ON THE CLASSIFICATION OF FIFTH-ORDER INTEGRABLE SYSTEMS

ON THE CLASSIFICATION OF FIFTH-ORDER INTEGRABLE SYSTEMS DARYOUSH TALATI

arXiv:1305.4567v1 [nlin.SI] 20 May 2013

No. 15, 22nd Alley, Jomhoori Eslami Boulevard, Salmas, West Azarbaijan, Iran [email protected], [email protected] Abstract. In this work we classify the fifth-order integrable symmetrically coupled systems of weight 0 that possess seventh-order symmetry. we obtained 2 new integrable systems that related bi-Hamiltonian formulations are constructed too.

1. Introduction Many approaches have been advocated to the classification of integrable evolution equations, all of wich have their benefit but the most successful ones are the generalized symmetry method and the conservation law method. In recent years studies on fifth-order systems of two-component nonlinear evolution equations have received considerable attention. Some completely integrable equations of this type were found by Mikhailov, Novikov and Wang [15, 14] and Talati [19]. It is inevitable to consider limited classes of this type equations for doing a complete classification. It is useful in classifying symmetric generalization of scalar evolution equations. A mileston in this direction is the work of Foursov on the classification of third-order systems with two-components [8]. Many integrable systems have been classified using this direct classification. In this paper we extend this approach to the case of fifth-order evolution systems of weight 0 and have found all coupled integrable equations of this class and have demonstrated that this equations possess infinitely higher generalized symmetries and conservation laws. Coupled two component integrable equations of form u A1 [u, v] = (1) v t B1 [u, v] 1

possess a hierarchy of higher symmetries u Ai [u, v] = i = 2, 3, 4, ... v t Bi [u, v]

(2)

i

and this property can be taken as a definition of integrability. System (2) is said to be generalized symmetry of System (1) if u u DAi DAv i DA1 DAv 1 u u − = 0. u v u v v DB D D D v t B1 Bi Bi 1 t 1

i

d Here DF denotes the Frechet drivative defined by DFu [u,v] H[u, v] = dǫ F [u + ǫH[u, v], v]|ǫ=0 and d v DF [u,v] H[u, v] = dǫ F [u, v + ǫH[u, v]]|ǫ=0. In all Known cases the existence of one higher order symmetry seems to be sufficient for the existence of infinitely generalized symmetry.

1

2

DARYOUSH. TALATI

In case B[u, v] = A[v, u], system (1) become u A[u, v] = v t A[v, u]

(3)

that is said symmetricaly coupled integrable system. The well-known Sasa-Satsuma system u u3x + u2 vx + 3uvux = (4) v t v3x + v 2 ux + 3uvvx is one example of this type systems. It is easy to see that the system(4) is homogeneous if we assign weightings of 1 to the dependent variables, while x− and t−differentiation have weights 1 and 4, respectively. It is known that the majority of integrable systems have infinitely conservation gradient.To prove the integrability of an equation suspected to be bi-Hamiltonian, one need to find an appropriate compatible pair of Hamilton operator J and K such that the Magri scheme uti = Fi [u] = KGi [u] = JGi+1 [u], i = −1, 0, 1, 2, 3, ... constructed by the operators contains the equation in hand. Here Fi [u] are characteristics of symmetries and Gi are the conserved gradients. A system of evolution equations, M[u, v] G= (5) N[u, v] ∗ is said to be a generalized conserved gradient( whose Frechet drivative is self-adjoint DG = DG ) of system (1) if and only if u ∗ u v DA DAv M DM D M A + = 0. u v u v DB DB N DN DN B

Let ρi be a conserved density of two-component evolution system. The relationship between conserved gradient and conserved density of evolution system can be written as ! P J ∂ρi (−D) Mi [u, v] ∂uJ PJ . = J ∂ρi Ni [u, v] (−D) J ∂v J

2. Statement of the problem The right-hand side of all integrable systems of evolution equations is a homogeneous differential polinomial under a suitable weighting scheme. Let us consider a symmetric system of two equations u A[ux , vx ] = . (6) v t A[vx , ux ] Here A[ux , vx ] = A(ux , vx , ux x, vx x, ...) denotes a differential polynomial function of x-derivatives of u and v. The system of diferential equation (6) is said to be λ-homogeneous if it admits the group of scaling symmetries [7] (x, u, v) −→ (a−1 x, aλ u, aλ v) , a ∈ ℜ+ .

