2.1. Determining equations for symmetries and adjoint-symmetries

To begin, the algebraic formulation of determining equations for symmetries and adjoint-symmetries will be stated for a general (regular) PDE system of order N consisting of M equations

(2.1) \begin{equation}G^A(x,u^{(N)}) =0,\quad A=1,\ldots,M\end{equation}

with independent variables $x^i$ , $i=1,\ldots,n$ , and dependent variables $u^\alpha$ , $\alpha=1,\ldots,m$ . ${{\mathcal{E}_{G}}}$ will denote the solution space of the PDE system. The coordinate space $(x^i,u^\alpha,u_j^\alpha,\ldots)$ is called the jet space ${\textrm{J}^{(\infty)}}$ .

The determining equation for symmetries is given by

(2.2) \begin{equation}G'(P)^A|_{{\mathcal{E}_{G}}} =0,\end{equation}

where $P^\alpha(x,u^{(k)})$ is a set of functions representing the components of the symmetry in evolutionary form. Geometrically speaking, a symmetry is a vector field ${\mathbf{X}}_P = P^\alpha\partial_{u^\alpha}$ whose prolongation is tangent to ${{\mathcal{E}_{G}}}$ in ${\textrm{J}^{(\infty)}}$ .

The adjoint of the symmetry equation (2.2) is the determining equation for adjoint-symmetries

(2.3) \begin{equation}G'{}^*(Q)_\alpha|_{{\mathcal{E}_{G}}} =0,\end{equation}

where $Q_A(x,u^{(k)})$ is a set of functions representing the components of the adjoint-symmetry. Its geometrical meaning is that the 1-form $Q_A{\mathbf{d}} G^A$ in ${\textrm{J}^{(\infty)}}$ functionally vanishes on ${{\mathcal{E}_{G}}}$ , as discussed in [Reference Anco and Wang9].

Off of the solution space ${{\mathcal{E}_{G}}}$ , these determining equations are respectively given by

(2.4) \begin{equation}G'(P)^A = R_P(G)^A\end{equation}

and

(2.5) \begin{equation}G'{}^*(Q)_\alpha = R_Q(G)_\alpha,\end{equation}

where $R_P=(R_P)^{A\,I}_{B} D_I$ and $R_Q=(R_Q)_{\alpha\,B}^{I} D_I$ are some linear differential operator in total derivatives whose coefficients $(R_P)^{A\,I}_{B}$ and $(R_Q)_{\alpha\,B}^{I}$ are functions that are non-singular on ${{\mathcal{E}_{G}}}$ .

Recall that a multiplier is a set of functions $\Lambda_A(x,u^{(k)})$ that are non-singular on ${{\mathcal{E}_{G}}}$ and satisfy $\Lambda_A G^A = D_i\Psi^i$ off of ${{\mathcal{E}_{G}}}$ , for some vector function $\Psi^i$ in ${\textrm{J}^{(\infty)}}$ . This total divergence condition is equivalent to

(2.6) \begin{equation}0 = E_{u^\alpha}(\Lambda_A G^A) = \Lambda'{}^*(G)_\alpha + G'{}^*(\Lambda)_\alpha ,\end{equation}

whereby $\Lambda_A$ is an adjoint-symmetry,

(2.7) \begin{equation}G'{}^*(\Lambda)_\alpha|_{{\mathcal{E}_{G}}} =0 .\end{equation}

These equations (2.4), (2.5), (2.6) play a key role in formulating the actions of symmetries on adjoint-symmetries.

2.2. Actions of symmetries on adjoint-symmetries

There are two basic different actions of symmetries on adjoint-symmetries [Reference Anco3, Reference Anco and Wang10]. One action arises geometrically from applying the Lie derivative with respect to a symmetry ${\mathbf{X}}_P$ to the determining equation for adjoint-symmetries, which yields

(2.8) \begin{equation}Q_A\overset{{{\mathbf{X}}_P}}{\longrightarrow} Q'(P)_A + R_P^*(Q)_A .\end{equation}

The other action comes from the adjoint relationship between the determining equation for infinitesimal symmetries and adjoint-symmetries, yielding

(2.9) \begin{equation}Q_A\overset{{{\mathbf{X}}_P}}{\longrightarrow} R_P^*(Q)_A - R_Q^*(P)_A .\end{equation}

Under this action, adjoint-symmetries are mapped into conservation law multipliers.

For adjoint-symmetries that are conservation law multipliers, these two actions coincide with the better known action of symmetries on multipliers [Reference Anco1, Reference Anco4, Reference Anco and Kara8]. Furthermore, the difference of the two actions produces a third action

(2.10) \begin{equation}Q_A\overset{{{\mathbf{X}}_P}}{\longrightarrow} Q'(P)_A + R_Q^*(P)_A ,\end{equation}

which has the property that it vanishes on multipliers, as seen from the multiplier determining equation (2.6).

2.3. Dual actions and a Noether operator

For any symmetry action $Q_A {\longrightarrow} S_P(Q)_A$ , there is a dual action

(2.11) \begin{equation}S_Q(P)_A\;:\!=\; S_P(Q)_A\end{equation}

that maps symmetries into adjoint-symmetries. The symmetry action (2.10) is distinguished from the other two by the property that its dual action can be expressed as a linear operator in total derivatives:

(2.12) \begin{equation}{\mathcal{J}} \;:\!=\; S_{Q} = Q' + R_Q^* .\end{equation}

This operator (2.12) maps symmetries into adjoint-symmetries and thus is a Noether (pre-symplectic) operator. Its formal inverse defines a pre-Hamiltonian (inverse Noether) operator which maps adjoint-symmetries into symmetries.

2.4. Lie Bracket for adjoint-symmetries

A dual symmetry action $S_Q$ can be used to construct an associated Lie bracket on the subspace of adjoint-symmetries given by the range of the action. This yields a homomorphism from the Lie algebra of symmetries into a Lie algebra of adjoint-symmetries.

Let ${\textrm{Symm}}_G$ and ${\textrm{AdjSymm}}_G$ denote the linear spaces of symmetries and adjoint-symmetries. The Lie bracket is given by

(2.13) \begin{equation}{}^Q[Q_1,Q_2]_A \;:\!=\; S_Q([S_Q^{-1}Q_1,S_Q^{-1}Q_2])_A\end{equation}

on the linear space $S_Q({\textrm{Symm}}_G)\subseteq{\textrm{AdjSymm}}_G$ . Note that, since $S_Q^{-1}$ is well-defined only modulo $\ker(S_Q)$ , the condition that $\ker(S_Q)$ is an ideal is necessary and sufficient for the bracket to be well-defined. This condition will select a set of adjoint-symmetries $Q_A$ that can be used in constructing the bracket. If there is more than one such adjoint-symmetry (up to scaling), then a natural choice will be to select a $Q_A$ such that $\textrm{ran}(S_Q)$ is maximal in ${\textrm{AdjSymm}}_G$ .

