Lie symmetries and similarity solutions for the generalized Zakharov equations
aa r X i v : . [ n li n . S I] J un Lie symmetries and similarity solutions for thegeneralized Zakharov equations
K. Krishnakumar , A. Durga Devi and A. Paliathanasis , 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. 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
The theory of Lie point symmetries is applied to study the generalizedZakharov system with two unknown parameters. The system reduces into athree-dimensional real value functions system, where we find that admitsfive Lie point symmetries. From the resulting point, we focus on thesewhich provide travel-wave similarity transformation. The reduced systemcan be integrated while we remain with a system of two second-ordernonlinear ordinary differential equations. The parameters of the lattersystem are classified in order the equations to admit Lie point symmetries.Exact travel-wave solutions are found, while the generalized Zakharovsystem can be described by the one-dimensional Ermakov-Pinney equation.
MSC Subject Classification:
Key Words and Phrases:
Symmetries; Travel-wave; Plasma; Zakharovsystem.
One side, it is usual practice for many years to develop a mathematicalmodel to study the nature of the real life phenomena. In which differen-tial equations are playing a crucial role to describe the phenomena veryclearly. Another side, many researchers are trying to solve the differentialequations by using various methods. Especially, Sophus Lie, during the pe-riod 1872 − ′ s theoryhas been given much attention on both theoretical and applied point of view[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].Algorithm of Lie’s theory to differential equations is completely algorith-mic involves tedious calculations even for linear differential equations withconstant coefficients. To overcome the difficulties we have access to powerfulComputer Algebra Systems (CAS) like Maple and Mathematica (commer-cial), etc. which are enabling us to do the calculations rapidly . In thepast decades, many types of symmetries have been proposed in the literaturesuch as approximate symmetries [31, 32], generalized symmetries [22, 10],and nonlocal symmetries [22, 15, 24, 33] to quote a few.The novelty of the Lie’s theory is that provides a systematic way to treatnonlinear differential equations. The main application of Lie’s theory aresummarized to: the determination of similarity transformations which areused to reduce the differential equation; to determine invariant solutions,also known as similarity solutions, construct conservation laws and writelinearize differential equations [22, 15, 24, 33].Applications of Lie symmetries cover a wide range of physical and naturalsystems, from classical mechanics [38, 39], fluid dynamics [40, 41], plasmaphysics [42], optics [43], gravitational physics [44, 45], financial mathematics[46, 47] and other areas of applied mathematics.In this work are interesting on the symmetry analysis for the generalizedZakharov equations [48] of plasma physics [49] iU t + U xx − U V − ( | U | n ) U = 0 (1.1) V tt − V xx − ( | U | m ) xx = 0 (1.2)where the complex function U ( t, x ) is the envelope of the high-frequencyelectric field and the real function V ( t, x ) plasma density measured from itsequilibrium value. Parameters n, m are arbitrary in our consideration, whilethey describe the nonlinear self-interaction in the high-frequency subsystemwhich corresponds to a self-focusing effect in plasma physics [50]. For thesystem (1.1), (1.2) travel-wave solutions were studied in [50].In the following we focus on the Lie point symmetries for the system (1.1),(1.2) while the free parameters n, m will be determined by Lie’s theory, suchthat the resulting system to admit additional symmetries, that approach is In this work, for the calculation of the symmetries we use the Mathematica add-onSym [35, 36, 37, 34].