ON THE CLASSIFICATION OF FIFTH-ORDER INTEGRABLE SYSTEMS

3

In this work we are restricting our attention to λ = 0. We determine all equations of the form (6) with A = γ1 u5x + γ2 v5x + α1 ux u4x + α2 vx u4x + α3 ux v4x + α4 vx v4x + α5 uxx u3x +α6 vxx u3x + α7 u2x u3x + α8 ux vx u3x + α9 vx2 u3x + α10 uxx v3x + α11 vxx v3x +α12 u2x v3x + α13 ux vx v3x + α14 vx2 v3x + α15 ux u2xx + α16 vx u2xx + α17 ux vxx uxx +α18 vx vxx uxx + α19 u3x uxx + α20 u2x vx uxx + α21 ux vx2 uxx + α22 vx3 uxx 2 2 +α23 ux vxx + α24 vx vxx + α25 u3x vxx + α26 u2x vx vxx + α27 ux vx2 vxx + α28 vx3 vxx +α29 u5x + α30 u4x vx + α31 u3x vx2 + α32 u2x vx3 + α33 ux vx4 + α34 vx5

(7)

possessing an admissible generator of form (6) with A = γ1 u7x + γ2 v7x + β1 ux u6x + β2 vx u6x + β3 ux v6x + β4 vx v6x + β5 uxx u5x +β6 vxx u5x + β7 u2x u5x + β8 ux vx u5x + β9 vx2 u5x + β10 uxx v5x + β11 vxx v5x +β12 u2x v5x + β13 ux vx v5x + β14 vx2 v5x + β15 u3x u4x + β16 v3x u4x +β17 ux uxx u4x + β18 vx uxx u4x + β19 ux vxx u4x + β20 vx vxx u4x + β21 u3x u4x +β22 u2x vx u4x + β23 ux vx2 u4x + β24 vx3 u4x + β25 u3x v4x + β26 v3x v4x +β27 ux uxx v4x uxx + β28 vx v4x + β29 ux vxx v4x + β30 vx vxx v4x + β31 u3x v4x +β32 u2x vx v4x + β33 ux vx2 v4x + β34 vx3 v4x + β35 ux u23x + β36 vx u23x + β37 ux u3x v3x +β38 vx u3x v3x + β39 u2xx u3x + β40 uxx vxx u3x + β41 u2x uxx u3x + β42 ux vx uxx u3x 2 +β43 vx2 uxx u3x + β44 vxx u3x + β45 u2x vxx u3x + β46 ux vx vxx u3x +β47 vx2 vxx u3x + β48 u4x u3x + β49 u3x vx u3x + β50 u2x vx2 u3x + β51 ux vx3 u3x 2 2 + β55 u2xx v3x + β56 uxx vxx v3x + β57 u2x uxx v3x + β54 vx v3x +β52 vx4 u3x + β53 ux v3x 2 +β58 ux vx uxx v3x + β59 vx2 uxx v3x + β60 vxx v3x + β61 u2x vxx v3x + β62 ux vx vxx v3x 2 4 3 +β63 vx vxx v3x + β64 ux v3x + β65 ux vx v3x + β66 u2x vx2 v3x + β67 ux vx3 v3x + β68 vx4 v3x +β69 ux u3xx + β70 vx u3xx + β71 ux vxx u2xx + β72 vx vxx u2xx + β73 u3x u2xx + β74 u2x vx u2xx 2 2 +β75 ux vx2 u2xx + β76 vx3 u2xx + β77 ux uxx vxx + β78 vx uxx vxx + β79 u3x uxx vxx +β80 u2x vx uxx vxx + β81 ux vx2 uxx vxx + β82 vx3 uxx vxx + β83 u5x uxx + β84 u4x vx uxx 3 3 +β85 u3x vx2 uxx + β86 u2x vx3 uxx + β87 ux vx4 uxx + β88 vx5 uxx + β89 ux vxx + β90 vx vxx 2 2 2 2 + β93 ux vx2 vxx + β94 vx3 vxx + β95 u5x vxx + β96 u4x vx vxx + β92 u2x vx vxx +β91 u3x vxx 3 2 2 3 4 5 +β97 ux vx vxx + β98 ux vx vxx + β99 ux vx vxx + β100 vx vxx + β101 u7x + β102 u6x vx +β103 u5x vx2 + β104 u4x vx3 + β105 u3x vx4 + β106 u2x vx5 + β107 ux vx6 + β108 vx7 . (8) The main matrix of these sistems is γ1 γ2 . γ= γ2 γ1

