Non-Noether symmetry of the modified Boussinesq equations

In Hamiltonian systems, conservation laws are closely related to symmetries of evolutionary equations. In case of modified Boussinesq hierarchy this relationship is especially tight as its entire infinite set of conservation laws forms a single involutive orbit of a simple one-parameter symmetry group. We discuss some geometric properties of this symmetry and show how its properties ensure involutivity of conservation laws.

Recall that the modified Boussinesq system is formed by the following set of partial differential equations $$\begin{gathered} u_t=c{v}_{xx}+u_{x}v+u{v}_x \\ {v}_t=-c{u}_{xx}+u{u}_x+kv{v}_x \end{gathered}$$ where $u=u(x,t),v=v(x,t)$ are smooth functions on ${\text{ℝ}}^2$ subjected to zero boundary conditions $u(\pm \mathrm{\infty },t)=v(\pm \mathrm{\infty },t)=0$, while $c$ and $k$ are some real constants. In cases $k=-1$ and $k=3$ modified Boussinesq system has non-trivial bi-Hamiltonian structure that drastically simplifies analysis of the system in these sectors. The first case is described in [2],[5],[6], while in the present paper we focus on the second sector and show that in case $k=3$ bi-Hamiltonian structure of modified Boussinesq system is related to non-Noether symmetry [1] of equations (1). Thus in case $k=3$ modified Boussinesq equations $$\begin{gathered} u_t=c{v}_{xx}+u_{x}v+u{v}_x \\ {v}_t=-c{u}_{xx}+u{u}_x+3v{v}_x \end{gathered}$$ can be rewritten in bi-Hamiltonian form $$\begin{gathered} u_t=W(dh\wedge du)=Ŵ(dĥ\wedge du) \\ {v}_t=W(dh\wedge dv)=Ŵ(dĥ\wedge dv) \end{gathered}$$ where $W$ and $Ŵ$ are compatible Poison bivector fields, i.e. $$[W,W]=[W,Ŵ]=[Ŵ,Ŵ]=0$$ defined as follows $$\begin{gathered} W=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\mathrm{½}(A\wedge A_x+B\wedge B_x)dx \\ Ŵ=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(uB\wedge A_x+vB\wedge B_x-c{A}_x\wedge B_x)dx \end{gathered}$$ Note that $A,B$ are vector fields that for every smooth functional $R=R\left(u\right)$ are defined via variational derivatives $$A\left(R\right)=\frac{\delta R}{\delta u},B\left(R\right)=\frac{\delta R}{\delta v}.$$ Corresponding Hamiltonians in bi-Hamiltonian realization (3) are $$\begin{gathered} h=\mathrm{½}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(u^{2}v+v^3+2cu{v}_x)dx \\ ĥ=\mathrm{½}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(u^2+v^2)dx \end{gathered}$$ This bi-Hamiltonian structure is related to symmetry of equations (2), but before we proceed let us remind that symmetry of evolutionary equations is given by the group of transformations $$(u,v)\mapsto \left(g\right(u),g(v\left)\right)$$ which commutes with time evolution $$\frac{d}{dt}g\left(u\right)=g\left(u_t\right),\frac{d}{dt}g\left(v\right)=g\left(v_t\right)$$ In case of continuous one-parameter groups of transformation $$\begin{gathered} g\left(u\right)=e^{z{L}_E}\left(u\right)=u+z{L}_{E}u+\mathrm{½}{z}^2(L_E{)}^{2}u+\cdots \\ g(v)=e^{z{L}_E}(v)=v+z{L}_{E}v+\mathrm{½}{z}^2(L_E{)}^{2}v+\cdots \end{gathered}$$ generated by some vector field $E$, relation (9) gives rise to the following conditions for the generator of symmetry $E$ $$\begin{gathered} E(u{)}_t=cE(v{)}_{xx}+E(u{)}_{x}v+uE(v{)}_x+u_{x}E\left(v\right)+E\left(u\right)v_x \\ E(v{)}_t=-cE(u{)}_{xx}+uE(u{)}_x+3vE(v{)}_x+E\left(u\right)u_x+3E\left(v\right)v_x \end{gathered}$$ Among solutions of equations (11) there is one important vector field — the generator of non-Noether symmetry which has the following form $$\begin{gathered} E=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\left\{\right[xuv+2t(u^3+3u{v}^2+6cv{v}_x-2c^2{u}_{xx})]A_x-cxv{A}_{xx} \\ +(xu{u}_x+xv{v}_x)B+[x{u}^2+2x{v}^2+2t(5v^3+3u^{2}v-6cv{u}_x-2c^2{v}_{xx})]B_x \\ +cxu{B}_{xx}\}dx \end{gathered}$$ Applying one-parameter group of transformations $$g\left(z\right)=e^{z{L}_E}$$ generated by the vector field $E$ to the centre of Poisson algebra which in our case is formed by functional $$J=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(ku+mv)dx$$ where $k,m$ are arbitrary constants, produces one-parameter family of functions $$J\left(z\right)=e^{z{L}_E}J=J+z{L}_{E}J+\mathrm{½}(z{L}_E{)}^{2}J+\cdots$$ (actually this is the orbit of non-Noether symmetry group that passes centre of Poisson algebra). It is interesting that the functionals $(L_E{)}^{m}J$ are in involution.

By calculating Lie derivatives of $J^{\left(0\right)}$ along the vector field $E$ one can get explicit form of the conservation laws of the modified Boussinesq system: $$\begin{gathered} J^{\left(0\right)}=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(ku+mv)dx \\ {J}^{\left(1\right)}=L_E{J}^{\left(0\right)}=\frac{m}{2}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(u^2+v^2)dx \\ {J}^{\left(2\right)}=(L_E{)}^2{J}^{\left(0\right)}=m\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(u^{2}v+v^3+2cu{v}_x)dx \\ {J}^{\left(3\right)}=(L_E{)}^3{J}^{\left(0\right)}=\frac{3m}{4}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(u^4+5v^4+6u^2{v}^2 \\ -12c{v}^2{u}_x+4c^2{u}_x^2+4c^2{v}_x^2)dx \\ {J}^{\left(m\right)}=(L_E{)}^m{J}^{\left(0\right)}=L_E{J}^{(m-1)} \end{gathered}$$