Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia

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 $C\infty (M)$ 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 $C\infty (M)$ algebrag_{z} = e^{zLE} where $L$_{E} denotes Lie derivative along the vector field $E$.To any smooth function $J\; \in \; C\infty (M)$ this group assigns orbit that goes through $J$J(z) = g_{z}(J) = e^{zLE}(J) = J + zL_{E}J + ½z^{2}(L_{E})^{2}J + ⋯ the orbit $J(z)$ is called involutive if {J(x) , J(y)} = 0 ∀x, y ∈ ℝ Involutive 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:

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:

Two samples discussed above are representatives of one interesting family of infinite dimensional Hamiltonian systems formed by $D$ partial differential equations of the following typeU_{t} = − 2FGU_{xx} + 〈U , GU_{x}〉C + 〈C , GU_{x}〉U + 〈C , GU〉U_{x}detG ≠ 0, G^{T} = G, F^{T} = − FF_{mn}C_{k} + F_{km}C_{n} + F_{nk}C_{m} = 0 where $U$ is vector with components $u$_{m}that are smooth functions on $\mathbb{R}2$ subjected to zero boundary conditionsu_{m} = u_{m}(x, t); u_{m}(±∞, t) = 0; m = 1 ... D $G$ is constant symmetric nondegenerate matrix, $F$ is constant skew-symmetric matrix,$C$ is constants vector that satisfies conditionF_{mn}C_{k} + F_{km}C_{n} + F_{nk}C_{m} = 0 and $\langle \; \xb7\; ,\; \xb7\; \rangle $ denotes scalar product 〈X , Y〉 = D ∑ m=1 X_{m}Y_{m}. System of equations (37) is Hamiltonian with respect to Poisson bivector equal toW = + ∞ ∫ − ∞ 〈A , G^{−1}A_{x}〉dx where $A$ is vector with components $A$_{m} that are vector fields definedfor every smooth functional $R(u)$ via variational derivatives $A$_{m}(R) = δR/δu_{m}.Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivectorŴ = + ∞ ∫ − ∞ {〈C , A〉〈U , A_{x}〉 + 〈A_{x} , FA_{x}〉}dx that is compatible with $W$ or in other words[W , W] = [W , Ŵ] = [Ŵ , Ŵ] = 0 Corresponding Hamiltonians that produce Hamiltonian realization d dt U = Ŵ(dĤ ∧ dU) = W(dH ∧ dU) of the evolution equations (37) areĤ = ½+ ∞ ∫ − ∞ 〈U , GU〉dx andH = ½+ ∞ ∫ − ∞ {〈C , GU〉〈U , GU〉 + 2〈FGU_{x} , GU〉}dx The 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 , A_{x}〉 + 〈U , GU〉〈C , A_{x}〉+ 〈U , GU_{x}〉〈C , A〉 + 2〈FGU , A_{xx}〉}xdx= + ∞ ∫ − ∞ {〈C , GU〉〈U , A〉 + 〈U , GU〉〈C , A〉 + 4〈FGU_{x} , A〉+ x (〈C , GU_{x}〉〈U , A〉 + 〈C , GU〉〈U_{x} , A〉+ 〈U , GU_{x}〉〈C , A〉 + 2〈FGU_{xx} , A〉)}dx Note that centre of Poisson algebra (with respect to bracket defined by $W$) is formed byfunctionals of the following typeJ = + ∞ ∫ − ∞ 〈K , U〉dx where $K$ is arbitrary constant vector and applying group of transformations generated by $E$to this functional $J$ yields the infinite sequence of functionals J^{(0)} = + ∞ ∫ − ∞ 〈K , U〉dxJ^{(1)} = L_{E}J^{(0)} = ½〈C , K〉+ ∞ ∫ − ∞ 〈U , GU〉dxJ^{(2)} = (L_{E})^{2}J^{(0)} = 〈C , K〉+ ∞ ∫ − ∞ {〈C , GU〉〈U , GU〉 + 2〈FGU_{x} , GU〉}dxJ^{(3)} = (L_{E})^{3}J^{(0)} = ¼〈C , K〉+ ∞ ∫ − ∞ {3〈C , GC〉〈U , GU〉^{2} + 12〈C , GU〉^{2}〈U , GU〉 + 32〈C , GU〉〈GU , FGU_{x}〉+ 24〈U , GC〉〈GU , FGU_{x}〉 + 48〈FGU_{x} , GFGU_{x}〉}dxJ^{(m)} = (L_{E})^{m}J^{(0)} = L_{E}J^{(m − 1)} One can check that the vector field $E$ satisfies condition[E , [E , W]] = 0 and 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, F_{12} = − F_{21} = ½c, C = K = (0 , 1), G = 1 model (37) reduces to modified Boussinesq system discussed above.Another choice of constants $C,\; F,\; G,\; K$ D = 2, F_{12} = − F_{21} = ½c, C = K = (0 , 1)G_{12} = G_{21} = 1, G_{11} = G_{22} = 0 gives rise to Broer-Kaup system described in previous sample.

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