(9)

By a linear change of variables, the matrix (9) can be reduced to following canonical Jordan form γ1 + γ2 0 Jγ = . (10) 0 γ1 − γ2 Because of properties of symmetric systems, we will restrict our attention to γ2 ≤ γ1 , γ1,2 = 0, 1. 1 0 1 0 Similarly we will deal with two canonical Jordan form and . 0 1 0 0

4

DARYOUSH. TALATI

3. Classification of 0-homogenouse symmetricaly coupled fifth-order integrable Systems of two-component evolution equations Theorem 1. A coupled fifth-order system of two-component evolution equations of form (6) and (7) that possesses a seventh-order generalized symmetry of (6) and (8) with γ1 = γ2 = 1 have a lower order symmetry or transform by a linear change of variables to one of the following two equations: 

u





                          =   v                    t



u





u5 + v5 + 2u1 u4 − 2v1 u4 + 6u1 v4 − 6v1 v4 − 16u2 u3 − 4v2 u3 − 54u21 u3 −20u1 v1 u3 − 6v12 u3 − 4u2 v3 − 16v2 v3 − 22u21 v3 − 52u1v1 v3 − 6v12 v3 −52u1 u22 − 4v1 u22 − 32u1 u2 v2 − 16v1 u2 v2 − 12u31u2 + 4u21 v1 u2 − 4u1 v12 u2 +12v13u2 − 44u1 v22 − 12v1 v22 − 36u31v2 + 12u21v1 v2 − 12u1 v12 v2 +36v13v2 + 72u51 + 96u41v1 + 176u31v12 + 96u21 v13 + 72u1 v14 u5 + v5 − 6u1 u4 + 6v1 u4 − 2u1 v4 + 2v1 v4 − 16u2u3 − 4v2 u3 − 6u21 u3 −52u1 v1 u3 − 22v12 u3 − 4u2 v3 − 16v2 v3 − 6u21 v3 − 20u1v1 v3 − 54v12 v3 −12u1 u22 − 44v1 u22 − 16u1u2 v2 − 32v1 u2v2 + 36u31 u2 − 12u21 v1 u2 +12u1v12 u2 − 36v13 u2 − 4u1 v22 − 52v1 v22 + 12u31v2 − 4u21 v1 v2 + 4u1 v12 v2 −12v13 v2 + 72u41 v1 + 96u31 v12 + 176u21v13 + 96u1v14 + 72v15

                

(11)

u5 + v5 + u1 u4 − v1 u4 + 3u1 v4 − 3v1 v4 + 7u2 u3 + 13v2 u3 − 36u21u3 −20u1 v1 u3 − 24v12 u3 + 13u2 v3 + 7v2 v3 − 28u21 v3 − 28u1 v1 v3 − 24v12v3 −28u1 u22 − 16v1 u22 − 8u1 u2v2 − 64v1 u2 v2 − 24u31 u2 + 8u21 v1 u2 − 8u1 v12 u2 +24v13u2 + 4u1 v22 − 48v1 v22 − 72u31 v2 + 24u21 v1 v2 − 24u1v12 v2 + 72v13 v2 +72u51 + 96u41 v1 + 176u31 v12 + 96u21v13 + 72u1v14

