2.1. Determining equations and identities

An infinitesimal symmetry of a PDE system (2.1) is a set of functions $P^\alpha(x,u^{(k)})$ that are non-singular on ${\mathcal{E}}$ and satisfy

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

This is the determining equation for $P^\alpha$ , called the characteristic functions of the symmetry.

Off of the solution space ${\mathcal{E}}$ , the symmetry determining equation is given by

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

(due to Lemma 2.1) where $R_P=(R_P)^{A\,I}_{B} D_I$ is some linear differential operator in total derivatives whose coefficients $(R_P)^{A\,I}_{B}$ are functions that are non-singular on ${\mathcal{E}}$ .

The determining equation for adjoint-symmetries is the adjoint of the symmetry determining equation (2.3). It is obtained by using the Frechet derivative identity

(2.5) \begin{equation}Q_A G'(P)^A = P^\alpha G'{}^*(Q)_\alpha + D_i\Psi^i(P,Q) .\end{equation}

There is an explicit expression for $\Psi^i$ in terms of $G^A$ (see [Reference Anco and Melnik3] and references therein).

An adjoint-symmetry of a PDE system (2.1) is a set of functions $Q_A(x,u^{(k)})$ that are non-singular on ${\mathcal{E}}$ and satisfy

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

Off of the solution space ${\mathcal{E}}$ , this determining equation is given by

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

(again due to Lemma 2.1) where $R_Q=(R_Q)_{\alpha\,B}^{I} D_I$ is some linear differential operator in total derivatives whose coefficients $(R_Q)_{\alpha\,B}^{I}$ are functions that are non-singular on ${\mathcal{E}}$ .

The geometrical meaning of symmetries is well-known. From the algebraic viewpoint, it comes from the relation $G'(P){}^A = ({\rm pr} P^\alpha\partial_{u^\alpha}) G^A$ whereby the symmetry determining equation (2.3) can be expressed as

(2.8) \begin{equation}(({\rm pr} P^\alpha\partial_{u^\alpha})G^A)|_{\mathcal{E}} =0 .\end{equation}

This is usually the starting point for defining symmetries, since it indicates that ${\textbf{X}}_P = P^\alpha\partial_{u^\alpha}$ is a vector field that is tangent to surfaces $G^A=0$ (and their prolongations $D^k G^A=0$ , $k=0,1,2,\ldots$ ) in jet space. A geometrical meaning for adjoint-symmetries has recently been developed in [Reference Anco and Wang10], based on evolutionary 1-forms $Q_A{\textbf{d}} G^A$ that functionally vanish on the solution space ${\mathcal{E}}$ .

The most common form encountered for symmetries is a Lie point symmetry [Reference Bluman, Cheviakov and Anco13, Reference Olver23], given by $P^\alpha = \eta^\alpha(x,u) -\xi^i(x,u) u^\alpha_i$ . Symmetries that have a general form $P^\alpha(x,u^{(k)})$ with $k\geq 1$ are sometimes called generalised symmetries or symmetries of order k.

The most common form for adjoint-symmetries is given by $Q_A(x,u^{(k)})$ with $k<N$ , where N is the differential order of a given PDE system (2.1). Such adjoint-symmetries are called low-order [Reference Anco and Melnik3, Reference Anco and Kara8].

A symmetry or an adjoint-symmetry is called higher-order if has a differential order $k>N$ . Existence of an infinite hierarchy with k being unbounded is typically an indicator of integrability [Reference Mikhailov, Shabat and Sokolov22, Reference Olver23].

Running example: For the p-gKdV equation (2.2), the symmetry and adjoint-symmetry determining equations are respectively given by

(2.9) \begin{equation}(D_t P + u_x^p D_x P + D_x^3 P)|_{\mathcal{E}} =0,\quad(\!-D_t Q -D_x(u_x^p Q) - D_x^3 Q)|_{\mathcal{E}} =0 .\end{equation}

These equations are adjoints of each other. Since they do not coincide when $P=Q$ with $p\neq0$ , this shows that p-gKdV adjoint-symmetries differ from p-gKdV symmetries. It is well-known that, for arbitrary $p>0$ , the Lie point symmetries are spanned by

(2.10) \begin{equation}P_1=1,\quad P_2=-u_x,\quad\;P_3=-u_t,\quad\; P_4=(p-2)u -3p t u_t -xp u_x,\end{equation}

which respectively generate shifts, space-translations, time-translations and scalings. They satisfy

(2.11) \begin{equation}R_{P_1} = 0,\quad\;R_{P_2} = -D_x,\quad\;R_{P_3} = -D_t,\quad\;R_{P_4} = -p x D_x -3p t D_t - 2(p+1)\end{equation}

off of ${\mathcal{E}}$ . The low-order adjoint-symmetries can be shown to be spanned by

(2.12) \begin{equation}Q_1=u_{xx},\quad\;Q_2=u_{tx},\quad\;Q_3=2u_x + 3p t u_{tx} + p x u_{xx}\end{equation}

where

(2.13) \begin{equation}R_{Q_1} =-D_x^2,\quad\;R_{Q_2} = -D_tD_x,\quad\;R_{Q_3} = -p x D_x^2 -3p t D_tD_x - (3p+2)D_x .\end{equation}

In the special cases, $p=1,2$ , a hierarchy of higher-order symmetries and adjoint-symmetries exist, corresponding to the integrability structure of the KdV and mKdV equations. (No integrability structure is known for any other values of $p\neq0$ .)

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

(2.14) \begin{equation}E_{u^\alpha}(\Lambda_A G^A)=0 .\end{equation}

It can be further reformulated through the product rule of the Euler operator, which yields the equivalent condition $\Lambda'{}^*(G)_\alpha + G'{}^*(\Lambda)_\alpha =0$ . Consequently, on ${\mathcal{E}}$ ,

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

whereby $\Lambda_A$ is an adjoint-symmetry. Off of ${\mathcal{E}}$ , the adjoint-symmetry determining equation (2.7) yields

(2.16) \begin{equation}G'{}^*(\Lambda)_\alpha = R_\Lambda(G)_\alpha\end{equation}

where $R_\Lambda$ is a linear differential operator in total derivatives. Hence, one sees that $\Lambda'{}^*(G)_\alpha =- G'{}^*(\Lambda)_\alpha =-R_\Lambda(G)_\alpha$ . Now suppose that $G^A=0$ does not obey any differential identities. Then one can conclude (from Lemma 2.2) that $\Lambda'{}^* =-R_\Lambda + S^{I,J} (D_I G) D_J$ where $S^{I,J}=-S^{J,I}$ holds off of ${\mathcal{E}}$ and $S^{I,J}$ is non-singular on ${\mathcal{E}}$ . Furthermore, suppose that $\Lambda_A$ contains no leading derivatives of $G^A=0$ and no differential consequences of any leading derivatives. Then, one can assume without loss of generality that $S^{I,J}=0$ . Therefore, in this situation, $\Lambda'{}^*=-R_\Lambda$ holds identically. The adjoint of this equation yields the relation

(2.17) \begin{equation}\Lambda'{}=-R_\Lambda^* .\end{equation}

Every multiplier $\Lambda_A(x,u^{(k)})$ of a PDE system determines a conservation law $(D_i\Psi^i)|_{\mathcal{E}}=0$ holding on the solution space ${\mathcal{E}}$ . The components $\Psi^i$ can obtained from $\Lambda_A$ by homotopy integral formulas [Reference Anco and Melnik3, Reference Bluman, Cheviakov and Anco13, Reference Olver23] or by an algebraic formula when the given PDE system possesses a scaling symmetry [Reference Anco and Melnik3, Reference Anco1]. When a PDE system is regular, all conservation laws will arise from multipliers [Reference Anco and Melnik3].