Suppose consider an equation φ ( 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)where t, x, y are the set of independent variables and u, v are dependentvariables. Infinitesimal point transformation for each variables is defined asin the following manner,˜ t ( t, x, y, ǫ ) = t + ǫξ ( t, x, y ) + ◦ ( ǫ ) = t + ǫXt + ◦ ( ǫ ) , ˜ x ( t, x, y, ǫ ) = x + ǫξ ( t, x, y ) + ◦ ( ǫ ) = x + ǫXx + ◦ ( ǫ ) , ˜ y ( t, x, y, ǫ ) = y + ǫξ ( t, x, y ) + ◦ ( ǫ ) = y + ǫXy + ◦ ( ǫ ) , ˜ u ( t, x, y, ǫ ) = u + ǫη ( t, x, y ) + ◦ ( ǫ ) = u + ǫXu + ◦ ( ǫ ) , ˜ v ( t, x, y, ǫ ) = v + ǫη ( t, x, y ) + ◦ ( ǫ ) = v + ǫXv + ◦ ( ǫ ) , where X is called infinitesimal generator which is denoted by 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 Based of the theory the invariant condition for (2.1) is given by φ ( t, x, y, u, v ) = φ (˜ t, ˜ x, ˜ y, ˜ u, ˜ v )It is well known that one can reduce the order of the differential equations aswell as number of independent variables by using the infinitesimal generatorwhich is known as symmetries of (2.1).Therefore, if X is a Lie point symmetry for equation φ ≡
0, then thefollowing condition is true X [ k ] φ = λφ , mod φ = 03here λ is an arbitrary function, and X [ k ] is the k-th extension of X in thejet-space. We write the dynamical system (1.1), (1.2) as follows − q t + p xx − pv − p ( p + q ) n = 0 p t + q xx − qv − q ( p + q ) n = 0 v tt − v xx − m ( p + q ) m − ( pp xx + qq xx + p x + q x ) − m ( m − p + q ) m − ( pp x + qq x ) = 0 (3.1)after substituting the transformations U = p + iq and V = v. The latter real valued system admits five Lie point symmetriesΓ = ∂ t , Γ = ∂ x , Γ = p∂ q − q∂ p , Γ = qt∂ p − pt∂ q + ∂ v , Γ = − qt ∂ p + pt ∂ q − t∂ v . (3.2)On the other hand, if we use the variables U = Re iθ , the Lie symmetriesare simplified as Γ = ∂ t , Γ = ∂ x , Γ = ∂ θ , Γ = t∂ θ + ∂ v , Γ = t ∂ θ − t∂ v . (3.3)In the following we continue our analysis by applying the Lie point sym-metries which provide travel-wave solutions. The traveling wave solution of (3.1) can be constructed by taking linearcombination Γ and Γ . The new canonical variable is r = x − ct. Therefore,the system of PDE (3.1) reduces to system of ODE as follows4 ′′ + cq ′ − p [ v + ( p + q ) n ] = 0 (3.4) q ′′ − cp ′ − q [ v + ( p + q ) n ] = 0 (3.5)( c − v ′′ − m ( p + q ) m − ( pp ′′ + qq ′′ + p ′ + q ′ ) − m ( m − p + q ) m − ( pp ′ + qq ′ ) = 0 (3.6)where prime represents the derivative with respect to r. The solution of (3.6) is given by v = ( p + q ) m c − c r + c ) (3.7)By substituting the expression of v in equations (3.4) and (3.5) we have p ′′ + cq ′ = p (cid:18) ( p + q ) m c − c r + c ) + ( p + q ) n (cid:19) (3.8) q ′′ − cp ′ = q (cid:18) ( p + q ) m c − c r + c ) + ( p + q ) n (cid:19) (3.9)The above equations in general are having a rotational symmetry whichis symmetry Γ . Now we have to solve only the equations (3.8) and (3.9).Divide equations (3.8) and (3.9) then we have p ′′ + cq ′ q ′′ − cp ′ = pq (3.10)This implies that qp ′′ − pq ′′ + c ( pp ′ + qq ′ ) = 0 (3.11) d ( qp ′ − pq ′ ) + c d ( p + q ) = 0 (3.12) qp ′ − pq ′ p + q + c d (arctan (cid:20) pq (cid:21) ) + c pq = tan (cid:16) c − c r (cid:17) . However, for specific values of the free parameters, c , c , n and m , thedynamical system (3.8), (3.9) admits additional Lie point symmetries.5 .1.1 Case I: c = 0If c = 0 then (3.8) and (3.9) is simplified as p ′′ + cq ′ = p (cid:18) ( p + q ) m c − c + ( p + q ) n (cid:19) (3.15) q ′′ − cp ′ = q (cid:18) ( p + q ) m c − c + ( p + q ) n (cid:19) (3.16)The solutions of (3.15) and (3.16) are expressed by the functions p = R ( r ) sin[ θ ( r )] (3.17) q = R ( r ) cos[ θ ( r )] (3.18)where θ ( r ) = Z c R ( t ) dt + ct