                        =       u5 + v5 − 3u1 u4 + 3v1 u4 − u1 v4 + v1 v4 + 7u2 u3 + 13v2 u3 − 24u21u3   v   −28u v u − 28v 2 u + 13u v + 7v v − 24u2 v − 20u v v − 36v   2    1 1 3 2 3 2 3 1 1 3 1 v3 1 3 1 3  −48u u2 + 4v u2 − 64u u v − 8v u v + 72u3u − 24u2v u + 24u v 2 u   1 2 1 2 1 2 2 1 2 2 1 1 2    1 2 1 1 2  −72v 3 u − 16u v 2 − 28v v 2 + 24u3 v − 8u2 v v + 8u v 2 v − 24v 3 v   1 2 1 2 1 1 2 1 2 1 2 1 1 2 1 2 4 3 2 2 3 4 5 +72u v + 96u v + 176u v + 96u v + 72v 1 1 1 1 1 1 1 1 1 t

                

(12)

Theorem 2. Every coupled fifth-order systems of two-component evolution equations of form (6) and (7) that possesses a seventh-order generalized symmetry of (6) and (8) with γ1 = 1, γ2 = 0 have a lower order symmetry. Very recently, He and Geng [17] introduced a 3 × 3 matrix spectral problem, from which they founded a hierarchy of new nonlinear evolution equations u wxx + 2wwx − 2wzx − 2zwx − 2zxx , (13) = v τ −zxx + 2zzx − 2wzx − 2zwx + 2wxx . The potential form of system (13) can be written as u −uxx + 2vxx − u2x − 2ux vx + 2vx2 = v t −2uxx + vxx − 2u2x + 2ux vx + vx2

(14)


5

and the potential form of second member of the hierarchy is the following symmetrycaly coupled 0-homogeneous fifth-order system: 

u



             v   

t

 u5x + 5uxx u3x − 5vxx u3x − 5u2x u3x + 5ux vx u3x − 5vx2 u3x − 5uxx v3x 2   −5ux u2xx + 5vx u2xx − 10vx uxx vxx + 5u2x vx uxx − 5ux vx2 uxx + 5ux vxx   2 2 5 3 2 2 3 4 −10ux vx vxx + 10ux vx vxx + ux + 5ux vx − 10ux vx + 5ux vx     =  (15)   2 2  v5x − 5vxx u3x − 5uxxv3x + 5vxx v3x − 5ux v3x + 5ux vx v3x − 5vx v3x    2 +5vx u2xx − 10ux uxx vxx + 10u2x vx uxx − 10ux vx2 uxx − 5vx vxx 2 2 2 4 3 2 2 3 5 −5ux vx vxx + 5ux vxx + 5ux vx vxx + 5ux vx − 10ux vx + 5ux vx + vx 

that can be reduced to the potential kupershmidt equation with v = 0. By a linear change of variables, one can show that the system (13) is not new, it is related to system (21) listed in [13]

u v

=

t

uxx + vvx −vxx + uux

(16)

.

it is easy to see that by change of variables w → 21 (v − u), z → 41 ((31/2 i − 1)u + (31/2 i + 1)v), τ → − 23 3−1/2 it.

(17)

system (13) can be reduced into the system (16). It is obvios that this transformation changes the system (18) in [17] into the fifth order symetry of system (16) as well. The system (16) is the well known modified Boussinesq system [1] that is the member of class of following systems of evolution equations

u v

=

t

uxx + A[u, v] −vxx + B[u, v]

.

(18)

All systems (18) possessing higher conservation laws were classified by Mikhailov, Shabat and Yamilov [2, 3].

3.1. Integrability of systems (11). By change of dependent variables Z 1 u→ (w − z)dx 2 Z 1 v→ (w + z)dx 2

6

DARYOUSH. TALATI

sistem(11) can be written in its canonical form



u





w5x − 2zz4x − 10wx w3x − 20w 2w3x − 2z 2 w3x − 8zx z3x − 8wzz3x 2 2 −10wxx − 80wwxwxx − 8zzx wxx − 6zxx − 12zwx zxx − 24wzx zxx 2 3 3 2 +8w zzxx + 4z zxx − 20wx − 12wx zx + 16wzwx zx + 80w 4wx +48w 2z 2 wx + 4z 4 wx + 8w 2zx2 + 12z 2 zx2 + 32w 3zzx + 16wz 3 zx

