them are Kortewegde Vries (KdV) Equation, nonlinear coupledKdV Equation [1], Camassa-Holm (CH) Equation [2], Modified Camassa-Holm (MCH) Equation [3, 4]. Numerous methods have been discussed tosolve above mentioned equations [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].In general, finding a general non-trivial solutions of a nonlinear differentialequation is very hard to achieve in most instances. However, Lie symmetryhas been used as an important tool that provide a systematic method to findthe solution of the given differential equations successfully [15, 16, 17, 18,19, 20, 21, 22]. By using the symmetries, one able to get algebraic equationsfrom the given ordinary or partial differential equations through performingthe reduction of order the differential equations [23]. Hence, Lie symmetriesplays a major role in it.The purpose of this work is to study the symmetry and integrability of allthe members of sequence of GMCH equation [24] is given by u t − u xxt + 2 nu x ( u − u x ) n − ( u − u xx ) + ( u − u x ) n ( u x − u xxx ) = 0 , (1.1)When n = 1 the above equation gives the first member of GMCH and it isalso exactly the modified Camassa-Holm Equation (MCH). This MCH equa-tion has been given much attention and it has been analyzed by researchersthoroughly [11, 25, 26, 27, 28]. Also the MCH equation has been discussedin the point of view of symmetries and integrability [29]. Interestingly, allthe other higher members of the sequence are having dynamically increasingnonlinear terms based on n values ( ie ) the nonlinear terms are increasingproportionally to each value of n . Therefore, these higher members of the se-quence can also be given considerable attention due to the common beautifulproperties on symmetries and Painlev´e Test.2n this paper, first we examine the Lie point symmetries and reductionsof order for (1.1) then we prove that the resultant equation is linearizable. Next, we apply the Painlev´e Test for equation (1.1) and we able to identifythat the equation (1.1) is integrable.
The generalized modified Camassa-Holm equation is u t − u xxt + 2 nu x ( u − u x ) n − ( u − u xx ) + ( u − u x ) n ( u x − u xxx ) = 0 (2.1)The first few members of the family are given by u t − u xxt + 2 u x ( u − u xx ) + ( u − u x )( u x − u xxx ) = 0 , (2.2) u t − u xxt + 4 u x ( u − u x )( u − u xx ) + ( u − u x ) ( u x − u xxx ) = 0 , (2.3) u t − u xxt + 6 u x ( u − u x ) ( u − u xx ) + ( u − u x ) ( u x − u xxx ) = 0 . (2.4)here we can observe that the nonlinearity is increasing as n increases. Thesymmetries of the general equation (2.1) are X = ∂ t , (2.5) X = ∂ x , (2.6) X = − nt∂t + u∂u. (2.7)Therefore for n = 1 , , , ... in (2.7) we have the symmetries for equations(2 . , (2 . , (2 . ... respectively. Though the family contains many nonlinearequations, they have similar form of symmetries. The Lie Brackets of the Liesymmetries of general equation (2.1) is[ X I , X J ] X X X X X X X − X A ⊕ A . Throughout this literature the Mathematica add-on Sym [30, 31, 32] is used to computethe symmetries.
3y using the linear combination of X and X , the traveling wave solutionof equation (2.1) can be derived as follows. The linear combination of thetwo symmetries X and X can be represented as X = ∂ t + c∂ x . Thecorresponding canonical variables are r = x − ct and u ( x, t ) = P ( r ). Throughthese canonical variables one can reduce the equation (2 .
