Involutive orbits of non-Noether symmetry groups

In Hamiltonian integrable models, conservation laws often form involutive orbit of one-parameter symmetry group. Such a symmetry carries important information about integrable model and its bi-Hamiltonian structure. The present paper is an attempt to describe class of one-parameter group of transformations of Poisson manifold that possess involutive orbits and may be related to Hamiltonian integrable systems.

Let $C^{\mathrm{\infty }}\left(M\right)$ be algebra of smooth functions on manifold $M$ equipped with Poisson bracket $$\{f,g\}=W(df\wedge dg)$$ 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 $C^{\mathrm{\infty }}\left(M\right)$ algebra $$g_z=e^{z{L}_E}$$ where $L_E$ denotes Lie derivative along the vector field $E$. To any smooth function $J\in C^{\mathrm{\infty }}\left(M\right)$ this group assigns orbit that goes through $J$ $$J\left(z\right)=g_z\left(J\right)=e^{z{L}_E}\left(J\right)=J+z{L}_{E}J+\mathrm{½}{z}^2(L_E{)}^{2}J+\cdots$$ the orbit $J\left(z\right)$ is called involutive if $$\left\{J\right(x),J(y\left)\right\}=0\forall x,y\in \text{ℝ}$$ Involutive orbits are often related to integrable models where $J\left(z\right)$ plays the role of involutive family of conservation laws.

Involutivity of orbit $J\left(z\right)$ depends on nature of vector field $E$ and function $J=J\left(0\right)$ and in general it is hard to describe all pairs $(E,J)$ that produce involutive orbits however 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 through centre 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 type $$\begin{gathered} U_t=-2FG{U}_{xx}+\text{〈}U,G{U}_x\text{〉}C+\text{〈}C,G{U}_x\text{〉}U+\text{〈}C,GU\text{〉}{U}_x \\ detG\ne 0,G^T=G,F^T=-F \\ {F}_{mn}{C}_k+F_{km}{C}_n+F_{nk}{C}_m=0 \end{gathered}$$ where $U$ is vector with components $u_m$ that are smooth functions on ${\text{ℝ}}^2$ subjected to zero boundary conditions $$u_m=u_m(x,t);u_m(\pm \mathrm{\infty },t)=0;m=1...D$$ $G$ is constant symmetric nondegenerate matrix, $F$ is constant skew-symmetric matrix, $C$ is constants vector that satisfies condition $$F_{mn}{C}_k+F_{km}{C}_n+F_{nk}{C}_m=0$$ and $\text{〈}·,·\text{〉}$ denotes scalar product $$\text{〈}X,Y\text{〉}=\sum _{m=1}^D{X}_m{Y}_m.$$ System of equations (37) is Hamiltonian with respect to Poisson bivector equal to $$W=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\text{〈}A,G^{-1}{A}_x\text{〉}dx$$ where $A$ is vector with components $A_m$ that are vector fields defined for every smooth functional $R\left(u\right)$ via variational derivatives $A_m\left(R\right)=\delta R/\delta u_m$. Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivector $$Ŵ=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{\text{〈}C,A\text{〉}\text{〈}U,A_x\text{〉}+\text{〈}{A}_x,F{A}_x\text{〉}\}dx$$ that is compatible with $W$ or in other words $$[W,W]=[W,Ŵ]=[Ŵ,Ŵ]=0$$ Corresponding Hamiltonians that produce Hamiltonian realization $$\frac{d}{dt}U=Ŵ(dĤ\wedge dU)=W(dH\wedge dU)$$ of the evolution equations (37) are $$Ĥ=\mathrm{½}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\text{〈}U,GU\text{〉}dx$$ and $$H=\mathrm{½}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{\text{〈}C,GU\text{〉}\text{〈}U,GU\text{〉}+2\text{〈}FG{U}_x,GU\text{〉}\}dx$$ 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-parameter group of transformations generated by the following vector field $$\begin{gathered} E=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{\text{〈}C,GU\text{〉}\text{〈}U,A_x\text{〉}+\text{〈}U,GU\text{〉}\text{〈}C,A_x\text{〉} \\ +\text{〈}U,G{U}_x\text{〉}\text{〈}C,A\text{〉}+2\text{〈}FGU,A_{xx}\text{〉}\}xdx \\ =\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{\text{〈}C,GU\text{〉}\text{〈}U,A\text{〉}+\text{〈}U,GU\text{〉}\text{〈}C,A\text{〉}+4\text{〈}FG{U}_x,A\text{〉} \\ +x(\text{〈}C,G{U}_x\text{〉}\text{〈}U,A\text{〉}+\text{〈}C,GU\text{〉}\text{〈}{U}_x,A\text{〉} \\ +\text{〈}U,G{U}_x\text{〉}\text{〈}C,A\text{〉}+2\text{〈}FG{U}_{xx},A\text{〉}\left)\right\}dx \end{gathered}$$ Note that centre of Poisson algebra (with respect to bracket defined by $W$) is formed by functionals of the following type $$J=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\text{〈}K,U\text{〉}dx$$ where $K$ is arbitrary constant vector and applying group of transformations generated by $E$ to this functional $J$ yields the infinite sequence of functionals $$\begin{gathered} J^{\left(0\right)}=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\text{〈}K,U\text{〉}dx \\ {J}^{\left(1\right)}=L_E{J}^{\left(0\right)}=\mathrm{½}\text{〈}C,K\text{〉}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\text{〈}U,GU\text{〉}dx \\ {J}^{\left(2\right)}=(L_E{)}^2{J}^{\left(0\right)}=\text{〈}C,K\text{〉}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{\text{〈}C,GU\text{〉}\text{〈}U,GU\text{〉}+2\text{〈}FG{U}_x,GU\text{〉}\}dx \\ {J}^{\left(3\right)}=(L_E{)}^3{J}^{\left(0\right)}=\mathrm{¼}\text{〈}C,K\text{〉}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\{3\text{〈}C,GC\text{〉}\text{〈}U,GU{\text{〉}}^2 \\ +12\text{〈}C,GU{\text{〉}}^2\text{〈}U,GU\text{〉}+32\text{〈}C,GU\text{〉}\text{〈}GU,FG{U}_x\text{〉} \\ +24\text{〈}U,GC\text{〉}\text{〈}GU,FG{U}_x\text{〉}+48\text{〈}FG{U}_x,GFG{U}_x\text{〉}\}dx \\ {J}^{\left(m\right)}=(L_E{)}^m{J}^{\left(0\right)}=L_E{J}^{(m-1)} \end{gathered}$$ One can check that the vector field $E$ satisfies condition $$[E,[E,W\left]\right]=0$$ and according to Theorem 2 the sequence $J^{\left(m\right)}$ is involutive. So $J^{\left(m\right)}$ 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 form $$D=2,F_{12}=-F_{21}=\mathrm{½}c,C=K=(0,1),G=1$$ model (37) reduces to modified Boussinesq system discussed above. Another choice of constants $C,F,G,K$ $$\begin{gathered} D=2,F_{12}=-F_{21}=\mathrm{½}c,C=K=(0,1) \\ {G}_{12}=G_{21}=1,G_{11}=G_{22}=0 \end{gathered}$$ gives rise to Broer-Kaup system described in previous sample.