An alternative way to have the bracket be well-defined arises when the symmetry Lie algebra contains a scaling symmetry. If $\ker(S_Q)$ can be characterised as a subspace by its scaling weight, then the symmetry Lie algebra possesses an extra structure of a direct-sum decomposition as a linear space

(2.14) \begin{equation}{\textrm{Symm}}_G = \ker(S_Q)\oplus\textrm{coker}(S_Q)\end{equation}

with $\textrm{coker}(S_Q)$ being defined by having a distinct scaling weight. In this situation, $S_Q^{-1}$ can be defined as belonging to the subspace $\textrm{coker}(S_Q)$ , and hence, the bracket will be well-defined. More generally, a scaling decomposition (2.14) in which both $\ker(S_Q)$ and $\textrm{coker}(S_Q)$ are each a direct sum of scaling homogeneous subspaces that have no scaling weights in common is sufficient.

2.5. Results for evolution PDEs

The preceding general results have a further development for evolution PDEs

(2.15) \begin{equation}u^\alpha_t = g^\alpha(x,u,\partial_x u,\ldots,\partial_x^N u),\end{equation}

where x now denotes the spatial independent variables $x^i$ , $i=1,\ldots,n$ , while t is the time variable. Note that, for such a PDE system,

(2.16) \begin{equation}G^\alpha(t,x,u^{(N)}) = u_t^\alpha - g^\alpha(x,u,\partial_x u,\ldots,\partial_x^N u)\end{equation}

with $A=\alpha$ for the indices.

On the solution space ${{\mathcal{E}_{G}}}$ , all t-derivatives of $u^\alpha$ can be eliminated in any expression through substituting the equation (2.15) and its spatial derivatives. Consequently, symmetries consist of a set of functions $P^\alpha(t,x,u,\partial_x u,\ldots,\partial_x^k u)$ satisfying

(2.17) \begin{equation}\partial_t P^\alpha + P'(g)^\alpha - g'(P)^\alpha= \partial_t P^\alpha + [g,P]^\alpha =0\end{equation}

which is the symmetry determining equation (2.2) in simplified form off of ${{\mathcal{E}_{G}}}$ . Hence,

(2.18) \begin{equation}R_P = P' .\end{equation}

Likewise, adjoint-symmetries consist of a set of functions $Q_\alpha(t,x,u,\partial_x u,\ldots,\partial_x^k u)$ satisfying the adjoint equation

(2.19) \begin{equation}-(\partial_t Q_\alpha +Q'(g)_\alpha + g'{}^*(Q)_\alpha)=0\end{equation}

which is a simplified form of the determining equation (2.19) off of ${{\mathcal{E}_{G}}}$ . Thus,

(2.20) \begin{equation}R_Q = -Q' .\end{equation}

An equivalent formulation is given by

(2.21) \begin{equation}\partial_t Q_\alpha +\{Q,g\}^*_\alpha=0\end{equation}

in terms of the anti-commutator $\{A,B\}=A'(B) + B'(A)$ , where $\{A,B\}^*=A'{}^*(B) + B'{}^*(A)$ .

The well-known necessary and sufficient condition for an adjoint-symmetry to be a conservation law multiplier is that its Frechet derivative is self-adjoint

(2.22) \begin{equation}Q'=Q'{}^* ,\end{equation}

which follows directly from equations (2.20) and (2.6). Self-adjointness (2.22) is equivalent to the property that $Q_\alpha$ is a variational derivative (gradient)

(2.23) \begin{equation}\Lambda_\alpha = E_{u^\alpha}(\Phi)\end{equation}

for some function $\Phi(x,u^{(k)})$ , $k\geq 0$ , where $E_{u^\alpha}$ is the Euler operator with respect to $u^\alpha$ .

The relations (2.18) and (2.20) give simplified expressions for the symmetry actions (2.8) and (2.9):

(2.24) \begin{align}& Q_\alpha \overset{{{\mathbf{X}}_P}}{\longrightarrow} Q'(P)_\alpha + P'{}^*(Q)_\alpha, \end{align}

(2.25) \begin{align} & Q_\alpha \overset{{{\mathbf{X}}_P}}{\longrightarrow} Q'{}^*(P)_\alpha + P'{}^*(Q)_\alpha ,= E_{u^\alpha}(P^\beta Q_\beta) , \end{align}

which coincide if $Q_\alpha$ is a conservation law multiplier. The symmetry action (2.10) is given by

(2.26) \begin{equation}Q_\alpha\overset{{{\mathbf{X}}_P}}{\longrightarrow} Q'(P)_\alpha -Q'{}^*(P)_\alpha ,\end{equation}

which is trivial if $Q_\alpha$ is a conservation law multiplier.

For the sequel, indices will be omitted for simplicity of notation wherever it is convenient.

2.6. Symplectic 2-form

The Noether operator defined by the symmetry action (2.26) is simply

(2.27) \begin{equation}{\mathcal{J}}=Q'-Q'{}^* = -{\mathcal{J}}^* ,\end{equation}

which is skew. It gives rise to a 2-form on the linear space of symmetries:

(2.28) \begin{equation}{\boldsymbol\omega}_Q(P_1,P_2) = \int_{{\mathbb{R}}^n} (P_1^\alpha Q'(P_2)_\alpha - P_2^\alpha Q'(P_1)_\alpha)\, d^nx .\end{equation}

This 2-form is symplectic, namely ${\mathbf{d}}{\boldsymbol\omega}_Q =0$ , as proven in [Reference Anco3].

The formal inverse of the Noether operator (2.27) defines a pre-Hamiltonian (inverse Noether) operator ${\mathcal{J}}^{-1}$ which maps adjoint-symmetries into symmetries. It also formally yields a Poisson bracket defined by

(2.29) \begin{equation}\{{F}_1,{F}_2\}_{{\mathcal{J}}^{-1}}\;:\!=\; \int_{{\mathbb{R}}^n} (\delta {F}_1/\delta u){\mathcal{J}}^{-1}(\delta {F}_2/\delta u) \, d^nx\end{equation}

for functionals ${F}=\int_{{\mathbb{R}}^n} f(x,u^{(k)})\,d^nx$ , where $\delta/\delta u$ denotes the variational derivative, namely $\delta{F}/\delta u^\alpha = E_{u^\alpha}(f)$ . In particular, the Jacobi identity for this bracket holds as a consequence of closure of the symplectic 2-form (see [Reference Anco3], and also [Reference Olver23] for a related general result).