1) into the followingform c ( P ′′′ − P ′ ) + ( P − P ′ ) n ( P ′ − P ′′′ ) + 2 nP ′ ( P − P ′ ) n − ( P − P ′′ ) = 0 , (2.8)where P is the function of the new independent variable r . The trivialsymmetry of autonomous differential equation (2.8) is ∂ r . This symmetryenable us to take the canonical variables P ( r ) = p and P ′ = W ( p ). By usingthese canonical variables the equation (2.8), can be reduced to(( p − W ) n − c )(1 − W ′ − W W ′′ ) +( p − W ) n − (2 np − npW W ′ + 2 nW W ′ ) = 0 , (2.9)where W is a function of p . When we find the symmetries of the equation,(2.9), we arrive the following two possible cases. Case:1 n = −
1, Then the symmetries of the equation (2.9) , areΓ = 1 W (( p − W ) n − c ) ∂ W (2.10)Γ = pW (( p − W ) n − c ) ∂ W (2.11)Γ = ∂ p + p ( p − W ) n W (( p − W ) n − c ) ∂ W (2.12)Γ = ( p − W )( c ( n + 1) − ( p − W ) n )( n + 1) W (( p − W ) n − c ) ∂ W (2.13)Γ = 4 p∂ p − c ( n + 1)(3 p + W ) − ( p − W ) n ((4 n + 3) p + W )( n + 1) W (( p − W ) n − c ) ∂ W (2.14)Γ = 2 p ∂ p (2.15) − p ( c ( n + 1)(3 p + W ) − ( p − W ) n ((4 n + 3) p + W ))( n + 1) W (( p − W ) n − c ) ∂ W Γ = − p − W ) n +1 + ( n + 1) cW n + 1 ∂ p − (cid:18) c ( n + 1)( p − W ) + 2( p − W ) n +1 ( n + 1) W (( p − W ) n − c ) (2.16) − c ( p − W ) n (3 p − (2 n + 5) W )( n + 1) W (( p − W ) n − c ) (cid:19) p∂ W Γ = 2( p − W )( c ( n + 1) − ( p − W ) n ) n + 1 p∂ p − (cid:18) c ( n + 1)( p − W )( c ( n + 1) − ( p − W ) n )( n + 1) W (( p − W ) n − c ) (2.17) − ( c ( n + 1) − ( p − W ) n )( p − W ) n +1 ((2 n + 1) p + W )( n + 1) W (( p − W ) n − c ) (cid:19) ∂ W When we use Γ , the solution of equation (2.9), is given by12 cW + ( p − W ) n +1 n + 1) + I = 0 , (2.18)where I is the constant of integration. From Γ , the solutions are W = ± pW = ± q p − ( c ( n + 1)) n (2.19) These symmetries can also be obtained for all n ∈ R − {− } , R − denotes the realnumber
5y using Γ , the solution is( p − W )(( p − W ) n − c ( n + 1)) p + I = 0 , (2.20)where I is the constant of integration. By using Γ , the solution is( n + 1) c ( p − W ) − p − W ) n +1 n + 1)( p − W ) ( c ( n + 1) − ( p − W ) n ) − I = 0 , (2.21)where I is the constant of integration. Case:2 If n = −
1, then the symmetries of the equation (2.9) areΓ = p − W W ( c ( p − W ) − ∂ W (2.22)Γ = p ( p − W ) W ( c ( p − W ) − ∂ W (2.23)Γ = p − W W ( c ( p − W ) − ∂ p + p ( p − W ) W ( c ( p − W ) − ∂ W (2.24)Γ = ( p − W )(log [ p − W ] − c ( p − W ) W ( c ( p − W ) − ∂ W (2.25)Γ = 2 p ∂ p (2.26)+ p ( p − W ) log [ p − W ] − p (2 p − c ( p − W )) W ( c ( p − W ) − ∂ W Γ = 4 p∂ p (2.27)+ ( p − W ) log [ p − W ] + c (3 p − W ) − p (2 + cW ) W ( c ( p − W ) − ∂ W Γ = (log [ p − W ] − c ( p − W )) ∂ p (2.28)+ p ( c ( p − W ) −
1) log [ p − W ] − p ( c ( p − W ) ) W ( c ( p − W ) − ∂ W Γ = 2 p (log [ p − W ] − c ( p − W )) ∂ p + (cid:18) W ( c ( p − W ) −
1) log [ p − W ] W ( c ( p − W ) − c ( p − W )(2 p − c ( p − W )) W ( c ( p − W ) − (cid:19) ∂ W (2.29) . or Γ , the solution of equation (2.9) is W = ± p. (2.30)From Γ , the solutions are W = ± pW = p I (2.31)By using Γ pc ( p − W ) − log [ p − W ] + I = 0 (2.32)By using Γ , the solution is14 (2 log [ p − W ]( c ( p − W ) − − c ( p + W ) − cp (1 − cW )) + I = 0 . (2.33)In the above I , I , I , I , I and I are the constants of integration. We are analyzing the behavior of the singularities of the resultant differentialequation by using singularity analysis. The first Sofia Kovalevskaya (1889)[33, 34, 35] analyzed the singularity structures for a set of six first-ordercoupled nonlinear integrable ODEs which is dynamically described the in-fluence of gravity on a heavy rigid body. The singularities are exhibited bythe general solution in terms of meromorphic (Jacobian elliptic) functions.Based on Kovalevskaya work, the Painlev´e Property was proposed for ODE’s[36, 37, 38, 39, 40]. If all the movable singularities are single valued then thesystem of ordinary differential equations said to possess the Painlev´e Prop-erty. The systems are expected to be integrable when they possess Painlev´eProperty. Many researchers are still using Painlev´e Property to identify newhigher-order integrable systems.Almost all the new integrable equations has been already discussed onsecond-order by considering the general second-order