Involutive orbits of non-Noether symmetry groups

George Chavchanidze
Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia
abstract. We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functionsis investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.
keywords. Non-Noether symmetry; Conservation laws; Modified Boussinesq system; Broer-Kaup system;
msc. 70H33; 70H06; 58J70; 53Z05; 35A30
J. Phys. A: Math. Gen. 38 (2005) 6517-6524
In Hamiltonian integrable models, conservation laws often form involutive orbit ofone-parameter symmetry group. Such a symmetry carries important information about integrable model and its bi-Hamiltonian structure. The present paper is an attempt todescribe class of one-parameter group of transformations of Poisson manifold that possess involutive orbits and may be related to Hamiltonian integrable systems.
Let C∞(M) be algebra of smooth functions on manifold M equipped with Poisson bracket{f , g} = W(df ∧ dg) where W is Poisson bivector satisfying property [W , W] = 0.Each vector field E on manifold M gives rise to one-parameter group of transformations of C∞(M) algebragz = ezLEwhere LE denotes Lie derivative along the vector field E.To any smooth function J ∈ C∞(M) this group assigns orbit that goes through JJ(z) = gz(J) = ezLE(J) = J + zLEJ + ½z2(LE)2J + ⋯the orbit J(z) is called involutive if {J(x) , J(y)} = 0          ∀x, y ∈ ℝInvolutive orbits are often related to integrable models where J(z)plays the role of involutive family of conservation laws.
Involutivity of orbit J(z) depends on nature of vector field E and function J = J(0) and in general it is hard to describe all pairs (E , J) that produce involutive orbitshowever one interesting class of involutive orbits can be outlined by the following theorem:
theorem. For any non-Poisson [E , W] ≠ 0 vector field E satisfying property [E , [E , W]] = 0 and any function J such thatW(dLEJ) = c[E , W](dJ)          c ∈ ℝ∖(0∪ℕ)one-parameter family of functions J(z) = ezLE(J) is involutive.
proof. By taking Lie derivative of property (6) along the vector field E we get[E , W](dLEJ) + W(d(LE)2J) = c[E,[E , W]](dJ) + c[E , W](dLEJ)where c is real constant which is neither zero nor positive integer.Taking into account (5) one can rewrite result as followsW(d(LE)2J) = (c − 1)[E , W](dLEJ)that after m iterations producesW(d(LE)m + 1J) = (c − m)[E , W](d(LE)mJ)Now using this property let us prove that functions J(m) = (LE)mJ are in involution.Indeed{J(k), J(m)} = W(dJ(k) ∧ dJ(m))Suppose that k > m and let us rewrite Poisson bracket as followsW(dJ(k) ∧ dJ(m)) = W(d(LE)kJ ∧ dJ(m)) = LW(d(LE)kJ)J(m)= (c − k + 1)L[E , W](d(LE)k − 1J)J(m) = (c − k + 1)[E , W](dJ(k − 1) ∧ dJ(m))= − (c − k + 1)L[E , W](d(LE)mJ)J(k − 1)= − c − k + 1c − mLW(d(LE)m + 1J)J(k − 1)= c − k + 1c − mW(dJ(k − 1) ∧ dJ(m + 1))Thus we have(c − m){J(k), J(m)} = (c − k + 1){J(k − 1), J(m + 1)} Using this property 2(m − k) times produces{J(k), J(m)} = {J(m), J(k)}and since Poisson bracket is skew-symmetric we finally get{J(k), J(m)} = 0So we showed that functions J(m) = (LE)mJ are in involution.In the same time orbit J(z) is linear combination of functions J(m)and thus it is involutive as well.
remark. Property (9) implies that vector field S = (c − m)E + t(c − m + 1)W(dJ(m + 1)) is non-Noether symmetry [1] of Hamiltonian dynamical systemddtf = {J(m), f}in other words non-Poisson vector field S commutes with time evolution defined by Hamiltonian vector fieldX = ∂∂t + W(dJ(m))This fact can be checked directly[S , X] = (c − m)[E , X] + t(c − m + 1)[W(dJ(m + 1)), W(dJ(m))] − (c − m + 1)W(dJ(m + 1)) = (c − m)[E , W](dJ(m)) + (c − m)W(dLEJ(m)) + t(c − m + 1)W(d{J(m + 1),J(m)}) − (c − m + 1)W(dJ(m + 1)) = W(dJ(m + 1)) + (c − m)W(dJ(m + 1)) − (c − m + 1)W(dJ(m + 1)) = 0In the same time property (9) means that functions J(m) = (LE)mJ form Lenard scheme with respect to bi-Hamiltonian structure formed by Poisson bivector fields W and [E , W] (see [1],[4]).
