Lax pair and first integrals for two of nonlinear coupled oscillators
aa r X i v : . [ n li n . S I] J a n Lax pair and first integrals for twoof nonlinear coupled oscillators
Nikolay A. Kudryashov
Abstract
The system of two nonlinear coupled oscillators is studied. As partial casethis system of equation is reduced to the Duffing oscillator which has manyapplications for describing physical processes. It is well known that the in-verse scattering transform is one of the most powerful methods for solving theCauchy problems of partial differential equations. To solve the Cauchy problemfor nonlinear differential equations we can use the Lax pair corresponding tothis equation. The Lax pair for ordinary differential or systems or for systemordinary differential equations allows us to find the first integrals, which alsoallow us to solve the question of integrability for differential equations. In thisreport we present the Lax pair for the system of coupled oscillators. Using theLax pair we get two first integrals for the system of equations. The consideredsystem of equations can be also reduced to the fourth-order ordinary differentialequation and the Lax pair can be used for the ordinary differential equation offourth order. Some special cases of the system of equations are considered.
Key words:
System of equations, Oscillator, Lax pair, First integral.
It is known that Gardner, Green. Kruskal and Miura first opened the inversescattering transform [1–3] for solving the Cauchy problem of the Korteweg-deVries equation [4]. Using a linear system of equations of the above-mentionedauthors Peter Lax in 1968 introduced a new concept [5–7] now called the Laxpair, which allows to solve the Cauchy problem by means of the inverse scatter-ing transform for a certain class of nonlinear evolution equations.Five years later, in 1973 four young graduates from the Potsdam UniversityMark Ablowitz, David Kaup, Alain Newell and Harvi Segur suggested to lookfor nonlinear evolution equations for which the Cauchy problems can be solvedby the inverse scattering transform taking into account the operator equation.Using the power dependencies of the matrix elements on the spectral param-eter and on the function and their derivatives, from the operator equation forthe AKNS scheme e dependencies of the matrix elements and the evolutionary Corresponding author: [email protected]
Let us consider the following system of nonlinear differential equations a q tt + b q t + c p q + d q = 0 (1) a p tt + b p t + c q p + d p = 0 , (2)where p(t) and q(t) are unknown functions and t is independent variable, a a b , b , c , c , d and d are parameters of mathematical models.Let us look for the Lax pairs for the system of equations in the form A ψ = λ ψ,ψ t = B ψ, (3)where ψ , A and B are matrices in the form [8, 9, 13] ψ = ψ ψ , A = a a a a , B = − i λ q(t)p(t) i λ (4)We look for the Lax pair for traveling wave reduction of the KdV hierarchytakin into account the equationd A dt = B A − A B . (5)Note that equation (5) similar to the Lax pair for the KdV hierarchy if we writethis one using the traveling wave solutions.2rom equation (5) we have four ordinary differential equation for matrixelements a , a , a and a in the formda dt = w a − w a , (6)da dt = − λ a + w a − w a , (7)da dt = w a − w a + 2 i λ a , (8)da dt = w a − w a . (9)Adding equations (6) and (9) we haveddt (a + a ) = 0 . (10)From the last equality follows that we geta = − a . (11)Taking into account equations (7) and (8) we obtainddt (a + a ) = 2 i λ (a − a ) . (12)Let us look for the dependence of elements a , a , a and a in the forma = n X k=0 a k (w , w z , . . . ) λ n − k , a = n − X k=0 b k (w , w z , . . . ) λ n − − k , a = n − X k=0 c k (w , w z , . . . ) λ n − − k . a = − a (13)The matrix elements a , a , a and a of the matrix A can be used forfinding the first integrals of the system of equations. It is known that if thematrix A satisfies equation (5) then the first integrals corresponding to theoriginal equation can be obtained by means of calculating of traces tr A k .We can use the consequence of this proposition. If the matrix elementsa = − a then tr A = − A . Let us note that in case a = − a we havethe following equalitytr A = a + 2 a a + a = (a + a ) ++2 a a − a = 0 + 2(a a − a a ) = − A . (14)So, to look for the first integrals of the system of equations (1) and (2)we haveto calculate the determinant of matrix A .3 Two nonlinear coupled oscillators and theirfirst integrals