ODE of the form d vdr = F (cid:18) r, v, dvdr (cid:19) , (3.1)where F is rational in v , algebraic in dvdr , locally analytic in r . Especially,there were only fifty equations possessing the Painlev´e Property. The solu-tions which is in terms of elementary functions including the elliptic functions7an be obtained only for forty four differential equations out of the fifty inte-grable equations. The remaining six are referred as Painlev´e transcendentalequations [40]. Recently both Right and Left Painlev´e Series constitutes thesolution of the given differential equation within a punctured disc centred onthe singularity has studied in [41, 42, 43, 44].Ablowitz et al. have given an algorithmic procedure which is calledAblowitz-Ramani-Segur (ARS) algorithm [45, 46, 47] to perform the Painlev´eTest. This algorithm has the following three steps1. Leading-order behavior.2. Determination of resonances.3. Arbitrariness of sufficient numbers of constants. We are very much interested to apply Painlev´e Test for a third-order resultantordinary differential equation which is obtained from the following partialdifferential equation (2.1) u t − u xxt + 2 nu x ( u − u x ) n − ( u − u xx ) + ( u − u x ) n ( u x − u xxx ) = 0 . (3.2)The following resultant third-order nonlinear autonomous ordinary differ-ential equation which obtain from to equation (3.2) by substituting r = x − ct is given by c ( P ′′′ − P ′ ) + ( P − P ′ ) n ( P ′ − P ′′′ ) + 2 nP ′ ( P − P ′ ) n − ( P − P ′′ ) = 0 , (3.3)where P is a function of r . This equation (3.3) does not pass the Painlev´eTest. But, a substitution, P ( r ) = 1 v [ r ] , for P ( r ) shall make the above resul-8ant equation as integrable. The resultant equation is given by c n v ′ − n + 3) v ′ n X k =0 ( − k (cid:18) nk (cid:19) v n − k v ′ k + 6 cv n +3 v ′ v ′′ − cv n +1 v ′ v ′′ − n + 3) v v ′ v ′′ n X k =0 ( − k (cid:18) nk (cid:19) v n − k v ′ k +2(4 n + 3) vv ′ v ′′ n X k =0 ( − k (cid:18) nk (cid:19) v n − k v ′ k + cv n +4 ( v ′ − v ′′′ ) − cv n +2 v ′ (7 v ′ − v ′′′ ) − v ((2 n + 1) v ′ − v ′′′ ) n X k =0 ( − k (cid:18) nk (cid:19) v n − k v ′ k + v v ′ ((8 n + 7) v ′ − nv ′′ − v ′ v ′′′ ) n X k =0 ( − k (cid:18) nk (cid:19) v n − k v ′ k = 0 . (3.4)According to the Painlev´e Test the dominant terms of (3.4) are given by6 cv n v ′ − ( − n n + 3) v ′ v ′ n − cv n +1 v ′ v ′′ +( − n n + 3) vv ′ n +3 v ′′ + cv n +2 v ′ v ′′′ − ( − n v v ′ n +1 (2 nv ′′ − v ′ v ′′′ ) (3.5)Here, v = a − w − , where w = ( r − r ) , is a leading-order behavior of theLaurent series in the neighborhood of the movable singular point r . Todetermine the resonances we substitute v ( w ) = a − w − + mw − s into (3.5)and by equating the coefficients of m to zero then we have s = − , w at whicharbitrary constants of the solution for (3.4) can enter into the Laurent seriesexpansion. In this series we have sufficient number of arbitrary constants.Hence, the equation (3.2) is integrable as consequence of Painlev´e propertyholds for equation (3.3). Even though the nonlinearity increasing as n increases, all members of thefamily which is obtained from the generalized modified Camassa-Holm (GMCH)Equation have similar form of symmetries, reductions and solutions. We haveobtained two sets of eight dimensional symmetries ( ie maximum number of The resultant equations of all members of the sequence can be obtained from (3.4) bytaking the values of n = 1 , , , ... n = − n ∈ R − {− } respectively. Therefore all membersof the family of second-order differential equation are linearizable.By the performance of Painlev´e Test for the resultant generalized third-order differential equation (3.3) we have observed that all members of thesequence of generalized modified Camassa-Holm (GMCH) Equation possesPainlev´e Property. In particular, leading order, resonances and arbitrarinessof constant of solutions are also having similar form for the entire membersof the sequence. Hence all members of the sequence of generalized modifiedCamassa-Holm (GMCH) Equation are integrable. Acknowledgements
KK and ADD thank Prof. Stylianos Dimas, S´ao Jos´e dos Campos/SP, Brasil,for providing a new version of the SYM-Package. KK thank Late Prof.K.M.Tamizhmani for his support and academic guidance during his Doctoralprogramme.