Running example: The low-order multipliers of the p-gKdV equation (2.2) consist of the span of a subset of the low-order adjoint-symmetries:

(2.18) \begin{equation}\Lambda_1 = Q_1 = u_{xx},\quad\Lambda_2 =Q_2 = u_{tx} .\end{equation}

In particular, the adjoint-symmetry $Q_3=2u_x + 3p t u_{t,x} + p x u_{xx}$ is not a multiplier. The conservation laws arising from the two multipliers are respectively given by

(2.19) \begin{equation}\left(\Psi^t,\Psi^x\right)=\left(-\frac{1}{2} u_x{}^2, \frac{1}{2}u_{xx}{}^2 + u_t u_x +\frac{1}{(p+1)(p+2)} u_x{}^{p+1}\right)\end{equation}

and

(2.20) \begin{equation}\left(\Psi^t,\Psi^x\right)=\left(-\frac{1}{2}u_{xx}{}^2 + \frac{1}{(p+1)(p+2)} u_x{}^{p+2},u_{tx}u_{xx}+\frac{1}{2} u_x{}^2\right) .\end{equation}

These describe continuity equations for momentum and energy, which can be seen from the form of the conserved densities $\Psi^t=\frac{1}{2}v^2,\frac{1}{2}v_x{}^2 -\frac{1}{(p+1)(p+2)} v{}^{p+2}$ (up to an overall sign) expressed in terms of the gKdV variable $v=u_x$ (see e.g. [Reference Anco, Nayeri and Recio9]).

3.1. Symmetry action on multipliers

The action of a symmetry vector field ${\textbf{X}}_P = P^\alpha\partial_{u^\alpha}$ on the multiplier equation $\Lambda_A G^A = D_i\Psi^i$ yields, for the righthand side,

(3.14) \begin{equation}{\rm pr}{\textbf{X}}_P D_i\Psi^i = D_i({\rm pr}{\textbf{X}}_P \Psi^i),\end{equation}

while for the lefthand side, ${\rm pr}{\textbf{X}}_P(\Lambda_A G^A) = \Lambda'(P)_A G^A + \Lambda_A G'(P)^A$ . The last term can be simplified by using the symmetry equation (2.4) off of ${\mathcal{E}}$ :

(3.15) \begin{equation}\Lambda_A G'(P)^A= \Lambda_A R_P(G)^A=R_P^*(\Lambda)_A G^A + D_i F^i .\end{equation}

Thus,

(3.16) \begin{equation}{\rm pr}{\textbf{X}}_P(\Lambda_A G^A)= (\Lambda'(P)_A + R_P^*(\Lambda)_A)G^A\quad\;\textrm{modulo total derivatives.}\end{equation}

Now, from equating expressions (3.16) and (3.14), one concludes that $(\Lambda'(P)_A + R_P^*(\Lambda)_A)G^A$ is a total derivative. Therefore, $\Lambda'(P)_A + R_P^*(\Lambda)_A$ is a multiplier.

This yields the following well-known action [Reference Anco1, Reference Anco and Kara8]:

(3.17) \begin{equation}\Lambda_A \overset{{\textbf{X}}_P}{\longrightarrow} \Lambda'(P)_A + R_P^*(\Lambda)_A .\end{equation}

Theorem 3.1 shows that this action extends from conservation law multipliers to adjoint-symmetries through the symmetry action (3.8) on adjoint-symmetries.

3.2. Action of Lie point symmetries

An explicit expression for the first symmetry action (3.8) in Theorem 3.1 can be derived in the case of Lie point symmetries.

A Lie point symmetry vector field has the form [Reference Bluman, Cheviakov and Anco13, Reference Olver23]

(3.18) \begin{equation}{\textbf{X}}_{\rm p} = P_{\rm p}^\alpha\partial_{u^\alpha} ,\quad\;P_{\rm p}^\alpha = \eta^\alpha(x,u) -\xi^i(x,u) u^\alpha_i,\end{equation}

which generates a point transformation group acting on the space (x,u), as given by exponentiation of the corresponding canonical vector field

(3.19) \begin{equation}{\textbf{Y}}_{\rm p} = \xi^i \partial_{x^i} + \eta^\alpha\partial_{u^\alpha} .\end{equation}

The prolongations of these vector fields are related by [Reference Bluman, Cheviakov and Anco13, Reference Olver23]

(3.20) \begin{equation}{\rm pr}{\textbf{Y}}_{\rm p} = \xi^i D_i +{\rm pr}{\textbf{X}} .\end{equation}

A function $F(x,u^{(k)})$ is symmetry invariant iff ${\rm pr}{\textbf{Y}}_{\rm p} F$ vanishes identically. More generally, a function $F(x,u^{(k)})$ is symmetry homogeneous iff ${\rm pr}{\textbf{Y}}_{\rm p} F = \sigma_F F$ holds identically for some function $\sigma_F(x,u)$ .

The symmetry determining equation (2.4) for Lie point symmetries can be expressed as

(3.21) \begin{equation}{\rm pr}{\textbf{Y}}_{\rm p}(G)= R_{\rm p}(G)\end{equation}

where $R_{\rm p}=(R_{\rm p})^{A\,I}_{B} D_I$ is some linear differential operator in total derivatives whose coefficients $(R_{\rm p})^{A\,I}_{B}$ are functions that are non-singular on ${\mathcal{E}}$ . When every PDE in the system $G^A=0$ has the same differential order, and the system has no differential identities, then $R_{\rm p}$ will be purely algebraic, namely $(R_{\rm p})^{A\,I}_{B}$ vanishes for $I\neq\emptyset$ .

Proposition 3.3. The first symmetry action (3.8) for a Lie point symmetry (3.18) on an adjoint-symmetry is given by

(3.22) \begin{equation}Q_A \overset{{{\textbf{X}}_{\rm p}}}{\longrightarrow}{\textbf{Y}}_p(Q)_A + R_{\rm p}^*(Q)_A +(D_i\xi^i)Q_A\end{equation}

where $R_{\rm p}^*$ is the adjoint of $R_{\rm p}$ .

The proof is a straightforward computation of the terms $Q'(P_{\rm p})_A +R_{P_{\rm p}}^*(Q)_A$ in the action (3.8). One has $Q'(P_{\rm p})_A = {\rm pr}{\textbf{Y}}_{\rm p}(Q)_A - \xi^iD_i Q_A$ and $R_{P_{\rm p}}(G)^A= R_{\rm p}(G)^A - \xi^i D_i G^A$ from identity (3.20). Hence, $R_{P_{\rm p}}^*(Q)_A= R_{\rm p}^*(Q)_A +D_i(\xi^i Q_A)$ , and thus after cancellation of terms, one obtains the action (3.22).