                  =     4zw4x + 4zx w3x − 16wzw3x − 8z 2 z3x − 40zwx wxx − 16wzx wxx  v   −16w 2 zw − 8z 3 w − 32zz z − 12w 2 z − 32wzw 2 − 16w 2w z    xx xx x xx x x   x x x  −24z 2 w z + 64w 3zw + 32wz 3 w − 8z 3 + 16w 4 z + 48w 2z 2 z x x x x x x x 4 t +20z zx

            

(19)

proposition The infinite hierarchy of system (19) can be write in two different way

wt zt

=J

δw δz

Z

ρ1 dx = K

δw δz

Z

ρ0 dx

(20)

with the campatible pair of Hamiltonian operators

J=

Dx 0 0 2Dx

,K =

K11 K12 K13 K14

where

K11 = Dx7 + ω1 Dx5 + Dx5 ω1 + ω2 Dx3 + Dx3 ω2 + ω3 Dx + Dx ω3 + 8wx Dx−1 wt + 8wt Dx−1 wx K21 = Dx6 ω4 + Dx5 ω5 + Dx4 ω6 + Dx3 ω7 + Dx2 ω8 + Dx ω9 + ω10 + 8wx Dx−1 zt + 8wt Dx−1 zx K31 = −ω4 Dx6 + ω5 Dx5 − ω6 Dx4 + ω7 Dx3 − ω8 Dx2 + ω9 Dx − ω10 + 8zx Dx−1 wt + 8zt Dx−1 wx k41 =

ω11 Dx5 + Dx5 ω11 + ω12 Dx3 + Dx3 ω12 + ω13 Dx + Dx ω13 + 8zx Dx−1 zt + 8zt Dx−1 zx

(21)


7

where the coefficients satisfy ω1 = ω2 = ω3 =

ω4 ω5 ω6 ω7 ω8

= = = = =

ω9 =

ω10 =

ω11 = ω12 = ω13 =

−6wx − 12w 2 − 2z 2 16w3x + 40wwxx + 8zzxx + 58wx2 + 24w 2wx + 12z 2 wx + 8zx2 + 16wzzx + 72w 4 +40w 2z 2 + 18z 4 2 −10w5x − 24ww4x − 4zz4x − 100wx w3x − 24w 2w3x − 12z 2 w3x − 16zx z3x − 84wxx 2 −64wwx wxx − 64zzx wxx − 128w 3wxx − 64wz 2 wxx − 12zxx − 56zwx zxx 2 3 3 2 2 2 2 −16w zzxx − 152z zxx − 48wx − 704w wx − 80z wx − 56wx zx2 − 288wzwx zx −96z 4 wx − 16w 2zx2 − 216z 2 zx2 − 64w 3 zzx + 96wz 3 zx − 128w 6 − 128w 4z 2 −96w 2 z 4 − 16z 6 −4z 4zx − 16wz +48zwx + 16wzx + 32w 2 z −72zwxx − 32wx zx − 160wzwx − 32w 2zx + 96z 2 zx + 128w 3z +40zw3x + 40zx wxx + 192wzwxx + 208zwx2 + 96wwx zx − 576w 2zwx − 320zzx2 −128w 3 zx − 64w 4z −8zw4x − 128wzw3x − 104z 2 z3x − 240zwx wxx − 96wzx wxx + 480w 2zwxx − 96z 3 wxx −152zzx zxx + 576wz 2 zxx − 112wx2 zx + 640wzwx2 + 192w 2wx zx − 192z 2 wx zx +384w 3zwx + 384wz 3wx + 160zx3 + 448wzzx2 + 64w 4zx − 768w 2z 2 zx − 96z 4 zx −256w 5 z +8zx w4x + 32wzw4x + 40z 2 z4x + 32wzx w3x − 128w 2zw3x + 48z 3 w3x + 224zzx z3x −288wz 2 z3x − 80wx zx wxx − 320wzwx wxx − 288w 2zx wxx − 128w 3zwxx − 192wz 3 wxx 2 +304z 2 zx wxx + 120zzxx + 64z 2 wx zxx + 192zx2 zxx − 1376wzzx zxx + 384w 2z 2 zxx +48z 4 zxx − 256wwx2 zx − 256w 2zwx2 + 64z 3 wx2 + 160zwx zx2 − 128w 3 wx zx −192wz 2 wx zx + 512w 4 zwx − 512wzx3 + 1216w 2zzx2 + 272z 3 zx2 + 256w 5zx −12z 2 +100zzxx − 72z 2 wx + 80zx2 − 96wzzx + 144w 2z 2 + 36z 4 2 −68zz4x + 8z 2 w3x − 292zx z3x + 176wzz3x + 232zzx wxx + 64wz 2 wxx − 264zxx 2 3 2 2 2 +512zwx zxx + 816wzx zxx − 896w zzxx − 88z zxx − 64z wx + 280wx zx −1056wzwx zx + 96z 4 wx − 752w 2zx2 − 380z 2 zx2 + 768w 3zzx + 96wz 3 zx − 384w 4z 2 −192w 2 z 4 − 32z 6