3.1. Adjoint-symmetry bracket

The adjoint-symmetry bracket (2.13) arising from this symmetry action is defined via the dual operator

(3.9) \begin{equation}S_Q(P)=(E_u(P Q^{\textrm{t}}),E_v(P Q^{\textrm{t}})) .\end{equation}

To obtain the maximal domain, namely the whole linear space $\textrm{span}(Q_1,Q_2)$ , one can choose $Q=Q_2$ , whence

(3.10) \begin{equation}S_{Q_2}(P)=(E_u(x(P^u+P^v)),E_v(x(P^u+P^v))) .\end{equation}

Thereby, one has $\ker(S_{Q_2}) = \textrm{span}(P_1)$ , which is an ideal, and $\textrm{ran}(S_{Q_2}^{-1}) =\textrm{span}(P_2,P_3)$ modulo $\ker(S_{Q_2})$ . From Table 1, one then obtains

(3.11) \begin{equation}S_{Q_2}^{-1}(Q_1) = -P_2,\quad S_{Q_2}^{-1}(Q_2) = \frac{1}{1-p} P_3 ,\end{equation}

and thus the adjoint-symmetry bracket can be directly computed by

(3.12) \begin{equation}{}^{Q_2}[Q_1,Q_2] = S_{Q_2}\left(\left[-P_2, \frac{1}{1-p}P_3\right]\right) = \frac{p}{2(1-p)} S_{Q_2}(P_2)= \frac{p}{2(p-1)} Q_1\end{equation}

through the symmetry commutator (3.6).

This bracket (3.12), shown in Table 2, is a non-trivial Lie bracket. It is isomorphic to the symmetry subalgebra $\mathcal A=\textrm{span}(P_2,P_3)$ , which is generated by space translation and scaling. This correspondence between symmetries and adjoint-symmetries exists in the absence of any local variational structure (Hamiltonian or Lagrangian) for the reaction-diffusion equations (3.1).

Since the third symmetry action is trivial, both the corresponding Noether operator (2.27) and symplectic 2-form (2.28) are trivial. This is expected, as reaction-diffusion systems are inherently dissipative (namely, the linearised system is parabolic).

4.1. Adjoint-symmetry bracket

To obtain a maximal domain on which an adjoint-symmetry bracket (2.13) can be defined via the symmetry action (4.9), one seeks a maximal range for the dual operator

(4.10) \begin{equation}S_Q(P)=(E_u(P Q^{\textrm{t}}),E_\rho(P Q^{\textrm{t}})) .\end{equation}

From Table 3, it is clear that the maximal range will be the whole linear space of adjoint-symmetries, which is obtained for the choice $Q=Q_3+c_2 Q_2 +c_1Q_1$ , where one has

(4.11) \begin{equation}\ker(S_{Q_3+c_2 Q_2 +c_1 Q_1}) = \textrm{span}(P_3-c_2 P_2) .\end{equation}

The subalgebra (4.11) is not an ideal, since $[P_1,P_3-c_2 P_2] = P_2$ , and consequently, the resulting adjoint-symmetry bracket will depend on how $\textrm{coker}(S_{Q_3+c_2Q_2+c_1Q_1})$ is chosen in $\textrm{span}(P_1,P_2,P_3,P_4)$ . However, the scaling symmetry $P_4$ can be utilised (cf Section 2.4) to fix a canonical choice of $\textrm{coker}(S_{Q_3+c_2Q_2+c_1Q_1})$ as follows. From the symmetry commutators (4.7), observe that $\textrm{span}(P_3)$ has a different scaling weight compared to $\textrm{span}(P_1)$ and $\textrm{span}(P_2)$ . Also observe that $\textrm{span}(Q_1)$ , $\textrm{span}(Q_2)$ , and $\textrm{span}(Q_3)$ have different scaling weights. Hence, one can take $c_2=c_1=0$ , whereby $\ker(S_{Q_3}) = \textrm{span}(P_3)$ and $\textrm{coker}(S_{Q_3}) = \textrm{span}(P_4)\oplus\textrm{span}(P_1)\oplus\textrm{span}(P_2)$ provides a well-defined decomposition (2.14) of the symmetry Lie algebra under scaling. This determines $\textrm{ran}(S_{Q_3}^{-1}) =\textrm{span}(P_1,P_2,P_4)$ .

Table 3. Navier-Stokes equations: symmetry action (4.9) on adjoint-symmetries

Table 4. Navier-Stokes equations: adjoint-symmetry bracket

From Table 3, one now obtains

(4.12) \begin{equation}S_{Q_3}^{-1}(Q_1) = P_2,\quad S_{Q_3}^{-1}(Q_2) = -P_1,\quad S_{Q_3}^{-1}(Q_3) = \frac{q+1}{2q} P_4 .\end{equation}

Hence, the adjoint-symmetry bracket (2.13) can be directly computed by

(4.13a) \begin{align} {}^{Q_3}[Q_1,Q_2] &= S_{Q_3}([P_2, -P_1]) = S_{Q_3}(0) = 0, \end{align}

(4.13b) \begin{align}{}^{Q_3}[Q_1,Q_3] &= S_{Q_3}\left(\left[P_2, \frac{q+1}{2q}P_4\right]\right) = \frac{q+1}{2q} S_{Q_3}(P_2) = \frac{q+1}{2q} Q_1 , \end{align}

(4.13c) \begin{align} {}^{Q_3}[Q_2,Q_3] &= S_{Q_3}([-P_1, \frac{q+1}{2q}P_4]) = -S_{Q_3}(P_1) =Q_2 , \end{align}

through the symmetry commutator (4.7).

This bracket (3.12), shown in Table 4, is a non-trivial Lie bracket on the whole linear space $\textrm{span}(Q_1,Q_2,Q_3)$ of adjoint-symmetries. In particular, from the inverse of the dual operator (4.12), one sees that this Lie bracket structure is isomorphic to the symmetry subalgebra $\mathcal A=\textrm{span}(P_1,P_2,P_4)$ , which is generated by time translation, space translation, and scaling.

Remarkably, this correspondence between symmetries and adjoint-symmetries exists despite the lack of any local variational structure (Hamiltonian or Lagrangian) for the Navier-Stokes equations (4.1).

Finally, since the third symmetry action is trivial, both the corresponding Noether operator (2.27) and symplectic 2-form (2.28) are trivial. This is expected, as viscous fluid flow is inherently dissipative.