In many infinite dimensional integrable Hamiltonian systems Poisson bivector has nontrivial kernel,and set of conservation laws belongs to orbit of non-Noether symmetry group that goes throughcentre of Poisson algebra. This fact is reflected in the following theorem:
theorem. If non-Poisson vector field E satisfies property[E, [E , W]] = 0 then every orbit derived from centre I of Poisson algebra C∞(M) is involutive.
proof. If function J belongs to centre J ∈ I of Poisson algebra C∞(M)then by definition W(dJ) = 0. By taking Lie derivative of this condition along vector field Eone gets W(dLEJ) = − [E , W](dJ)that according to Theorem 1 ensures involutivity of J(z) orbit.
example. The theorems proved above may have interesting applications in theory of infinite dimensionalHamiltonian models where they provide simple way to construct involutive family of conservation laws.One non-trivial example of such a model is modified Boussinesq system [2],[5],[6] described by the followingset of partial differential equationsut = cvxx + uxv + uvxvt = − cuxx + uux + 3vvxwhere u = u(x, t), v = v(x, t) are smooth functions on ℝ2subjected to zero boundary conditions u(±∞, t) = v(±∞, t) = 0This system can be rewritten in Hamiltonian formddtf = {h, f} = W(dh ∧ df)with the following Hamiltonianh = ½+ ∞∫− ∞ (u2v + v3 + 2cuvx)dxand Poisson bracket defined by Poisson bivector fieldW = ½+ ∞∫− ∞ (A ∧ Ax + B ∧ Bx)dxwhere A, B are vector fields that for every smooth functional R = R(u) are definedvia variational derivatives A(R) = δR/δu and B(R) = δR/δv.For Poisson bivector (24) there exist vector field E such that[E,[E,W]] = 0this vector field has the following formE = + ∞∫− ∞ (uvAx − cvAxx + (uux + vvx)B + (u2 + 2v2)Bx + cuBxx)xdx= − + ∞∫− ∞ [(uv + 2cvx + x((uv)x + cvxx))A+ (u2 + 2v2 − 2cux + x(uux + 3vvx − cuxx))B]dxApplying one-parameter group of transformations generated by this vector field to centre of Poisson algebra which in our case is formed by functional J = + ∞∫− ∞ (ku + mv)dxwhere k, m are arbitrary constants, produces involutive orbit that recovers infinite sequence of conservation laws of modified Boussinesq hierarchyJ(0) = + ∞∫− ∞ (ku + mv)dxJ(1) = LEJ(0)= m2+ ∞∫− ∞(u2 + v2)dxJ(2) = (LE)2J(0) = m+ ∞∫− ∞ (u2v + v3 + 2cuvx)dxJ(3) = (LE)3J(0) = 3m4+ ∞∫− ∞ (u4 + 5v4 + 6u2v2 − 12cv2ux + 4c2ux2 + 4c2vx2)dxJ(m) = (LE)mJ(0) = LEJ(m − 1)
example. Another interesting model that has infinite sequence of conservation laws lying on singleorbit of non-Noether symmetry group is Broer-Kaup system [3],[5],[6], or more precisely special caseof Broer-Kaup system formed by the following partial differential equationsut = cuxx + 2uuxvt = − cvxx + 2uvx + 2uxvwhere u = u(x, t), v = v(x, t) are again smooth functions on ℝ2subjected to zero boundary conditions u(±∞, t) = v(±∞, t) = 0Equations (29) can be rewritten in Hamiltonian formddtf = {h, f} = W(dh ∧ df)with the Hamiltonian equal toh = + ∞∫− ∞ (u2v + cuxv)dxand Poisson bracket defined byW = + ∞∫− ∞ A ∧ BxdxOne can show that the following vector field E E = + ∞∫− ∞(u2Ax − cuAxx + (uv)xB + 3uvBx + cvBxx)xdx= − + ∞∫− ∞ [(u2 + 2cux + x(2uux + cuxx))A + (3uv − 2cvx + x(2(uv)x − cvxx))B]dxhas property [E,[E,W]] = 0and thus group of transformations generated by this vector field transforms centre of Poisson algebra formed by functional J = + ∞∫− ∞ (ku + mv)dxinto involutive orbit that reproduces well known infinite set of conservation laws of modified Broer-Kaup hierarchyJ(0) =+ ∞∫− ∞ (ku + mv)dxJ(1) = LEJ(0) = m+ ∞∫− ∞ uvdxJ(2) = (LE)2J(0) = 2m+ ∞∫− ∞ (u2v + cuxv)dxJ(3) = (LE)3J(0) = 3m+ ∞∫− ∞ (2u3v − 3cu2vx − 2c2uxvx)dxJ(m) = (LE)mJ(0) = LEJ(m − 1)