By straigthforward calculation It is easily seen that the functional trivector of linear combination K + λJ with constant λ, vanishes independently from the value of λ [4]. The first few conserved densities of the hierarchy are listed below. ρ0 = α ρ1 = 2w 2 + z 2 ρ2 = +3ww4x + 10w 2 w3x + 6z 2 w3x − 20w 3 wxx − 6wz 2 wxx − 18w 2 zzxx − 4z 3 zxx − 18w 2zx2 +16w 3 zzx + 24wz 3 zx + 16w 6 + 24w 4z 2 + 12w 2 z 4 + 2z 6 ρ = −45ww6x − 126w 2 w5x − 90z 2 w5x + 210w 3w4x + 270wz 2 w4x + 270w 2zz4x − 30z 3 z4x +360wzzx w3x + +280w 4w3x + 840w 2z 2 w3x + 180z 4 w3x + 1080w 2zx z3x − 640w 3zz3x 2 −360wz 3 z3x + 630w 2wxx + +1440w 2zzx wxx − 2016w 5wxx − 1920w 3z 2 wxx − 360wz 4 wxx 2 2 +810w 2zxx + 270z 2 zxx + −1920w 3zx zxx − 1080wz 2 zx zxx − 1200w 4zzxx − 1440w 2z 3 zxx 5 4 2 −216z zxx − 1200w zx + −2160w 2 z 2 zx2 + 1152w 5zzx + 1920w 3z 3 zx + 1440wz 5zx +960w 8 + 1920w 6z 2 + 1440w 4z 4 + 480w 2z 6 + 60z 8 ˙. ˙.

8

DARYOUSH. TALATI

These densities suffice to write two magri shemes with same hamiltonian operators that one of them contains the new system proving integrability of the system (19).

3.2. Integrability of systems (12). By change of dependent variables Z 1 u→ (w − z)dx 2 Z 1 (w + z)dx v→ 2 sistem(12) can be written in its canonical form    w5x − zz4x + 10wx w3x − 20w 2 w3x − 8z 2 w3x − 4zx z3x − 2wzz3x u 2 2  +10wxx − 80wwx wxx − 32zzx wxx − 3zxx − 18zwx zxx − 6wzx zxx    2 3 3 2  +16w zzxx + 8z zxx − 20wx − 18wx zx + 32wzwx zx + 80w 4wx       +48w 2z 2 wx + 4z 4 wx + 16w 2 zx2 + 24z 2 zx2 + 32w 3zzx + 16wz 3 zx        =     2zw4x + 2zx w3x − 4wzw3x − 2z 2 z3x + 20zwx wxx − 4wzx wxx  v     −32w 2 zwxx − 16z 3 wxx − 8zzx zxx + 12wx2 zx − 64wzwx2     −48z 2 w z + 64w 3zw − 32w 2 w z + 32wz 3 w − 2z 3 x x

t

x

x x

+16w 4zx + 48w 2 z 2 zx + 20z 4 zx

x

x

            

(22)

proposition The infinite hierarchy of system (22) can be write innot just one but two different way Z Z wt δw δw =J ρ1 dx = K ρ1 dx (23) zt δz δz with the campatible pair of Hamiltonian operators 2 Dx 0 K1 K22 2 J= ,K = 0 2Dx K23 K24 where K12 = Dx7 + ψ1 Dx5 + Dx5 ψ1 + ψ2 Dx3 + Dx3 ψ2 + ψ3 Dx + Dx ψ3 + 8wx Dx−1 wt + 8wt Dx−1 wx K22 = Dx6 ψ4 + Dx5 ψ5 + Dx4 ψ6 + Dx3 ψ7 + Dx2 ψ8 + Dx ψ9 + ψ10 + 8wx Dx−1 zt + 8wt Dx−1 zx (24) K32