5.1. Symplectic 2-form and Hamiltonian operator

The third symmetry action encodes a Noether operator ${\mathcal{J}}$ (cf Section 2.3) for the Boussinesq system (5.2). Specifically, one has

(5.11) \begin{equation}{\mathcal{J}} = \frac{1}{3} (Q'_{\!\!6} +R^*_{Q_6}) = \begin{pmatrix} 0\;\;\;\; & -1 \\[5pt] 1\;\;\;\; & 0 \end{pmatrix} ,\end{equation}

which is obtained from

(5.12) \begin{equation}Q'_6 = \begin{pmatrix} 0\;\;\;\; & -2 -t D_t\;\;\;\; & \\[5pt] 1 + t D_t \end{pmatrix},\quad R^*_{Q_6} = \begin{pmatrix} 0\;\;\;\; & t D_t -1 \\[5pt] -t D_t +2\;\;\;\; & 0 \end{pmatrix} .\end{equation}

The factor $\tfrac{1}{3}$ has been inserted here as a convenient normalisation for the sequel.

There is a bilinear form (2.28) associated with the Noether operator (5.11):

(5.13) \begin{equation}{\boldsymbol\omega}_{Q_6}(P,\tilde P) = \int \tilde P {\mathcal{J}}(P)^{\textrm{t}} \, dx= \int (\tilde P^h P^u - \tilde P^u P^h)\, dx ,\end{equation}

where $P^u\partial_u +P^h\partial_h$ and $\tilde P^u\partial_u +\tilde P^h\partial_h$ are any pair of symmetries. This bilinear form is skew and closed (cf Section 2.6), whence it defines a symplectic 2-form. In particular, ${\mathcal{J}}$ is a symplectic operator.

The inverse of ${\mathcal{J}}$ defines a Hamiltonian operator

(5.14) \begin{equation}{\mathcal{H}} = {\mathcal{J}}^{-1} = \begin{pmatrix} 0\;\;\;\; & 1 \\[5pt] -1\;\;\;\; & 0 \end{pmatrix} .\end{equation}

This indicates that the Boussinesq system (5.2) has a Hamiltonian formulation. If the time-derivative terms in each equation in the system are combined on one side, then

(5.15) \begin{equation}(u_t - \Delta u_t,h_t -\Delta h_t)^{\textrm{t}}= {\mathcal{H}}(\delta \mathcal{E}/\delta u,\delta \mathcal{E}/\delta h)^{\textrm{t}}\end{equation}

yields a Hamiltonian formulation in terms of the conserved energy, $\mathcal E$ . This formulation can be expressed in the equivalent form

(5.16) \begin{equation}\begin{pmatrix} u \\[5pt] h \end{pmatrix}_t= (1-\Delta)^{-1}{\mathcal{H}}\begin{pmatrix} \delta \mathcal{E}/\delta u \\[5pt] \delta \mathcal{E}/\delta h\end{pmatrix},\end{equation}

where $(1-\Delta)^{-1}{\mathcal{H}}$ is also a Hamiltonian operator.

Thus, the symmetry action (6.17) in Table 8 involving the non-multiplier adjoint-symmetry directly encodes the Hamiltonian structure of the Boussinesq system (5.2). Moreover, because the Hamiltonian operator (5.14) is algebraic, it directly yields a Lagrangian structure:

(5.17) \begin{equation}{\mathcal{H}} \begin{pmatrix} G^u \\[5pt] G^h \end{pmatrix}= \begin{pmatrix} E_u(L) \\[5pt] E_h(L) \end{pmatrix},\quad L = (h - \Delta h) u_t +\frac{1}{2}( h^2 + h(u_x^2 +u_y^2),\end{equation}

where $G^u$ and $G^h$ denote the respective PDEs in the Boussinesq system (5.2). Here, $E_u$ and $E_h$ are the Euler operators with respect to u and h.

6.1. A nonlocal adjoint-symmetry and associated adjoint-symmetry brackets

From the symmetry action shown in Table 9, the dual action has $\ker S_Q(P)=\textrm{span}(P_1,P_2)$ for any choice of adjoint-symmetry Q. Hence, the cokernel, which is given by $\textrm{span}(P_3)$ , is only 1-dimensional. Since the dimension of the cokernel gives the dimension of the maximal domain on which a corresponding adjoint-symmetry bracket can be defined, the resulting bracket is trivial.

However, the situation changes when one considers the possibility of nonlocal adjoint-symmetries arising from potentials obtained via the mass conservation laws. These potentials are given by $u=U_x$ and $v=V_x$ , and they satisfy the coupled system

(6.7) \begin{equation}U_t + U_x V_x +U_{xxx} =0,\quad V_t + \frac{1}{2} U_x^2 + \kappa V_{xxx} =0 .\end{equation}

Adjoint-symmetries $(Q^U,Q^V)$ of this system are determined by the equations

(6.8a) \begin{align} ( {-}D_t Q^U -D_x( V_x Q^U + U_x Q^V) -D_x^3 Q^U )|_{{\mathcal{E}_{G}}} =0, \end{align}

(6.8b) \begin{align} ( {-}D_t Q^V - D_x(U_x Q^U) -\kappa D_x^3 Q^V )|_{{\mathcal{E}_{G}}} =0 . \end{align}

Note the relation

(6.9) \begin{equation}(Q^U,Q^V)=-D_x(Q^u,Q^v)\end{equation}

holds directly from the adjoint-symmetry equations (6.3) and (6.8).

The linear space of low-order adjoint-symmetries $(Q^U,Q^V)$ is generated by three adjoint-symmetries, two of which are inherited from the adjoint-symmetries $Q_3$ and $Q_4$ of the original system (6.1) for u, v, through the relation (6.9). The other low-order adjoint-symmetry is given by

(6.10) \begin{equation}(Q^U,Q^V) = -( 2U_x + xU_{xx} +3t U_{tx}, 2V_x +xV_{xx} + 3t V_{tx} ) .\end{equation}

Applying the inverse relation

(6.11) \begin{equation}(Q^u,Q^v)=-D_x^{-1}(Q^U,Q^V) ,\end{equation}

one obtains the nonlocal adjoint-symmetry

(6.12) \begin{equation}Q_5 =( U + xU_x +3t U_t, V +xV_x + 3t V_t )= \left( U + x u -3t(u v +\kappa u_{xx}), V +x v - 3t\left(\frac{1}{2} u^2 + v_{xx}\right) \right)\end{equation}

admitted by the system (6.1) for u, v. One can straightforwardly show that this adjoint-symmetry is not a multiplier.