Similar explicit expressions can be obtained for the other two symmetry actions (3.11), (3.12) in Theorem 3.1 in the case of adjoint-symmetries with a first-order linear form

(3.23) \begin{equation}Q_A = \kappa_A(x,u) + \rho^i_{A\alpha}(x,u) u^\alpha_i .\end{equation}

This form is a counterpart of Lie point symmetries (more generally, first-order linear symmetries). The adjoint-symmetry determining equation (2.7) implies that

(3.24) \begin{equation}G'{}^*(Q)_\alpha = \rho^i_{A\alpha} D_i G^A + K_{A\alpha} G^A\end{equation}

for some functions $K_{A\alpha}$ that are non-singular on ${\mathcal{E}}$ , when every PDE in the system $G^A=0$ has the same differential order, and the system has no differential identities.

This leads to the following result.

Proposition 3.4. For a Lie point symmetry (3.18), the second and third symmetry actions (3.11) and (3.12) on a first-order linear adjoint-symmetry (3.23)–(3.24) are given by

(3.25) \begin{align} Q_A \overset{{{\textbf{X}}_{\rm p}}}{\longrightarrow}R_{\rm p}^*(Q)_A+ u^\alpha_j D_i( 2\xi^{[i} \rho^{j]}_{A\alpha})+D_i(\xi^i\kappa_A+\rho^i_{A\alpha} \eta^\alpha)-K_{A\alpha}(\eta^\alpha - \xi^i u^\alpha_i),\end{align}

(3.26) \begin{align} Q_A \overset{{{\textbf{X}}_{\rm p}}}{\longrightarrow}{\textbf{Y}}_{\rm p}(Q)_A +(D_i\xi^i) Q_A-u^\alpha_j D_i(2\xi^{[i} \rho^{j]}_{A\alpha})-D_i(\xi^i\kappa_A +\rho^i_{A\alpha} \eta^\alpha)+ K_{A\alpha}(\eta^\alpha - \xi^i u^\alpha_i) ,\end{align}

where $R_{\rm p}^*$ is the adjoint of $R_{\rm p}$ .

The proof is similar to that for the action (3.8). One has $R_{P_{\rm p}}^*(Q)_A= R_{\rm p}^*(Q)_A +D_i(\xi^i Q_A)$ , where $D_i(\xi^i Q_A) = D_i(\xi^i\kappa_A) + D_i(\xi^i\rho^j_{A\alpha}) u^\alpha_j+ \xi^i\rho^j_{A\alpha} u^\alpha_{ij}$ . Next, from relation (3.24), one obtains $R_Q^*(P_{\rm p})_A = K_{A\alpha} P_{\rm p}^\alpha -D_i(\rho^i_{A\alpha} P_{\rm p}^\alpha)$ where $D_i(\rho^i_{A\alpha} P_{\rm p}^\alpha) =D_i(\rho^i_{A\alpha} \eta^\alpha) - D_i(\rho^i_{A\alpha} \xi^j) u^\alpha_j- \rho^i_{A\alpha} \xi^j u^\alpha_{ij}$ and $K_{A\alpha} P_{\rm p}^\alpha = K_{A\alpha}(\eta^\alpha - \xi^i u^\alpha_i)$ . Then, combining the terms $R_{P_{\rm p}}^*(Q)_A -R_Q^*(P_{\rm p})_A$ , one gets expression (3.25). Likewise, combining the terms $Q'(P_{\rm p})_A+ R_Q^*(P_{\rm p})_A$ yields expression (3.26).

Two basic types of Lie point symmetries which appear in numerous applications are translations ${\textbf{Y}}_{\rm trans.} = a^i\partial_{x^i}$ and scalings ${\textbf{Y}}_{\rm scal.} = w_{(i)} x^i \partial_{x^i} + w_{(\alpha)} u^\alpha \partial_{u^\alpha}$ . Here, the vector $a^i$ represents the direction of the translation; the scalars $w_{(\alpha)}, w_{(i)}$ represent the scaling weights of $u^\alpha$ and $x^i$ . The corresponding evolutionary form of these symmetries is given by

(3.27) \begin{equation}P_{\rm trans.}^\alpha = -a^i u^\alpha_i\end{equation}

and

(3.28) \begin{equation}P_{\rm scal.} = w^{(\alpha)} u^\alpha -w^{(i)} x^i u^\alpha_i .\end{equation}

Their action on adjoint-symmetries has a very simple form, which is an immediate consequence of Propositions 3.3 and 3.4.

Corollary 3.5. (i) Suppose $Q_A$ and $G^A$ are translation invariant: ${\textbf{Y}}_{\rm trans.}(Q)_A=0$ and ${\textbf{Y}}_{\rm trans.}(G)^A=0$ . Then, the three symmetry actions respectively consist of

(3.29) \begin{align} Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}0 ,\end{align}

(3.30) \begin{align} Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}2 u^\alpha_j a^{[i} D_i \rho^{j]}_{A\alpha}+ a^i D_i \kappa_A+a^i u^\alpha_i K_{A\alpha} ,\end{align}

(3.31) \begin{align} Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}-2 u^\alpha_j a^{[i} D_i \rho^{j]}_{A\alpha}-a^i D_i \kappa_A- a^i u^\alpha_i K_{A\alpha} .\end{align}

(ii) Suppose $Q_A$ and $G^A$ are scaling homogeneous: ${\textbf{Y}}_{\rm scal.}(Q)_A=w^{(A)} Q_A$ and ${\textbf{Y}}_{\rm scal.}(G)^A=\omega^{(A)} G^A$ . Then, the three symmetry actions respectively consist of

(3.32) \begin{align} \begin{aligned}Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}\ &(\omega^{(A)} + w^{(A)} + \textstyle\sum_i w^{(i)}) Q_A ,\end{aligned} \end{align}

(3.33) \begin{align} \begin{aligned}Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}\ &\omega^{(A)} Q_A+ u^\alpha_j w^{(i)} D_i( 2x^{[i} \rho^{j]}_{A\alpha})+ w^{(i)} D_i(x^i\kappa_A)+ w^{(\alpha)} D_i(\rho^i_{A\alpha} u^\alpha)\\[5pt]&\qquad-K_{A\alpha}(w^{(\alpha)} u^\alpha - w^{(i)} x^i u^\alpha_i),\end{aligned} \end{align}

(3.34) \begin{align} \begin{aligned}Q_A \overset{{\boldsymbol{X}_{\rm p}}}{\longrightarrow}\ &(w^{(A)} + \textstyle\sum_i w^{(i)}) Q_A-u^\alpha_j w^{(i)} D_i(2x^{[i} \rho^{j]}_{A\alpha})- w^{(i)} D_i(x^i\kappa_A) + w^{(\alpha)} D_i(\rho^i_{A\alpha} u^\alpha)\\[5pt]&\qquad+ K_{A\alpha}(w^{(\alpha)} u^\alpha - w^{(i)} xi^i u^\alpha_i) .\end{aligned} \end{align}

For both translations and scalings, the second and third symmetry actions here are considered only for first-order linear adjoint-symmetries (3.23)–(3.24).