c and R ( t ) can be found from the following equation R ′′ = − c R + c R + R m +1 c − R n +1 + c R (3.19)where ′ represents the derivative with respect to r. The trivial symmetry of (3.19) is ∂ r . Therefore the above equation reducedas follows. Let R ( r ) = R and R ′ = φ ( R ) then (3.19) φ dφdR = − c R + c R + R m +1 c − R n +1 + c R (3.20)where φ is a function of R. The solution of (3.20) is given by φ = ± s(cid:18) c − c (cid:19) R + R m +2 ( c − m + 1) + R n +2 n + 1 − c R + c (3.21)that is Z dR q(cid:0) c − c (cid:1) R + R m +2 ( c − m +1) + R n +2 n +1 − c R + c = r − r . (3.22)6 .1.2 Case II: c = c = 0 and m = n = − c = c = 0 and m = n = − p ′′ + cq ′ = (cid:18) c c − (cid:19) p ( p + q ) (3.23) q ′′ − cp ′ = (cid:18) c c − (cid:19) q ( p + q ) (3.24)The symmetries of the above equations are given byΓ = ∂ r Γ = p∂ q − q∂ p Γ = ( cp sin[ cr ] + cq cos[ cr ]) ∂ p + ( cq sin[ cr ] − cp cos[ cr ]) ∂ q − cr ] ∂ r Γ = ( cq sin[ cr ] − cp cos[ cr ]) ∂ p + ( cp sin[ cr ] + cq cos[ cr ]) ∂ q + 2 sin[ cr ] ∂ r (3.25)Surprisingly, the Lie point symmetries Γ , Γ and Γ for the SL (2 , R ) Liealgebra, which means the resulting system can be reduced to the Ermakov-Pinney system [51, 52, 53].Indeed under, the change of variables p = R ( r ) sin[ Z c R ( t ) dt + ct c ] (3.26) q = R ( r ) cos[ Z c R ( t ) dt + ct c ] (3.27)we find the analytic solution which is expressed in closed form functions R ( t ) = ± p c (( c − c + c c − c exp[ − ict ] + 2 cc c exp[ − ict ] − c c c ) cc exp[ − ic t ] . m = n = − m = n = − p ′′ + cq ′ = (cid:18) c c − (cid:19) p ( p + q ) + ( c r + c ) p (3.28) q ′′ − cp ′ = (cid:18) c c − (cid:19) q ( p + q ) + ( c r + c ) q (3.29)The solution of (3.28) and (3 . = R ( r ) sin[ θ ( r )] (3.30) q = R ( r ) cos[ θ ( r )] (3.31)where θ ( r ) = Z c R ( t ) dt + ct c and R ( t ) can be found from the following equation R ′′ = ( c r + c − c R + (cid:18) c ( c −
1) + c (cid:19) (cid:18) R (cid:19) the latter equation is nothing else than the Ermakov-Pinney equation, whileits symmetries areΓ = c AiryAi h c r + c ) − c c i AiryAi ′ h c r + c ) − c c i R∂ R + AiryAi h c r + c ) − c c i ∂ r Γ = c AiryBi h c r + c ) − c c i AiryBi ′ h c r + c ) − c c i R∂ R + AiryBi h c r + c ) − c c i ∂ r Γ = c (cid:16) AiryAi h c r + c ) − c c i AiryBi ′ h c r + c ) − c c i + AiryBi h c r + c ) − c c i AiryAi ′ h c r + c ) − c c i(cid:17) R∂ R +2 AiryAi h c r + c ) − c c i AiryBi h c r + c ) − c c i ∂ r Finally, the solution of (3.29) is expressed in terms of the Airy functionas follows R ( r ) = c AiryAi (cid:20) c r + c ) − c c (cid:21) (3.32)with a condition c = ± c √ − c . The algebraic properties of the generalized Zakharov equations was the sub-ject of this study. We wrote the Zakharov equations as a system of three8ifferential equations of real values functions. For the latter system we deter-mined the infinitesimal transformations which leave invariants the Zakharovequations. The admitted Lie point symmetries are five. From the Lie symme-tries we found a similarity transformation which provides reductions whichlead to travel-wave solutions. The reduced system has been classified accord-ing to the admitted Lie point symmetries, where we found surprisingly thatthe Ermakov-Pinney equation can describe the generalized Zakharov equa-tions. This work demonstrate the application of the Lie point symmetriesin applied mathematics and in example in plasma physics. In a future workwe plan to investigate the nature of the conservation laws provided by thesymmetry vectors.
Acknowledgements
KK thank Prof. Dr. Stylianos Dimas, S´ao Jos´e dos Campos/SP, Brasil forproviding us new version of SYM-Package.
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