Let us assume in (13) n = 2. In this case we havea (t) = a (t) + a (t) λ + a (t) λ , a = − a , a (t) = b (t) + b (t) λ, a (t) = c (t) + c (t) λ. (15)Substituting (15) into equations (6), (7), (8) and (9) we have after calculationsthe following values of the matrix elementsa = − α p(t) q(t) − C − i β λ − α λ , a = − a , (16)a = α q t + β q − α q λ, a = − α p t + β p − α p λ. (17)We also have the system of equations in the form α q tt + β q t − α p q − q = 0 (18)and α p tt − β p t − α p q − p = 0 , (19)where p(t) and q(t) are unknown functions and α , β and C are parameters ofthe system of equations.To look for the first integrals for the system of equations from the Lax pairwe have to calculate the determinant of matrix A . Determinant of matrix A takes the formdet A = α p t q t − α p q − α p q − C − β p q++ α β (q p t − p q t ) + 2 i (cid:0) α β p q − β C + α p q t − α q p t (cid:1) λ ++( β − α C ) λ − i α β λ − α λ . (20)From expression (20) we obtain two first integral for the system of equations(19) and (18) in the formI = α β p q − β C + α p q t − α q p t (21)and I = α p t q t − α p q − α pq − β pq + αβ (qp t − pq t ) − C . (22)Now let us consider the partial cases of the system of equations (19) and (18)with obtained lax pair. 4ssuming α = 1, β = 0 and p = q we have the well-known second-ordernonlinear differential equationq tt − − q = 0 . (23)As this takes place integral (21) is generated and integral (22) is transformedto the well-known integral for equation (23) in the formI (1)2 = q − q − C q = C . (24)The general solution of equation (24) is expressed by mens of the Jacobi ellipticfunction.Equation (24) can be written in the formq = (q − α ) (q − β ) (q − γ ) (q − δ ) . (25)where α , β , γ and δ ( α ≥ β ≥ γ ≥ δ ) are real roots of the algebraic equationq + 12 C q + C + 0 . (26)Equation (25) can be transformed to the following formv = (1 − v )(1 − k v ) , v = ( β − δ )(q − α )( α − δ )(y − β ) , k = ( β − γ )( α − δ )( α − γ )( β − δ ) (27)The general solution of (25) is expressed via the elliptic function in the form[17–21] v(t) = sn ( χ t , k) , χ = 14 ( β − δ ) ( α − γ ) (28)where sn ( χ t , k) is the elliptic sine.The general solution of equation (28) takes the formy(t) = β ( α − δ ) sn ( χ t , k) − α ( β − δ )( α − δ ) sn ( χ t , k) − β + δ . (29)From equation (18) we getp = q tt + β q t α q − C α q . (30)Substituting p from (30) into equation (23) we have the fourth-order differentialequation in the form α (cid:0) q q tttt − z q ttt − + 6 q q tt (cid:1) ++ α (cid:0) β q − β q q t q tt + 4 C q q tt − q q (cid:1) + β q q − β q q tt = 0 . (31)5rom (22) and (21) we have the first integrals for (31) in the formI = α (q q ttt − t q tt ) + α (cid:0) qq t − β q (cid:1) + 4 β C q − β q q t (32)and I = α (cid:0) t q ttt − q tt − q q (cid:1) + 4 β C q − β q q t ++ α (cid:0) β q q ttt − β qq t q tt − β q + 4C qq (cid:1) + αβ (cid:0) q q t − β qq (cid:1) . (33)Assuming β = 0 and α = 1 we obtain from (18) and (19) the system of equationsfor description in the form p tt − q − p = 0 (34)and q tt − p − q = 0 (35)with Hamiltonian that follows from the first integral (22) in the formH = p t q t − q p − q p = 0 . (36)The first integrals I and I at lpha = 1 and β take the formI = − p q t − q p t (37)and I = p t q t − p q − pq − C . (38)Assuming q = q , p = p t , q = p , p = q t (39)we obtain that system equations (34) and (35) are the Hamilton system ofequations ˙q i = ∂ H ∂ p i , ˙p i = − ∂ H ∂ q i , (i = 1 , . (40)We also obtain that the first integrals (21) and (22) satisfy to the involution,As this take we have { I , I } = 0 (41)At β = 0 the system equations (18) and (19) is the Hamilton system too.hamiltonian for this system of equation can be found form the first integrals(21) and (22) using the same variables q i and p i , where ( i = 1 , α = 1 and β = 0 the following integrable differential equation offourth orderq q tttt − t q ttt − + 6 q q tt + 4 C q q tt − q q = 0 (42)with two first integrals in the formI = q q ttt − t q tt + 4 C q q t (43)6nd I = 2 q q t q ttt − tt q − q q + 4 C q q . (44)Equation (43) can be interated with respect to z. it takes the formq q tt − + 2 C q = I t + I (45)Taking into account the new variable q = − we have from (45) the secondorder differential equation in the formV tt − V + (I + I t) V = 0 , (46)where I and I are arbitrary constants.At I = 0 we obtain after integration the equation for the eliiptic functionJacobi in the form V − V + 12 I V = C , (47)where C is arbitrary constant.Let us use the new variable q(t) = − again in equation (44). We have2 V V tt − V V t V ttt + 12 V V − V V − I V = 0 (48)Substituting V tt from (46) we get the first-order nonlinear equation in the form(I t + I ) V + I V V t − − (2 I C t + I C ) V −−
12 I V + (cid:18)
12 I + I I t + 12 I t (cid:19) V = 0 . (49)From (49) at I = 0 we obtain the equationI V + 2 C − C V −
12 I V + 12 I V = 0 .. (50)The general solution of equation (50) is expressed via the Jacobi elliptic function. In this report we have considered the system of two nonlinear differential equa-tions. We have found the Lax pair for this system. Using this one we haveobtained the first integrals for the system of equations.
Acknowledgements
The reported study was funded by RFBR according to the research Project No.18-29-10025. 7 eferences [1]
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