The first symmetry actions (3.29) and (3.32) are a generalisation of the same actions derived on multipliers in [Reference Anco1, Reference Anco2]. The other results are new.

5.1. Adjoint-symmetry commutator brackets from symmetry actions

The construction of the first bracket goes as follows.

Proposition 5.1. Fix an adjoint-symmetry $Q_A$ in ${\mathrm{AdjSymm}}_G$ , and let $S_Q$ be the dual linear operator (4.9) associated with a symmetry action $S_P$ on ${\mathrm{AdjSymm}}_G$ . If the kernel of $S_Q$ is an ideal in ${\mathrm{Symm}}_G$ , then

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

defines a bilinear bracket on the linear space $S_Q({\mathrm{Symm}}_G)\subseteq{\mathrm{AdjSymm}}_G$ . This bracket can be expressed as

(5.2) \begin{equation}{}^Q[Q_1,Q_2]_A = Q'_{\!\!2}(S_Q^{-1}Q_1) - Q'_{\!\!1}(S_Q^{-1}Q_2) - S'_{\!\!Q}(S_Q^{-1}Q_2)(S_Q^{-1}Q_1) + S'_{\!\!Q}(S_Q^{-1}Q_1)(S_Q^{-1}Q_2)\end{equation}

where $S'_{\!\!Q}$ denotes the Frechet derivative of $S_Q$ .

Any one of the symmetry actions (3.8), (3.11), (3.12) can be used to write down formally a corresponding bracket (5.1). However, $S_Q^{-1}$ is well-defined only modulo $\ker\!(S_Q)$ , and so in the absence of any extra structure to fix this arbitrariness, the condition that $\ker\!(S_Q)$ is an ideal is necessary and sufficient for the bracket to be well defined (namely, invariant under $S_Q^{-1}\rightarrow S_Q^{-1}+\ker\!(S_Q)$ ). This condition will select a set of adjoint-symmetries $Q_A$ that can be used in constructing the bracket. When $\ker\!(S_Q)$ is an ideal, so is $\ker\!(S_{\lambda Q}) = \lambda \ker\!(S_{Q})$ , for any constant $\lambda$ . Hence, the set of adjoint-symmetries $Q_A$ for which $\ker\!(S_Q)$ is an ideal comprises a projective subspace in ${\mathrm{AdjSymm}}_G$ . In the case when the dimension of this subspace is larger than 1, it is natural to select $Q_A$ such that ${\rm ran}(S_Q)$ is maximal in ${\mathrm{AdjSymm}}_G$ .

For the linear space $\ker\!(S_Q)\subseteq{\mathrm{Symm}}_G$ to be an ideal, it must be a subalgebra that is preserved by the action of ${\mathrm{Symm}}_G$ given by the Lie bracket (3.1). The subalgebra condition

(5.3) \begin{equation}[\ker\!(S_Q),\ker\!(S_Q)] \subseteq \ker\!(S_Q)\end{equation}

states that $S_Q([P_1,P_2])=0$ is required to hold for all pairs of symmetries ${\textbf{X}}_{P_1}=P_1^\alpha\partial_{u^\alpha}$ and ${\textbf{X}}_{P_2}=P_2^\alpha\partial_{u^\alpha}$ such that $S_Q(P_1)_A= S_{P_1}(Q)_A = 0$ and $S_Q(P_2)_A= S_{P_2}(Q)_A = 0$ . The question of whether this condition (5.3) is satisfied by each of the three symmetry actions will now be addressed.

For the first symmetry action (3.8), consider

(5.4) \begin{equation}0= S_{1\,Q}(P_1)_A = Q'(P_1)_A + R_{P_1}^*(Q)_A ,\quad\;0= S_{1\,Q}(P_2)_A = Q'(P_2)_A + R_{P_2}^*(Q)_A .\end{equation}

Applying the symmetries ${\textbf{X}}_{P_2}$ and ${\textbf{X}}_{P_1}$ respectively to these two equations and subtracting them yields

(5.5) \begin{equation}\begin{aligned}0 &= {\rm pr}{\textbf{X}}_{P_2}(Q'(P_1)_A + R_{P_1}^*(Q)_A) - {\rm pr}{\textbf{X}}_{P_1}(Q'(P_2)_A + R_{P_2}^*(Q)_A) \\[5pt] &= Q'({\rm pr}{\textbf{X}}_{P_2}(P_1) -{\rm pr}{\textbf{X}}_{P_1}(P_2))_A+ {\rm pr}{\textbf{X}}_{P_2}(R_{P_1}^*)(Q)_A - {\rm pr}{\textbf{X}}_{P_1}(R_{P_2}^*)(Q)_A\\[5pt]&\quad+ R_{P_1}^*({\rm pr}{\textbf{X}}_{P_2}(Q))_A - R_{P_2}^*({\rm pr}{\textbf{X}}_{P_1}(Q))_A\end{aligned}\end{equation}

using ${\rm pr}{\textbf{X}}_{P_2}Q'(P_1) - {\rm pr}{\textbf{X}}_{P_1}Q'(P_2) = Q''(P_2,P_1) - Q''(P_1,P_2) = 0$ . The first term in equation (5.5) reduces to a commutator expression

(5.6) \begin{equation}Q'({\rm pr}{\textbf{X}}_{P_2}(P_1) -{\rm pr}{\textbf{X}}_{P_1}(P_2))_A = Q'([P_2,P_1])_A .\end{equation}

The middle two terms can be expressed as

(5.7) \begin{equation}{\rm pr}{\textbf{X}}_{P_2}(R_{P_1}^*)(Q)_A -{\rm pr}{\textbf{X}}_{P_1}(R_{P_2}^*)(Q)_A= R_{[P_2,P_1]}^*(Q)_A - [R_{P_2}^*,R_{P_1}^*](Q)_A\end{equation}

by use of the identity

(5.8) \begin{equation}R_{[P_2,P_1]}^* = [R_{P_2}^*,R_{P_1}^*] + {\rm pr}{\textbf{X}}_{P_2}(R_{P_1}^*) -{\rm pr}{\textbf{X}}_{P_1}(R_{P_2}^*)\end{equation}

which can derived straightforwardly from the symmetry determining equation (2.4) off of ${\mathcal{E}}$ . Next, the last term in equation (5.7) can be combined with the last two terms in equation (5.5), yielding

(5.9) \begin{equation}R_{P_1}^*(Q'(P_2) +R_{P_2}^*(Q))_A - R_{P_2}^*(Q'(P_1) +R_{P_1}^*(Q))_A=0\end{equation}

due to equations (5.4). Hence, after these simplifications, equation (5.5) becomes $0=Q'([P_2,P_1])_A + R_{[P_2,P_1]}^*(Q)_A = S_{1\,Q}([P_2,P_1])_A$ . This establishes the following result.