When the first symmetry action, as given by the linear operator (3.8), is applied to $Q_5$ , one obtains

(6.13) \begin{equation}S_{P_1}(Q_5) = 5 Q_4,\quad S_{P_2}(Q_5) = -3 Q_3,\quad S_{P_3}(Q_5) = 0,\end{equation}

by using the variational derivative relations $E_u = -D_x^{-1} E_U$ and $E_v = -D_x^{-1} E_V$ . Consequently, if one chooses

(6.14) \begin{equation}Q=Q_5+c_2 Q_2 +c_1Q_1 \;:\!=\; Q_{5'}\end{equation}

with at least one of $c_1,c_2$ being non-zero, then $\ker S_Q(P)$ is empty, and so the cokernel consists of the whole linear space of Lie point symmetries, $\textrm{span}(P_1,P_2,P_3)$ . This choice of Q produces a maximal domain for defining the adjoint-symmetry bracket (2.13).

From equation (6.13) and Table 9, one obtains

(6.15) \begin{equation}\begin{aligned}&S_{P_1}(Q_{5'}) = S_{Q_{5'}}(P_1) = 5Q_4,\quad S_{P_2}(Q_{5'}) = S_{Q_{5'}}(P_2) = -3 Q_3 ,\\[5pt] &S_{P_3}(Q_{5'}) = S_{Q_{5'}}(P_3) = c_1 Q_1+c_2 Q_2 ,\end{aligned}\end{equation}

which yields

(6.16) \begin{equation}S_{Q_{5'}}^{-1}(c_1 Q_1+c_2 Q_2) = P_3,\quad S_{Q_{5'}}^{-1}(Q_3) = -\frac{1}{3} P_2,\quad S_{Q_{5'}}^{-1}(Q_4) = \frac{1}{5} P_1 .\end{equation}

The resulting adjoint-symmetry bracket (2.13) on the linear subspace $\textrm{span}(c_1 Q_1 + c_2 Q_2,Q_3,Q_4)$ is shown in Table 10. This bracket is a Lie bracket which is isomorphic to the symmetry algebra (6.5).

Table 10. Coupled solitary wave equations: adjoint-symmetry bracket from symmetry action (3.8) with $Q=Q_5 +c_2 Q_2 +c_1Q_1$

Since $Q_5$ is not a multiplier, the third symmetry action (2.26) is now non-trivial. In terms of components $Q=(Q^u,Q^v)$ and $P=(P^u,P^v)$ , the form of this symmetry action is given by the linear operator

(6.17) \begin{equation}S_P(Q) = \big( \textrm{pr}{\mathbf{X}}_f(Q^u) - E_u(Q f^{\textrm{t}}),\textrm{pr}{\mathbf{X}}_f(Q^v) - E_v(Q f^{\textrm{t}}) \big)\big|_{f=P}\end{equation}

with ${\mathbf{X}}_f = f^u(t,x)\partial_u + f^v(t,x)\partial_v$ , where $\textrm{pr}{\mathbf{X}}$ , $E_u$ , $E_v$ are regarded as operators in total derivatives when $f=(f^u(t,x),f^v(t,x))$ is replaced by $P=(P^u,P^v)$ . For $Q=Q_5$ , the resulting action is summarised in Table 11.

Table 11. Coupled solitary wave equations: symmetry action (6.17) on the nonlocal adjoint-symmetry (6.12)

The range of this action is the linear subspace of adjoint-symmetries $\textrm{span}(Q_3,Q_4,Q_5)$ , which provides a maximal domain for the adjoint-symmetry bracket (2.13) with $Q=Q_5$ , as shown in Table 12. The resulting bracket is a Lie bracket which is isomorphic to the symmetry algebra (6.5).

Table 12. Coupled solitary wave equations: adjoint-symmetry bracket from symmetry action (6.17) with $Q=Q_5$

6.2. Symplectic 2-form and Hamiltonian operator

The symmetry action (6.17) constructed in terms of the nonlocal adjoint-symmetry (6.12) encodes a Noether operator ${\mathcal{J}}$ (cf Section 2.3) for the coupled KdV equations (6.1). Specifically, one has ${\mathcal{J}}(P) = S_P(Q_5) = \big( \textrm{pr}{\mathbf{X}}_f Q_5^u - E_u(Q_5 f^{\textrm{t}}),\textrm{pr}{\mathbf{X}}_f Q_5^v - E_v(Q_5 f^{\textrm{t}}) \big)\big|_{f=P}$ , where

(6.18) \begin{equation}\begin{aligned}& ( \textrm{pr}{\mathbf{X}}_f Q_5^u, \textrm{pr}{\mathbf{X}}_f Q_5^v )|_{f=P} \\[5pt] & =\big( D_x^{-1}P^u + x P^u -3t(v P^u+u P^v +\kappa D_x^2 P^u),D_x^{-1} P^v +x P^v - 3t(u P^u + D_x^2 P^v) \big)\end{aligned}\end{equation}

and

(6.19) \begin{equation}\begin{aligned}& ( E_u(Q_5 f^{\textrm{t}}), E_v(Q_5 f^{\textrm{t}}) )|_{f=P} \\[5pt] & =\big( {-}D_x^{-1}P^u + x P^u -3t(v P^u+u P^v +\kappa D_x^2 P^u),-D_x^{-1} P^v +x P^v - 3t(u P^u + D_x^2 P^v) \big) .\end{aligned}\end{equation}

This yields, after scaling by a convenient normalisation factor $\tfrac{1}{2}$ ,

(6.20) \begin{equation}{\mathcal{J}} = \begin{pmatrix} D_x^{-1}\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & D_x^{-1} \end{pmatrix}\end{equation}

which actually is a symplectic operator. In particular, there is a bilinear form (2.28) associated with this operator,

(6.21) \begin{equation}{\boldsymbol\omega}_{Q_5}(P,\tilde P) = \int \tilde P {\mathcal{J}}(P)^{\textrm{t}} \, dx= \int (\tilde P^u D_x^{-1} P^u + \tilde P^v D_x^{-1} P^v)\, dx ,\end{equation}

where $P^u\partial_u +P^v\partial_v$ and $\tilde P^u\partial_u +\tilde P^v\partial_v$ are any pair of symmetries, and ${\textrm{t}}$ denotes the transpose. Modulo boundary terms, this bilinear form is skew and closed (cf Section 2.6), and hence it defines a symplectic 2-form.

The inverse of ${\mathcal{J}}$ defines a Hamiltonian operator

(6.22) \begin{equation}{\mathcal{H}} = {\mathcal{J}}^{-1} =\begin{pmatrix} D_x\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & D_x \end{pmatrix} .\end{equation}

As a consequence, the coupled KdV equations (6.1) have the Hamiltonian formulation

(6.23) \begin{equation}(u_t,v_t)^{\textrm{t}} = {\mathcal{H}}(\delta \mathcal{E}/\delta u,\delta \mathcal{E}/\delta v)^{\textrm{t}}\end{equation}

in terms of the conserved energy, $\mathcal E$ . From this formulation, one has $u_t = D_x(\delta \mathcal{E}/\delta u)$ and $v_t = D_x(\delta \mathcal{E}/\delta v)$ , which are analogous to the well-known first Hamiltonian structure of the KdV equation.