Two samples discussed above are representatives of one interesting family of infinite dimensional Hamiltonian systems formed by D partial differential equations of the following typeUt = − 2FGUxx + 〈U , GUx〉C + 〈C , GUx〉U + 〈C , GU〉UxdetG ≠ 0,          GT = G,       FT = − FFmnCk + FkmCn + FnkCm = 0where U is vector with components umthat are smooth functions on ℝ2 subjected to zero boundary conditionsum = um(x, t);          um(±∞, t) = 0;          m = 1 ... D G is constant symmetric nondegenerate matrix, F is constant skew-symmetric matrix,C is constants vector that satisfies conditionFmnCk + FkmCn + FnkCm = 0and 〈 · , · 〉 denotes scalar product 〈X , Y〉 = D∑m=1XmYm.System of equations (37) is Hamiltonian with respect to Poisson bivector equal toW = + ∞∫− ∞〈A , G−1Ax〉dxwhere A is vector with components Am that are vector fields definedfor every smooth functional R(u) via variational derivatives Am(R) = δR/δum.Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivectorŴ = + ∞∫− ∞{〈C , A〉〈U , Ax〉 + 〈Ax , FAx〉}dxthat is compatible with W or in other words[W , W] = [W , Ŵ] = [Ŵ , Ŵ] = 0Corresponding Hamiltonians that produce Hamiltonian realization ddtU = Ŵ(dĤ ∧ dU) = W(dH ∧ dU)of the evolution equations (37) areĤ = ½+ ∞∫− ∞〈U , GU〉dxandH = ½+ ∞∫− ∞{〈C , GU〉〈U , GU〉 + 2〈FGUx , GU〉}dxThe most remarkable property of system (37) is that it possesses set of conservation laws that belong to single orbit obtained from centre of Poisson algebra via one-parametergroup of transformations generated by the following vector fieldE = + ∞∫− ∞{〈C , GU〉〈U , Ax〉 + 〈U , GU〉〈C , Ax〉+ 〈U , GUx〉〈C , A〉 + 2〈FGU , Axx〉}xdx= + ∞∫− ∞{〈C , GU〉〈U , A〉 + 〈U , GU〉〈C , A〉 + 4〈FGUx , A〉+ x (〈C , GUx〉〈U , A〉 + 〈C , GU〉〈Ux , A〉+ 〈U , GUx〉〈C , A〉 + 2〈FGUxx , A〉)}dxNote that centre of Poisson algebra (with respect to bracket defined by W) is formed byfunctionals of the following typeJ = + ∞∫− ∞〈K , U〉dxwhere K is arbitrary constant vector and applying group of transformations generated by Eto this functional J yields the infinite sequence of functionals J(0) = + ∞∫− ∞〈K , U〉dxJ(1) = LEJ(0) = ½〈C , K〉+ ∞∫− ∞〈U , GU〉dxJ(2) = (LE)2J(0) = 〈C , K〉+ ∞∫− ∞{〈C , GU〉〈U , GU〉 + 2〈FGUx , GU〉}dxJ(3) = (LE)3J(0) = ¼〈C , K〉+ ∞∫− ∞{3〈C , GC〉〈U , GU〉2 + 12〈C , GU〉2〈U , GU〉 + 32〈C , GU〉〈GU , FGUx〉+ 24〈U , GC〉〈GU , FGUx〉 + 48〈FGUx , GFGUx〉}dxJ(m) = (LE)mJ(0) = LEJ(m − 1)One can check that the vector field E satisfies condition[E , [E , W]] = 0and according to Theorem 2 the sequence J(m) is involutive. So J(m) are conservation laws of bi-Hamiltonian dynamical system (37)and vector field E is related to non-Noether symmetries of evolutionary equations(see Remark 1).
Note that in special case when C, F, G, K have the following formD = 2,      F12 = − F21 = ½c,      C = K = (0 , 1),      G = 1model (37) reduces to modified Boussinesq system discussed above.Another choice of constants C, F, G, K D = 2,     F12 = − F21 = ½c,      C = K = (0 , 1)G12 = G21 = 1,      G11 = G22 = 0gives rise to Broer-Kaup system described in previous sample.
summary. Groups of transformations of Poisson manifold that possess involutive orbits play importantrole in some integrable models where conservation laws form orbit of non-Noether symmetry group. Therefore classification of vector fields that generate such a groups would create good backgroundfor description of remarkable class of integrable system that have interesting geometric origin.The present paper is an attempt to outline one particular class of vector fields that arerelated to non-Noether symmetries of Hamiltonian dynamical systems and produce involutive families of conservation laws.
acknowledgements. The research described in this publication was made possible in part byAward No. GEP1-3327-TB-03 of the Georgian Research and Development Foundation (GRDF) and the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF).

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