To continue, consider the third symmetry action (3.12). Similar steps will now be carried out, starting from

(5.10) \begin{equation}0= S_{3\,Q}(P_1)_A = Q'(P_1)_A + R_Q^*(P_1)_A ,\quad\;0= S_{3\,Q}(P_2)_A = Q'(P_2)_A + R_Q^*(P_2)_A .\end{equation}

Respectively applying the symmetries ${\textbf{X}}_{P_2}$ and ${\textbf{X}}_{P_1}$ to these two equations and subtracting, one obtains

(5.11) \begin{equation}0 = Q'([P_2,P_1])_A +R_Q^*([P_2,P_1])_A+ {\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A -{\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A .\end{equation}

Hence, one sees that $S_{3\,Q}([P_2,P_1]) = Q'([P_2,P_1])_A +R_Q^*([P_2,P_1])_A= {\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A -{\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A$ does not vanish in general. This represents an obstruction to the bracket being well-defined. A useful remark is that if $Q=\Lambda$ is a conservation law multiplier for a PDE system with no differential identities (Lemma 2.2), then the relation (2.17) shows that

(5.12) \begin{equation}\begin{aligned}{\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A -{\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A& = {\rm pr}{\textbf{X}}_{P_2}(Q')(P_1)_A -{\rm pr}{\textbf{X}}_{P_1}(Q')(P_2)_A \\[5pt]& = Q''(P_1,P_2) - Q''(P_2,P_1) =0\end{aligned}\end{equation}

whereby the obstruction vanishes.

A similar obstruction arises for the bracket given by the second symmetry action (3.11). Specifically, by the same steps used for the first and third symmetry actions, one obtains $S_{2\,Q}([P_2,P_1]) = R_{[P_2,P_1]}^*(Q)_A -R_Q^*([P_2,P_1])_A= {\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A -{\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A+R_{P_2}^*(S_{1\,Q}(P_1))_A - R_{P_1}^*(S_{1\,Q}(P_2))_A$ . This expression contains the same obstruction terms as for the third symmetry action, as well as terms that involve the first symmetry action itself. If $Q=\Lambda$ is a conservation law multiplier for a PDE system with no differential identities (Lemma 2.2), then this obstruction vanishes.

Consequently, the following two results have been established.

Lemma 5.3. For the third symmetry action (3.12), $\ker\!(S_{3\,Q})$ is a subalgebra in ${\mathrm{Symm}}_G$ iff the condition

(5.13) \begin{equation}{\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A - {\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A =0\end{equation}

holds for all symmetries ${\textbf{X}}_{P_1}=P_1^\alpha\partial_{u^\alpha}$ and ${\textbf{X}}_{P_2}=P_2^\alpha\partial_{u^\alpha}$ in $\ker\!(S_{3\,Q})$ .

Lemma 5.4. For the second symmetry action (3.11), $\ker\!(S_{2\,Q})$ is a subalgebra in ${\mathrm{Symm}}_G$ iff the condition

(5.14) \begin{equation}{\rm pr}{\textbf{X}}_{P_2}(R_Q^*)(P_1)_A -{\rm pr}{\textbf{X}}_{P_1}(R_Q^*)(P_2)_A+R_{P_2}^*(S_{1\,Q}(P_1))_A - R_{P_1}^*(S_{1\,Q}(P_2))_A=0\end{equation}

holds for all symmetries ${\textbf{X}}_{P_1}=P_1^\alpha\partial_{u^\alpha}$ and ${\textbf{X}}_{P_2}=P_2^\alpha\partial_{u^\alpha}$ in $\ker\!(S_{2\,Q})$ .

The preceding developments can be summarised as follows.

An alternative way to have the bracket be well defined is if the quotient ${\mathrm{Symm}}_G/\ker\!(S_Q)$ can be naturally identified with a subspace in ${\mathrm{Symm}}_G$ . This is equivalent to requiring that the symmetry Lie algebra admits an extra structure of a direct sum decomposition as a linear space

(5.15) \begin{equation}{\mathrm{Symm}}_G = \ker\!(S_Q)\oplus{\rm coker}(S_Q)\end{equation}

such that the decomposition is independent of a choice of basis. Then $S_Q^{-1}$ can be defined as belonging to the subspace ${\rm coker}(S_Q)$ , and hence the bracket will be well defined.

It will now be shown that the extra structure (5.15) will typically exist for a symmetry Lie algebra that contains a scaling symmetry (3.28).

Every symmetry in ${\mathrm{Symm}}_G$ can be decomposed into a sum of symmetries that are scaling homogeneous. Consequently, there will exist a basis for ${\mathrm{Symm}}_G$ consisting of $P_{\rm scal.}$ and $\{P_k\}_{k=1,\ldots,\dim\!({\mathrm{Symm}}_G)-1}$ , such that $[P_{\rm scal.},P_k] = r^{(k)} P_k$ where the constant $r_{(k)}$ is the scaling weight of the symmetry $P_k$ . Then there exists a direct sum decomposition

(5.16) \begin{equation}{\mathrm{Symm}}_G = {\rm span}(P_{\rm scal.}) \oplus\sum_k{}^{\textstyle\oplus}{\rm span}(P_k) .\end{equation}

which is basis independent. This will provide the extra structure (5.15) if the subspaces $\ker\!(S_Q)$ and ${\rm coker}(S_Q)$ can be uniquely characterised in terms of their scaling weights.

This result can be generalised if $\ker\!(S_Q)$ is a direct sum of scaling homogeneous subspaces that have no scaling weights in common with any scaling homogeneous subspace in ${\rm coker}(S_Q)$ .

Now, the basic properties of the general adjoint-symmetry commutator bracket (5.1) will be studied. Recall that the underlying symmetry commutator bracket is antisymmetric and obeys the Jacobi identity. This implies that the same properties are inherited by the bracket (5.1).

Theorem 5.7. The adjoint-symmetry commutator bracket (5.1) is a Lie bracket, namely it is antisymmetric

(5.17) \begin{equation}{}^Q[Q_1,Q_2]_A +{}^Q[Q_2,Q_1]_A=0\end{equation}

and obeys the Jacobi identity

(5.18) \begin{equation}{}^Q[Q_1,{}^Q[Q_2,Q_3]]_A + {}^Q[Q_2,{}^Q[Q_3,Q_1]]_A + {}^Q[Q_3,{}^Q[Q_1,Q_2]]_A =0 .\end{equation}

Hence, the linear subspace $S_Q({\mathrm{Symm}}_G)\subseteq{\mathrm{AdjSymm}}_G$ of adjoint-symmetries acquires a Lie algebra structure which is homomorphic to the symmetry Lie algebra. If there exists an adjoint-symmetry $Q_A$ such that $S_Q({\mathrm{Symm}}_G)={\mathrm{AdjSymm}}_G$ where $\ker\!(S_Q)$ satisfies the conditions in either of Propositions 5.5 and 5.6, then the whole space ${\mathrm{AdjSymm}}_G$ will be a Lie algebra.