The symmetry action (6.17) involving the nonlocal adjoint-symmetry (6.12) thereby directly encodes the Hamiltonian structure of the coupled KdV equations (6.1).

7.1. Adjoint-symmetry brackets arising from a potential system

Symmetries (in evolutionary form) $P^v\partial_v$ of the potential equation (7.13) are determined by the equation

(7.14) \begin{equation}( D_t^2 P^v -2\beta r^{-2} D_t(v_t D_t P^v) -\alpha D_t^3 P^v -D_r^2 P^v+ 2D_r(r^{-1} P^v) )|_{{\mathcal{E}_{G^v}}} =0,\end{equation}

where ${{\mathcal{E}_{G^v}}}$ denotes the solution space of equation (7.13). Adjoint-symmetries $Q^v$ are determined by the adjoint equation

(7.15) \begin{equation}( D_t^2 Q^v -2\beta r^{-2} D_t(v_t D_t Q^v) +\alpha D_t^3 Q^v -D_r^2 Q^v - 2r^{-1} D_r Q^v) )|_{{\mathcal{E}_{G^v}}} =0 .\end{equation}

The linear space of Lie point symmetries of the potential equation (7.13) is generated by two shifts, in addition to the three point symmetries that are inherited from the acoustic wave equation (7.1) via the relation

(7.16) \begin{equation}P=r^{-2}D_t P^v .\end{equation}

In particular, these symmetries are given by

(7.17) \begin{equation}P^v_1 = r^2,\quad P^v_2= r,\quad P^v_3= v_t,\quad P^v_4 =2 t v_t + 3 r v_r - 6 v - \frac{1}{\beta}t\, r^2 ,\end{equation}

and, when $\alpha=0$ ,

(7.18) \begin{equation}P^v_5 =t v_t + r v_r - 3 v.\end{equation}

Their algebra has the non-zero commutators

(7.19) \begin{equation}\begin{aligned}& [P^v_1,P^v_5] = - P^v_1,\quad [P^v_2,P^v_4] = -3 P^v_2,\quad [P_2^v,P_5^v] = -2 P_2^v,\\[5pt] & [P_3^v,P_4^v] = \frac{1}{\beta} P_1^v -2 P_3^v ,\quad [P^v_3,P^v_5] = - P^v_3 .\end{aligned}\end{equation}

Note that the shifts $P^v_1$ and $P^v_2$ project onto trivial symmetries $P=0$ of the acoustic wave equation (7.1).

The linear space of low-order adjoint-symmetries is given by the basis

(7.20) \begin{equation}Q^v_1 = 1,\quad Q^v_2= r^{-1} ,\end{equation}

and, when $\alpha =0$ ,

(7.21a) \begin{align} &Q^v_3 =r^{-2} v_t,\quad Q^v_4 =r^{-2}(3 t v_t - 9 v) +7 r^{-1} v_r -4\frac{1}{\beta} t, \end{align}

(7.21b) \begin{align} &Q^v_5 = r^{-1} v_r - \frac{1}{\beta} t . \end{align}

One can easily check that the adjoint-symmetry (7.21b) is not a conservation law multiplier. The two adjoint-symmetries (7.21a) are multipliers for conservation of energy

(7.22) \begin{equation}\mathcal E = \int_0^\infty \left( \frac{1}{2} p^2 -\frac{2}{3}\beta p^3 +\frac{1}{2} r^{-4} v_r^2 - r^{-6} v^2 \right)\, r^2\,dr,\end{equation}

and a dilation energy

(7.23) \begin{align}\mathcal W & = \int_0^\infty \left( t\left(\frac{11}{2} p^2 -2\beta p^3 -4\frac{1}{\beta} p +\frac{3}{2}r^{-2} v_r^2 -3 r^{-6} v^2\right) \right. \nonumber \\ & \quad \left. +7 r^{-1}(p -\beta p^2) -\left(9p -9\beta p^2 -4\frac{1}{\beta}\right)v \right)\, r^2\,dr,\end{align}

respectively. Finally, the first two adjoint-symmetries (7.20) are inherited from $Q_3$ and $Q_4$ via the relation

(7.24) \begin{equation}Q^v =-r^{-2}D_t Q\end{equation}

which follows directly from the adjoint-symmetry determining equations (7.6) and (7.15). These inherited adjoint-symmetries are thus the respective multipliers for conservation of $\mathcal X$ and $\mathcal X^r$ . (Note that $\mathcal P$ and $\mathcal P^r$ are locally trivially when they are evaluated on ${{\mathcal{E}_{G^v}}}$ in terms of the potential v.)

The three actions of the symmetries (7.17)–(7.18) on the adjoint-symmetries (7.20)–(7.21) give rise to adjoint-symmetry brackets at the level of the potential equation (7.13). This structure will not be preserved under projection (7.16) and (7.24) back to the acoustic wave equation, since two of the potential symmetries are lost, in contrast to the example of the coupled solitary wave equations considered in the previous section.

The first and second actions (2.8), (2.9) here differ only on the non-multiplier $Q_5$ . They are computed from the symmetry and adjoint-symmetry expressions by use of:

(7.25) \begin{equation}R_{P^v_1} = R_{P^v_2} = 0,\quad R_{P^v_3} = D_t ,\quad R_{P^v_4} = 2t D_t + 3r D_r ,\quad R_{P^v_5} = t D_t + r D_r - 1 ,\end{equation}

and

(7.26) \begin{equation}R_{Q^v_1} =R_{Q^v_2} = 0,\quad R_{Q^v_3} = r^-2 D_t,\quad R_{Q^v_4} = r^{-2}( 3t D_t + 7r D_r + 5 ),\quad R_{Q^v_5} = r^{-2} (r D_r + 2) .\end{equation}

This leads to the results shown in Tables 13 and 14.

Table 13. Acoustic potential equation: symmetry action (2.8) on adjoint-symmetries

Table 14. Acoustic potential equation: symmetry action (2.9) on adjoint-symmetries

The third action (2.10), which is shown in Table 15, is non-trivial only on the non-multiplier $Q_5$ .

Table 15. Acoustic potential equation: symmetry action (2.10) on the non-multiplier adjoint-symmetry

Now, the adjoint-symmetry brackets arising from the dual actions (2.11) will be considered. The maximal domain for defining a bracket is given by having a maximal range for the dual action, and, therefore, a minimal dimension for the kernel which also must be an ideal in the symmetry algebra. From the symmetry commutators (7.19), the ideals with smallest dimension consist of $\textrm{span}(P^v_1)$ , $\textrm{span}(P^v_2)$ , which are 1-dimensional.