=

−ψ4 Dx6

+

ψ5 Dx5

−

ψ6 Dx4

+

ψ7 Dx3

−

ψ8 Dx2

+ ψ9 Dx − ψ10 +

8zx Dx−1 wt

+

8zt Dx−1 wx

K42 = ψ11 Dx5 + Dx5 ψ5 + ψ12 Dx3 + Dx3 ψ12 + ψ13 Dx + Dx ψ13 + 8zx Dx−1 zt + 8zt Dx−1 zx where the coefficients satisfy ψ1 = 6wx − 12w 2 − 5z 2 ψ2 = −16w3x + 40wwxx + 26zzxx + 58wx2 − 24w 2wx − 12z 2 wx + 26zx2 + 20wzzx +72w 4 + 52w 2 z 2 + 18z 4


ψ3 =

ψ4 ψ5 ψ6 ψ7 ψ8

= = = = =

ψ9 =

ψ10 =

ψ11 = ψ12 = ψ13 =

9

10w5x − 24ww4x − 16zz4x − 100wx w3x + 24w 2w3x + 12z 2 w3x − 64zx z3x − 12wzz3x 2 2 −84wxx + 64wwx wxx + 28zzx wxx − 128w 3wxx − 40wz 2 wxx − 48zxx − 16zwx zxx 2 3 3 2 2 2 2 −36wzx zxx − 88w zzxx − 134z zxx + 48wx − 704w wx − 104z wx − 16wx zx2 −288wzwx zx + 24z 4 wx − 88w 2zx2 − 216z 2 zx2 − 128w 3zzx − 24wz 3 zx − 128w 6 −128w 4z 2 − 96w 2 z 4 − 16z 6 −2z 2zx − 4wz −24zwx + 4wzx + 40w 2z +36zwxx + 28wx zx − 200wzwx − 40w 2 zx + 120z 2 zx + 80w 3z −20zw3x − 8zx wxx + 192wzwxx + 104zwx2 + 120wwx zx − 144w 2zwx − 80w 3zx −424zzx2 − 128w 4z +4zw4x + 12zx w3x − 88wzw3x − 154z 2 z3x − 120zwx wxx − 72wzx wxx + 96w 2zwxx +48z 3 wxx − 238zzx zxx + 360wz 2 zxx + 16wx2 zx − 128wzwx2 − 96w 2wx zx +312z 2 wx zx + 768w 3zwx − 96wz 3 wx + 224zx3 + 232wzzx2 + 128w 4zx −672w 2z 2 zx − 192z 4 zx − 256w 5z +8zx w4x + 16wzw4x + 62z 2 z4x + 16wzx w3x − 32w 2 zw3x − 24z 3 w3x − 180wz 2 z3x +364zzx z3x + 80wx zx wxx + 160wzwx wxx − 296z 2 zx wxx − 256w 3zwxx − 192w 2zx wxx 2 +48wz 3 wxx + 186zzxx − 176z 2 wx zxx + 348zx2 zxx − 812wzzx zxx + 336w 2z 2 zxx +96z 4 zxx − 64wwx2 zx − 512w 2 zwx2 + 128z 3 wx2 − 656zwx zx2 − 256w 3wx zx +512w 4zwx + 912wz 2 wx zx − 272wzx3 + 1136w 2zzx2 + 544z 3 zx2 + 256w 5zx −12z 2 +190zzxx − 144z 2 wx + 74zx2 − 120wzzx + 144w 2z 2 + 36z 4 2 −146zz4x + 88z 2 w3x − 514zx z3x + 244wzz3x + 560zzx wxx − 224wz 2 wxx − 480zxx 2 3 2 2 2 +904zwx zxx + 1092wzx zxx − 824w zzxx − 160z zxx + 512wx zx − 520z wx −1632wzwx zx + 864w 2 z 2 wx + 192z 4 wx − 632w 2 zx2 − 464z 2 zx2 + 672w 3 zzx +192wz 3zx − 384w 4 z 2 − 192w 2z 4 − 32z 6

By straigthforward calculation It is easily seen that the functional trivector of linear combination K + λJ with constant λ, vanishes independently from the value of λ. By constructing trivial compositions

J+

m X n=0

λ n K0 J

−1 n

K0

!