Since $S_Q$ is a linear mapping, the condition $S_Q({\mathrm{Symm}}_G)={\mathrm{AdjSymm}}_G$ can be expressed equivalently as

(5.19) \begin{equation}\dim{\mathrm{AdjSymm}}_G +\dim\ker\!(S_Q) = \dim{\mathrm{Symm}}_G .\end{equation}

Hence, $\dim{\mathrm{Symm}}_G \geq \dim{\mathrm{AdjSymm}}_G$ is a necessary condition. This version is most useful when the dimensions are finite.

Running example: For the p-gKdV equation (2.2), the three dual linear maps $S_Q$ with $Q=c_1Q_1+c_2Q_2+c_3Q_3$ have a 4-dimensional domain ${\rm span}(P_1,P_2,P_3,P_4)$ , while their range is at most 3-dimensional, since it belongs to ${\rm span}(Q_1,Q_2,Q_3)$ . Hence, the kernel of each $S_Q$ is at least 1-dimensional. The specific ranges and kernels, which depend on the choice of the constants $c_1,c_2,c_3$ , can be found from Table 2 as follows.

Firstly, for $S_{3\,Q}$ , there is no loss of generality in taking $c_3=1$ , $c_1=c_2=0$ . The range of $S_{3\,Q}$ is ${\rm span}(Q_1,Q_2,Q_3)$ , while the kernel is ${\rm span}(P_1)$ . This subspace in ${\mathrm{Symm}}_\textrm{p-gKdV}$ is an ideal, as shown by the symmetry commutators (3.13). The resulting adjoint-symmetry bracket ${}^{Q_3}[\ \cdot\ ,\ \cdot\ ]$ is thereby well-defined. It is computed by: $S_{3\,Q_3}^{-1}(Q_1) = \frac{1}{2(p-2)}P_2$ , $S_{3\,Q_3}^{-1}(Q_2) = \frac{1}{2(p-2)}P_3$ , $S_{3\,Q_3}^{-1}(Q_3) = \frac{1}{2(p-2)}P_4$ , modulo ${\rm span}(P_1)$ , and thus

(5.20a) \begin{align} {}^{Q_3}[Q_1,Q_2] = S_{3\,Q_3}\!\left(\left[\frac{1}{2(p-2)}P_2,\frac{1}{2(p-2)}P_3\right]\right)=\frac{1}{4(p-2)^2}S_{3\,Q_3}(0)=0, \end{align}

(5.20b) \begin{align} {}^{Q_3}[Q_1,Q_3] = S_{3\,Q_3}\!\left(\left[\frac{1}{2(p-2)}P_2,\frac{1}{2(p-2)}P_4\right]\right)= \frac{1}{4(p-2)^2} S_{3\,Q_3}(pP_2)= \frac{p}{2(p-2)}Q_1,\end{align}

(5.20c) \begin{align} {}^{Q_3}[Q_2,Q_3] = S_{3\,Q_3}\!\left(\left[\frac{1}{2(2-p)}P_3,\frac{1}{2(p-2)}P_4\right]\right)= \frac{1}{4(p-2)^2} S_{3\,Q_3}(3pP_3)= \frac{3p}{2(p-2)}Q_2,\end{align}

using the symmetry commutators (3.13). This bracket is a Lie bracket such that ${\rm span}(Q_1,Q_2,Q_3)$ is homomorphic to the symmetry algebra ${\rm span}(P_1,P_2,P_3,P_4)$ and isomorphic to the subalgebra ${\rm span}(P_2,P_3,P_4)$ . In particular, the bracket can be expressed in terms of the scaled Noether operator $D_x$ (cf (4.17)):

(5.21) \begin{equation}{}^{Q_3}[\ \cdot\ ,\ \cdot\ ] = D_x([D_x^{-1}\ \cdot\ ,D_x^{-1}\ \cdot]) .\end{equation}

Secondly, for $S_{1\,Q}$ , the range is maximal if $c_3\neq0$ with any values of $c_2,c_3$ . The simplest choice is $c_1=c_2=0$ , $c_3=1$ , namely $Q=Q_3$ . Then, the range and the kernel are the same as for $S_{3\,Q}$ , and so the resulting adjoint-symmetry bracket is well-defined. It is computed by: $S_{1\,Q_3}^{-1}(Q_1) = \frac{1}{p}P_2$ , $S_{1\,Q_3}^{-1}(Q_2) = \frac{1}{3p}P_3$ , $S_{1\,Q_3}^{-1}(Q_3) = \frac{1}{2(p-2)}P_4$ , modulo ${\rm span}(P_1)$ , and thus

(5.22a) \begin{align} {}^{Q_3}[Q_1,Q_2] = S_{1\,Q_3}\!\left(\left[\frac{1}{p}P_2,\frac{1}{3p}P_3\right]\right)=\frac{1}{3p^2}S_{1\,Q_3}(0)=0,\end{align}

(5.22b) \begin{align} {}^{Q_3}[Q_1,Q_3] = S_{1\,Q_3}\!\left(\left[\frac{1}{p}P_2,\frac{1}{2(p-2)}P_4\right]\right)= \frac{1}{2p(p-2)} S_{1\,Q_3}(pP_2)= \frac{p}{2(p-2)}Q_1,\end{align}

(5.22c) \begin{align} {}^{Q_3}[Q_2,Q_3] = S_{1\,Q_3}\!\left(\left[\frac{1}{3p}P_3,\frac{1}{2(p-2)}P_4\right]\right)= \frac{1}{6p(p-2)} S_{1\,Q_3}(3pP_3)= \frac{3}{2(p-2)}Q_2.\end{align}

This yields the same bracket as for $S_{3\,Q}$ .

Thirdly, for $S_{2\,Q}$ , the maximal range is ${\rm span}(Q_1,Q_2)$ , which arises if $c_3\neq0$ with any values for $c_1,c_2$ . The kernel is spanned by $P_1$ , $P_4 +\frac{c_1}{c_3}P_2 +\frac{c_2}{c_3}P_3$ . However, this space is not an ideal in ${\mathrm{Symm}}_\textrm{p-gKdV}$ , as shown by the symmetry commutators (3.13). Hence, a corresponding adjoint-symmetry bracket cannot be defined without the use of extra structure. The scaling symmetry $P_4$ is available to provide a direct sum decomposition ${\rm span}(P_1,P_2,P_3,P_4) = \ker\!(S_{2\,Q})\oplus{\rm coker}(S_{2\,Q})$ where, for the choice $c_1=c_2=0$ and $c_3=1$ , $\ker\!(S_{2\,Q}) = {\rm span}(P_4)\oplus{\rm span}(P_1)$ and ${\rm coker}(S_{2\,Q}) = {\rm span}(P_2)\oplus{\rm span}(P_3)$ are characterised by their distinct scaling weights with respect to ${\mathrm{ad}}(P_4)$ : $(0,2-p)$ and $(-p,-3p)$ . Then, an adjoint-symmetry bracket can be defined via $S_{2\,Q_3}^{-1}(Q_1) = \frac{1}{4-p} P_2$ , $S_{2\,Q_3}^{-1}(Q_2) = \frac{1}{p+4} P_3$ , in ${\rm coker}(S_{2\,Q})$ , and thus

(5.23) \begin{equation}{}^{Q_3}[Q_1,Q_2] = S_{2\,Q_3}\!\left(\left[\frac{1}{4-p}P_2,\frac{1}{p+4}P_3\right]\right)= \frac{1}{16-p^2} S_{2\,Q_3}(0)= 0.\end{equation}

This yields an abelian Lie bracket on the subspace ${\rm span}(Q_1,Q_2)\subset {\mathrm{AdjSymm}}_\textrm{p-gKdV}$ . It coincides with the previous Lie bracket restricted to this subspace.