For the first symmetry action, shown in Table 13, the dual action $S_{Q^v}(P^v)$ given by $Q^v=\sum_{i=1..5} c_i Q^v_i$ will have $\textrm{span}(P^v_1)$ as the kernel if $c_5=1$ , $c_4= -\tfrac{2}{5}$ , and $\textrm{span}(P^v_2)$ as the kernel if $c_5=1$ , $c_4= \tfrac{1}{2}$ . In both cases, the range of $S_{Q^v}(P^v)$ is contained in $\textrm{span}(Q^v_1,Q^v_2,Q^v_3,Q^v_5 +c_4 Q^v_4)$ , which will be the maximal domain for the resulting adjoint-symmetry bracket. This yields the brackets shown in Tables 16 and 17.

Table 16. Acoustic potential equation: adjoint-symmetry bracket from symmetry action (2.8) with $Q^v=Q^v_5 -\tfrac{2}{5} Q^v_4 +c_3 Q^v_3+c_2 Q^v_2 +c_1 Q^v_1$ , where $Q^v_{4'}= \lambda(Q^v_5 -\tfrac{2}{5} Q^v_4)$ , $Q^v_{3'}=\lambda Q^v_3$ , $Q^v_{2'}=\lambda Q^v_2$ , $Q^v_{1'} = \lambda(Q^v_1 -2\beta Q^v_3)$ , $\lambda = c_3 +2\beta c_1$

Table 17. Acoustic potential equation: adjoint-symmetry bracket from symmetry action (2.8) with $Q^v=Q^v_5 +\tfrac{1}{2} Q^v_4 +c_3 Q^v_3+ c_2 Q^v_2 +c_1 Q^v_1$ , where $Q^v_{4'}= Q^v_5 +\tfrac{1}{2} Q^v_4$ , $Q^v_{3'}= Q^v_3$ , $Q^v_{2'}= c_2 Q^v_2$ , $Q^v_{1'} = Q^v_1$

For the second symmetry action, shown in Table 14, the dual action $S_{Q^v}(P^v)$ given by $Q^v=\sum_{i=1..5} c_i Q^v_i$ never has $\textrm{span}(P^v_1)$ or $\textrm{span}(P^v_2)$ as the kernel. In particular, in the first case, one needs $c_5=1$ , $c_4= \tfrac{1}{5}$ , but then the kernel also contains $\textrm{span}(P^v_5 -\tfrac{3}{5} P^v_4 -\tfrac{c_3}{3}P^v_3 -\tfrac{c_2}{3}P^v_2)$ ; and in the second case, one needs $c_5=1$ , $c_4= -1$ , whereby the kernel then also contains $\textrm{span}(P^v_3 -\tfrac{1}{2\beta}P^v_1,P^v_5 -P^v_4+\tfrac{c_3+2\beta c_1}{6\beta}P^v_1)$ . In both cases, the kernel is at least 2-dimensional, and hence, the maximal domain on which an adjoint-symmetry bracket can be defined is at most 3-dimensional, which is no larger than the domain arising at the level of the acoustic wave equation.

Finally, for the third symmetry action, which appears in Table 15, the dual action with $Q^v=Q^v_5$ has an empty kernel, whence an adjoint-symmetry bracket is obtained on the whole space of adjoint-symmetries $\textrm{span}(Q^v_1,Q^v_2,Q^v_3,Q^v_4,Q^v_5)$ . This bracket is shown in Table 18, and one sees that it is isomorphic to symmetry algebra through the correspondence $Q^v_1 \leftrightarrow P^v_1$ , $Q^v_2 \leftrightarrow P^v_2$ , $Q^v_3 \leftrightarrow P^v_3$ , $Q^v_4 \leftrightarrow 4 P^v_4 - 5 P^v_5$ , $Q^v_5 \leftrightarrow P^v_4 - 2 P^v_5 $ .

Table 18. Acoustic potential equation: adjoint-symmetry bracket from symmetry action (2.10) with $Q^v=Q^v_5$

7.2. Noether operator and variational structure

The third symmetry action (2.10) encodes a Noether operator ${\mathcal{J}}$ for the acoustic potential equation (7.12) with $\alpha=0$ , namely when there is no damping.

This operator is given by ${\mathcal{J}}^v = Q^v_5{}' +R^*_{Q^v_5} = 3 r^{-2}$ , since one has

(7.27) \begin{equation}Q^v_5{}' = r^{-1} D_r,\quad R^*_{Q^v_5} = r^{-2}(3 - rD_r) .\end{equation}

For convenience, a factor $\tfrac{1}{3}$ will be inserted in hereafter so that

(7.28) \begin{equation}{\mathcal{J}}^v = r^{-2} .\end{equation}

The Noether operator (7.28) provides a mapping of symmetries into adjoint-symmetries:

(7.29) \begin{equation}{\mathcal{J}}^v(P^v) = r^{-2} P^v = Q^v .\end{equation}

Specifically, in terms of the respective symmetry basis (7.17)–(7.18) and adjoint-symmetry basis (7.20)–(7.21), one sees that ${\mathcal{J}}^v(P^v_1)=Q^v_1$ , ${\mathcal{J}}^v(P^v_2)=Q^v_2$ , ${\mathcal{J}}^v(P^v_3)=Q^v_3$ , ${\mathcal{J}}^v(P^v_4)=\tfrac{2}{3}Q^v_4 -\tfrac{5}{3}Q^v_5$ , ${\mathcal{J}}^v(P^v_5)=\tfrac{1}{3}Q^v_4 -\tfrac{4}{3}Q^v_5$ .

As the Noether operator is algebraic, it yields a Lagrangian structure:

(7.30) \begin{equation}{\mathcal{J}}^v(G^v) = E_v(L),\end{equation}

where $G^v$ denotes the potential equation (7.12) with $\alpha =0$ , and where the Lagrangian is straightforwardly found to be

(7.31) \begin{equation}L = \frac{1}{2} r^{-2}(v_r^2 -v_t^2) +r^{-4}\left(\frac{1}{3}\beta v_t^3 - v^2\right) .\end{equation}

Here, $E_v$ is the Euler operator with respect to v.