J −1

J+

m X n=0

¯ n K0 J λ

−1 n

K0

!

,

K0 J −1

of the successive partial sums m = 0, 1, 2, . . . of linear combinations J +

m P

0

= 1,

λn (K0 J −1 )n K0

n=0

¯ n and by induction on m that all Kn , n = 0, 1, 2, . . . are mutually with arbitrary constants λn , λ compatible HO’s if so is the HO’s K0 and (formally) invertible J [16, 18]. The first few conserved densities of system (19) are listed below. ρ0 = α ρ1 = 2w 2 + z 2 ρ2 = +3ww4x − 10w 2w3x + 3z 2 w3x − 20w 3wxx − 24wz 2 wxx + 18w 2 zzxx − z 3 zxx + 18w 2 zx2 +32w 3 zzx + 48wz 3 zx + 16w 6 + 24w 4z 2 + 12w 2 z 4 + 2z 6

10

DARYOUSH. TALATI

ρ = −90ww6x + 252w 2w5x − 90z 2 w5x + 420w 3w4x − 180w 2zz4x + 1080wz 2w4x − 15z 3 z4x +1440wzzx w3x − 560w 4w3x − 1680w 2z 2 w3x + 360z 4 w3x − 160w 3 zz3x − 720w 2zx z3x 2 −720wz 3 z3x + 1260w 2wxx − 7200w 2zzx wxx − 4032w 5wxx − 5280w 3z 2 wxx 2 2 −2880wz 4 wxx − 540w 2 zxx + 135z 2 zxx − 480w 3zx zxx − 2160wz 2 zx zxx + 480w 4zzxx 2 3 5 4 2 +3600w z zxx − 216z zxx + 480w zx + 12960w 2z 2 zx2 + 4608w 5zzx + 7680w 3z 3 zx +5760wz 5 zx + 1920w 8 + 3840w 6z 2 + 2880w 4z 4 + 960w 2z 6 + 120z 8 ˙. ˙. These densities suffice to write two magri shemes with same hamiltonian operators that one of them contains the new system proving integrability of the system (19).

4. conclusions It is obvious that it is essential to consider limited classes of equations for doing a complete classification of complicated integrable equations. Most of the classified such systems are generalization of the KdV and Burgers equations or equations related to them (see [9, 5, 6, 12, 8, 11] and references therein). Motivated by existing some examples of bi-Hamiltonian two-component generalization of fifth-order equations, we considered a class of fifth-order two-component systems for integrability. The approach outlined in this paper is to to classify two-component integrable systems of fifth order symmetric evolution equations that depends only to x−drivatives of dependent variables. Applying this method, we found two integrable systems which suffice to write Magri shemes contains the new systems proving complete integrability of them.

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[14] A. V. Mikhailov, V. S. Novikov and Jing Ping Wang. On classification of integrable non-evolutionary equations. Stud. Appl. Math., 118:419–457, (2007). [15] A. V. Mikhailov, V. S. Novikov, and J. P. Wang. Symbolic representation and classification of integrable systems, inAlgebraic Theory of Differential Equations edited by Mikhailov, A. V. and MacCallum, M. A. H. (Cambridge University Press ), pp. 156–216 ; e-print arXiv:0712.1972 (2009). [16] D. Talati and R. Turhan. On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators. SIGMA 7, 081 (2011). [17] He. Guo-Liang and Xian-Guo. Geng. An extension of the modified Sawada—Kotera equation and conservation laws. Chinese Physics B 21, 7: 070205 (2012). [18] A. De Sole and V. G. Kac. Non-local Poisson structures and applications to the theory of integrable systems. arXiv:1302.0148v1. (2013). [19] D. Talati, A fifth-order bi-Hamiltonian system. arXiv preprint arXiv:1304.1987 (2013).