These three Lie brackets are summarised in Table 3.

Table 3. p-gKdV equation: adjoint-symmetry Lie brackets

5.2. Adjoint-symmetry commutators associated with symmetry subalgebras

The Lie algebra structure identified in Theorem 5.7 motivates a related construction of adjoint-symmetry commutator brackets given by a pull-back of Lie subalgebras in ${\mathrm{Symm}}_G$ under $S_Q^{-1}$ .

As the starting point, the linear subspace $S_Q({\mathrm{Symm}}_G)$ will be replaced by $S_Q({\mathcal A})$ where ${\mathcal A}$ is any Lie subalgebra in ${\mathrm{Symm}}_G$ and where $Q_A$ is chosen such that $\ker\!(S_Q)\cap {\mathcal A}$ is empty. The set of such adjoint-symmetries will, as before, be a projective subspace in ${\mathrm{AdjSymm}}_G$ .

Then, the construction of the commutator bracket given in Proposition 5.1 is modified as follows.

In particular, $S_Q^{-1}$ provides an isomorphism under which the commutator bracket (5.1) on $S_Q({\mathrm{Symm}}_G)$ is the pull-back of the Lie bracket on $\mathcal A$ . The condition

(5.24) \begin{equation}\ker\!(S_Q)\cap {\mathcal A}=\emptyset\end{equation}

will select the adjoint-symmetries $Q_A$ that can be used in constructing this bracket. If this condition fails to be satisfied by all adjoint-symmetries, then it implies that there is no subspace in ${\mathrm{AdjSymm}}_G$ on which the bracket produces a Lie algebra isomorphic to $\mathcal A$ .

The question of which Lie subalgebras $\mathcal A$ in ${\mathrm{Symm}}_G$ have counterparts in ${\mathrm{AdjSymm}}_G$ for a PDE system $G^A=0$ thereby becomes an interesting algebraic classification problem.

5.3. Adjoint-symmetry non-commutator brackets from symmetry actions

The construction of the second bracket disregards the symmetry commutator but lacks the attendant properties.

Proposition 5.9. Fix an adjoint-symmetry $Q_A$ in ${\mathrm{AdjSymm}}_G$ , and let $S_Q$ be the dual linear operator (4.9) associated with a symmetry action $S_P$ on ${\mathrm{AdjSymm}}_G$ . If the kernel of $S_Q$ satisfies

(5.25) \begin{equation}S_P=0 \textrm{ for all } P\in\ker\!(S_Q),\end{equation}

then a bilinear bracket from ${\mathrm{AdjSymm}}_G\times S_Q({\mathrm{Symm}}_G)$ into $S_Q({\mathrm{Symm}}_G)\subseteq{\mathrm{AdjSymm}}_G$ is defined by

(5.26) \begin{equation}{}^Q(Q_1,Q_2)_A \;:\!=\; S_{Q_1}(S_Q^{-1}Q_2)_A .\end{equation}

Any one of the symmetry actions (3.11), (3.8), (3.12) can be used to write down formally a corresponding bracket (5.26). Note that, unlike the situation for the commutator bracket (5.1), the condition (5.25) only involves the properties of the symmetry action $S_Q$ and does not depend on the Lie algebra structure of ${\mathrm{Symm}}_G$ . This condition can be by-passed when a scaling symmetry (3.28) is contained in the symmetry Lie algebra.

In contrast to the commutator bracket (5.1), the bracket (5.26) is non-symmetric. Its only general property is that

(5.27) \begin{equation}{}^Q(Q,Q_2)= Q_2\end{equation}

for all $Q_2$ in the linear subspace $S_Q({\mathrm{Symm}}_G) \subseteq {\mathrm{AdjSymm}}_G$ .

There are two worthwhile remarks that can be made.

Remark 5.11. (i) The bracket (5.26) can be viewed as arising from the property that $S_{Q_1}S_{Q_2}^{-1}$ is a recursion operator on adjoint-symmetries in $S_Q({\mathrm{Symm}}_G)$ . (ii) A symmetric version and a skew-symmetric version of the bracket (5.26) can be defined by respectively symmetrising and antisymmetrising on the pair $Q_1$ and $Q_2$ :

(5.28) \begin{equation}{}^Q(Q_1,Q_2)^{-}_A \;:\!=\; \frac{1}{2}\big( S_{Q_1}(S_Q^{-1}Q_2)_A - S_{Q_2}(S_Q^{-1}Q_1)_A \big)\end{equation}

and

(5.29) \begin{equation}{}^Q(Q_1,Q_2)^{+}_A \;:\!=\; \frac{1}{2}\big( S_{Q_1}(S_Q^{-1}Q_2)_A + S_{Q_2}(S_Q^{-1}Q_1)_A \big) .\end{equation}

The recursion operator $S_{Q_1}S_{Q_2}^{-1}$ was derived originally for scalar PDE systems in [Reference Anco and Wang11].

Running example: For the p-gKdV equation (2.2), the condition (5.25) holds when $Q=Q_3$ where $\ker\!(S_Q)={\rm span}(P_1)$ for the symmetry actions $S_1$ and $S_3$ , as seen from Tables 1 and 2. The resulting brackets (5.28) and (5.29) on the linear space ${\rm span}(Q_1,Q_2,Q_3)$ have the form

(5.30a) \begin{align} {}^{Q_3}(Q_1,Q_1)^\pm = {}^{Q_3}(Q_2,Q_2)^\pm = 0,\quad{}^{Q_3}(Q_3,Q_3)^+ = Q_3,\quad{}^{Q_3}(Q_3,Q_3)^- = 0,\end{align}

(5.30b) \begin{align} {}^{Q_3}(Q_1,Q_2)^\pm = 0,\quad{}^{Q_3}(Q_1,Q_3)^\pm = \frac{1}{2} \lambda_1^\pm Q_1,\quad{}^{Q_3}(Q_2,Q_3)^\pm = \frac{1}{2} \lambda_2^\pm Q_2,\end{align}

where, for $S_1$ : $\lambda_1^+=\frac{3p-8}{2p-4}$ , $\lambda_1^-=\frac{p}{4-2p}$ , $\lambda_2^+=\frac{p-8}{2p-4}$ , $\lambda_2^-=\frac{3p}{4-2p}$ ; and for $S_3$ : $\lambda_1^\pm=\pm 1$ , $\lambda_2^\pm=\pm 1$ .