The preceding structure can be lifted to the acoustic wave equation (7.1) through the relations (7.24) and

(7.32) \begin{equation}r^{-2}D_t G^v = G,\end{equation}

where G denotes the PDE (7.1) with $\alpha=0$ . These relations imply that

(7.33) \begin{equation}{\mathcal{J}} = -r^2 D_t^{-1} {\mathcal{J}}^v r^2 D_t^{-1} = -r^2 D_t^{-2}\end{equation}

defines a Noether operator for the acoustic wave equation (7.1) with no damping, $\alpha=0$ . In particular, using the form of the potential $p = r^{-2} v_t$ , one has

(7.34a) \begin{align} {\mathcal{J}}(P_1) & =-r^2 D_t^{-2}(p_t) = -v , \end{align}

(7.34b) \begin{align} {\mathcal{J}}(P_2) & =-r^2 D_t^{-2}\left(2p + 2t p_t +3r p_r -\frac{1}{\beta}\right)= \frac{1}{\beta} t^2 r^2 -2 t v_t - 3r \partial_t^{-1}v_r +8 \partial_t^{-1}v , \end{align}

(7.34c) \begin{align} {\mathcal{J}}(P_3) & =-r^2 D_t^{-2}(t p_t +r p_r)= v - \partial_t^{-1}(t v) -r\partial_t^{-1}v_r + 2\partial_t^{-1}v ,\end{align}

all of which can be verified to be nonlocal adjoint-symmetries Q.

The Lagrangian structure becomes

(7.35) \begin{equation}{\mathcal{J}}(G) = E_p(L)\end{equation}

through the variational derivative relation $\delta/\delta v = -r^{-2}D_t \delta/\delta p$ , where L is a nonlocal expression (7.31) in terms of p:

(7.36) \begin{equation}L = r^2 \left( \frac{1}{2} \left(\partial_t^{-1} p_r\right)^2 -\frac{1}{2} p^2 + \frac{1}{3}\beta p^3 \right) -(\partial_t^{-1} p)^2 .\end{equation}

Thus, the symmetry action in Table 15 involving the non-multiplier adjoint-symmetry directly encodes a variational structure for the undamped acoustic wave equation.

7.3. Hamiltonian structure

The variational structure can also be lifted to the level of the first-layer potential system (7.10). Denote the PDEs in this system by $G^1= p_r - r^{-2} u_t$ and $G^2=w_r - r^2(p-\beta p^2)_t$ , which satisfy the relation

(7.37) \begin{equation}D_r (r^2 G^1) + D_t G^2 = -D_t G^v = -r^2 G .\end{equation}

The determining equations for an adjoint-symmetry $(Q^1,Q^2)$ of the potential system $(G^1,G^2)=0$ are given by

(7.38) \begin{equation}({-}D_x Q^1 + r^2 (1-2\beta p) D_t Q^2 )|_{{\mathcal{E}_{(G^1,G^2)}}} =0,\quad ({-}D_x Q^2 + r^{-2} D_t Q^1 )|_{{\mathcal{E}_{(G^1,G^2)}}} =0 .\end{equation}

From the PDE relation (7.37), one can show that

(7.39) \begin{equation}Q^1 = -r^2 D_r D_t^{-1} Q^v = r^2 D_r (r^{-2} Q),\quad Q^2 = -Q^v = r^2 D_t Q .\end{equation}

Similarly, a symmetry $P\partial_p + P^u\partial_u$ of the potential system satisfies

(7.40) \begin{equation}P = r^{-2} D_t P^v,\quad P^u = r^2 D_r (r^{-2} P^v) .\end{equation}

Now consider the non-multiplier adjoint-symmetry (7.21b). The corresponding adjoint-symmetry $(Q^1,Q^2)$ of the potential system $(G^1,G^2)=0$ is given by

(7.41) \begin{equation}\begin{aligned}Q^1 & = -r^2 D_r D_t^{-1}\left(r^{-1} v_r - \frac{1}{\beta} t \right)= -(\partial_t^{-1} u + r^3 (p-\beta p^2)),\\[5pt] Q^2 & = -\left(r^{-1} v_r - \frac{1}{\beta} t \right)= -(r^{-1} u + 2 \partial_t^{-1} p) + \frac{1}{\beta} t,\end{aligned}\end{equation}

with the use of equations (7.10) and (7.11) for the potentials. A straightforward computation then yields

(7.42) \begin{equation}Q^1{}'(P,P^u) = -(D_t^{-1} P^u +r^3(1-2\beta p)P),\quad Q^2{}'(P,P^u) = -(r^{-1} P^u + 2 D_t^{-1} P) ,\end{equation}

and

(7.43) \begin{equation}R_{Q^1}^*(P,P^u) = r^3 (1-2\beta p) P,\quad R_{Q^2}^*(P,P^u) = D_t^{-1} P +r^{-1} P^u .\end{equation}

Hence, one obtains

(7.44) \begin{equation}\begin{pmatrix}Q^1{}'(P,P^u) +R_{Q^1}^*(P,P^u) \\[5pt] Q^2{}'(P,P^u) +R_{Q^2}^*(P,P^u)\end{pmatrix}= \begin{pmatrix} -D_t^{-1} P^u \\[5pt] -D_t^{-1} P \end{pmatrix}= {\mathcal{J}}\begin{pmatrix} P \\[5pt] P^u \end{pmatrix} ,\end{equation}

giving the Noether operator

(7.45) \begin{equation}{\mathcal{J}} = -\begin{pmatrix} 0\;\;\;\; & D_t^{-1} \\[5pt] D_t^{-1}\;\;\;\; & 0 \end{pmatrix} .\end{equation}

The equations in the potential system have the following variational structure in terms of the inverse of the Noether operator (7.45),

(7.46) \begin{equation}{\mathcal{J}}^{-1} = - \begin{pmatrix} 0\;\;\;\; & D_t \\[5pt] D_t\;\;\;\; & 0 \end{pmatrix} .\end{equation}

One sees that

(7.47) \begin{equation}(p_r,u_r)^{\textrm{t}}= -{\mathcal{J}}^{-1}(\partial_p E,\partial_u E)^{\textrm{t}},\end{equation}

where

(7.48) \begin{equation}E = r^2 \left(\frac{1}{2}p^2 -\frac{1}{3}\beta p^3\right) + r^{-2}\frac{1}{2} u^2 .\end{equation}

This structure is, formally, a Hamiltonian formulation in which ${\mathcal{J}}^{-1}$ and E respectively play the roles of the Hamiltonian operator and the Hamiltonian density, where r represents the ‘time’ coordinate for the evolution and t represents the ‘spatial’ coordinate with respect to which ${\mathcal{J}}^{-1}$ is skew-adjoint. Also, E is in fact the density of the energy integral (7.22) expressed in terms of p and u: $\mathcal E = \int_0^\infty \big( r^2 (\tfrac{1}{2}p^2 -\tfrac{1}{3}\beta p^3) + r^{-2}\tfrac{1}{2} u^2 \big)\,dr$ .