References [1] Korteweg DJ & de Vries G (1895) On the change of form of long wavesadvancing in a rectangular canal and on a new type of long stationarywaves
Philosophical Magazine, 5th series Physics Review Letter American Journal of Computational Mathematics Archive for RationalMechanics and Analysis
Physical Review Letters Journal of Mathematical Physics Journal of MathematicalPhysics Journal of the Physical Society ofJapan Progress in Theoretical Physics Communications in MathematicalPhysics
PhysicaD Acta Ap-plicandae Mathematicae Com-munications in Nonlinear Science and Numerical Simulation Journalof Differential Equations
Mathemathischen An-nallen Applications of Lie Groups to Differential Equations (Springer, New York)[17] Leach PGL, Feix MR & Bouquet S (1988) Analysis and solution of a non-linear second-order differential equation through rescaling and through adynamical point of view
Journal of Mathematical Physics Journal ofMathematical Analysis and Application
Journal of Applied Mathematics
Article ID890171, (doi:10.1155/2011/890171)[20] Andriopoulos K, Dimas S, Leach PGL & Tsoubelis D (2009) On the sys-tematic approach to the classification of differential equations by grouptheoretical methods
Journal of Computational and Applied Mathematics
Applied Mathematics and Computation
Afrika Matematika
Indian Journal of Pure and Applied Mathematics Journal of Differential equations
Physics Review E Journal of Mathematical Physics Nonlinear Analysis Applicable Analysis http://dx.doi.org/10.1080/00036811.2017.1359565.[29] Durga Devi A, Krishnakumar K, Sinuvasan R & PGL Leach
Symmetriesand integrability of modified Camassa-Holm Equation with an arbitraryparameter (Submitted) 1230] Dimas S & Tsoubelis D (2005) SYM: A new symmetry-finding packagefor Mathematica
Group Analysis of Differential Equations (Avignon, France)[32] Dimas S (2008)
Partial Differential Equations, Algebraic Computing andNonlinear Systems (Thesis: University of Patras, Patras, Greece)[33] Kovalevskaya S (1889) Sur le probleme de la rotation d´un corps solideautour d´un point fixe
Acta Mathematica Acta Mathematica The Mathematics of Sonya Kovalevskaya (Springer, NewYork).[36] Painlev´e P (1973) Le¸cons sur la th´eorie analytique des ´equationsdiff´erentielles. (Le¸cons de Stockholm, 1895) (Hermann, Paris, 1897).Reprinted, Oeuvres de Paul Painlev´e, vol. I, ´Editions du CNRS, Paris.[37] Painlev´e P (1900) M´emoire sur les ´equations diff´erentielles du secondordre dont l’int´egrale g´en´erale est uniforme.
Bulletin of the MathematicalSociety of France Acta Mathemat-ica Ordinary Differential Equations in the Complex Domain ( Wiley Interscience, New York).[40] Ince EL (1956)
Ordinary Differential Equation (Dover, New York).[41] Feix MR, Geronimi C, Cair´o L, Leach PGL, Lemmer RL & Bouquet S ´E(1997) Right and left Painlev´e series for ordinary differential equationsinvariant under time translation and rescaling
Journal of Physics A:Mathematical and General Physics Letters A y ′′ + yy ′ + ky = 0 Journal of Physics A: Mathematicaland General A study of symmetries, reductions and solutionsof certain classes of differential equations (Thesis: Pondicherry CentralUniversity, Puducherry, India)[45] Ablowitz MJ, Ramani A & Segur H (1980) A connection between non-linear evolution equations and ordinary differential equations of P-typeI
Journal of Mathematical Physics Journal of Mathematical Physics PhysicsReports108