5.4. Properties and computational aspects

As emphasised already, the definition of the two brackets (5.1) and (5.26) involves the dual linear map $S_Q$ defined by a symmetry action (4.8), which can be chosen to be any one of the three symmetry actions given in Theorem 3.1. The different properties of these actions imply corresponding properties for the brackets.

In the case of the second symmetry action (3.11), since it maps any fixed adjoint-symmetry $Q_A$ into a multiplier, the resulting brackets will be defined on the linear (sub) space of multipliers given by the range of the symmetry action, namely ${\rm ran}(S_{2\,Q}) \subseteq {\mathrm{Multr}}_G \subseteq {\mathrm{AdjSymm}}_G$ . Thus, each of the two brackets will implicitly define a bracket structure on conservation laws of the given PDE system $G^A=0$ and thereby will constitute a generalised Poisson bracket on the conserved integrals associated with the conservation laws. Further development of this structure will be left to subsequent work.

If $Q_A$ is chosen to be a multiplier itself, then since the first symmetry action (3.8) coincides with the second symmetry action, the two brackets defined using the dual linear map $S_{1\,Q}$ will be the same as the preceding two brackets. Moreover, use of the third symmetry action (3.12) when $Q_A$ is a multiplier will produce trivial brackets, since $S_{3\,Q}$ vanishes in this case.

Both of the brackets (5.1) and (5.26) are constructed explicitly in terms of $S_Q$ and its inverse $S_Q^{-1}$ . For the two symmetry actions (3.8) and (3.11), $S_Q$ viewed as an operator involves total derivatives $D_I$ and partial derivatives $\partial_{u^\alpha_I}$ . This means that $S_Q^{-1}$ will involve an integral (operator) with respect to the variables $u^\alpha_I$ in jet space, whereby the brackets are essentially nonlocal in jet space. Nevertheless, as an alternative, $S_Q$ can be represented in terms of structure constants that are defined with respect to any fixed basis of the linear spaces ${\mathrm{Symm}}_G$ and ${\mathrm{AdjSymm}}_G$ . With such a representation, the pre-image of any given adjoint-symmetry can be found directly in terms of these structure constants. The resulting brackets thus should be viewed as an a posteriori structure on the linear space ${\mathrm{AdjSymm}}_G$ .

In contrast, for the third symmetry action (3.12), $S_{3\,Q} = Q' + R_Q^*$ is a linear operator in total derivatives, where $Q_A$ is any adjoint-symmetry that is not multiplier. Consequently, $S_{3\,Q}^{-1}$ only involves the inverse total derivatives $D_I^{-1}$ , and thus, the two brackets (5.1) and (5.26) are local in jet space and thereby constitute an a priori structure, just like the symmetry commutator.

The same considerations pertain to the corresponding pre-symplectic and pre-Hamiltonian (Noether) structures shown in Theorem 4.2.

6.1. Symmetry actions on adjoint-symmetries

The symmetry actions in Theorem 3.1 can be simplified by use of the relations (6.4) and (6.6). Combined with the condition (6.8) characterising multipliers, this yields the following result.

Theorem 6.1. The actions (3.8) and (3.11) of symmetries on the linear space of adjoint-symmetries are respectively given by

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

(6.18) \begin{align} Q_\alpha \overset{{\boldsymbol{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 action (3.12) given by the difference of these two actions consists of

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

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

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

6.2. Adjoint-symmetry brackets

For evolution PDEs (6.1), the dual linear map $S_Q$ in the form of the adjoint-symmetry commutator bracket (5.1) and the non-commutator bracket (5.26) is given by any of the three symmetry actions in Theorem 6.1.

Recall that the commutator bracket is well defined when $\ker\!(S_Q)$ satisfies the conditions in either of Propositions 5.5 and 5.6. The conditions in the first Proposition can be expressed entirely in terms of Q and a pair of symmetries $P_1$ , $P_2$ , by means of the relations (6.6) and (6.4). In particular, condition (5.13) takes the form

(6.20) \begin{equation}{\rm pr}{\textbf{X}}_{P_1}(Q'{}^*)(P_2) -{\rm pr}{\textbf{X}}_{P_2}(Q'{}^*)(P_1) =0\end{equation}

while condition (5.14) takes the form

(6.21) \begin{equation}{\rm pr}{\textbf{X}}_{P_1}(Q'{}^*)(P_2) -{\rm pr}{\textbf{X}}_{P_2}(Q'{}^*)(P_1)+P'_{\!\!2}{}^*(Q'(P_1)-Q'{}^*(P_1)) - P'_{\!\!1}{}^*(Q'(P_2)-Q'{}^*(P_2))=0\end{equation}

for all symmetries ${\textbf{X}}_{P_1}=P_1^\alpha\partial_{u^\alpha}$ and ${\textbf{X}}_{P_2}=P_2^\alpha\partial_{u^\alpha}$ in $\ker\!(S_Q)$ when $\dim\ker\!(S_Q)>1$ . When Q is a conservation law multiplier, each condition is identically satisfied, which can be seen from the properties (6.8) and $Q''(P_1,P_2) = Q''(P_2,P_1)$ .

It is worth emphasising that the existence of these adjoint-symmetry brackets does not rely on a PDE system having any variational structure. Indeed, examples of non-trivial brackets for dissipative PDE systems will be given in a subsequent paper.

6.3. A Noether operator and a symplectic 2-form

The third symmetry action (6.19) yields

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

which is the form of the Noether operator in Theorem 4.2 specialised to evolution PDEs through the relation (6.6). Note that it will be non-trivial if, and only if, Q is a non-variational (non-gradient) adjoint-symmetry. This operator is skew, ${\mathcal{J}}^* = -{\mathcal{J}}$ .

From Proposition 4.3, there is an associated integral bilinear form (4.16) on the linear space of symmetries $P^\alpha\partial_{u^\alpha}$ . Its explicit form for evolution equations is obtained by taking the integration domain $\Omega$ to be the spatial domain ${\mathbb{R}}^n$ , substituting expression (6.12), and integrating by parts to get

(6.23) \begin{equation}{\boldsymbol\omega}_Q(P_1,P_2) = \int_{{\mathbb{R}}^n} \Psi^t(P_1,{\mathcal{J}}(P_2)) \, d^nx= \int_{{\mathbb{R}}^n} (P_1^\alpha Q'(P_2)_\alpha - P_2^\alpha Q'(P_1)_\alpha)\, d^nx\end{equation}

which is manifestly skew. Hence, this defines a 2-form on the linear space of symmetries. As discussed in Remark 4.1, a 2-form is symplectic if it is closed. The closure condition, ${\textbf{d}}{\boldsymbol\omega}_Q =0$ , can be formulated as

(6.24) \begin{equation}{\rm pr}{\textbf{X}}_{f_3}{\boldsymbol\omega}_Q(f_1,f_2) +\textrm{ cyclic } =0\end{equation}

which must hold for all functions $f_1^\alpha(t,x)$ , $f_2^\alpha(t,x)$ , $f_3^\alpha(t,x)$ .

Proof. Consider

(6.25) \begin{equation}\begin{aligned}{\rm pr}{\textbf{X}}_{f_3}{\boldsymbol\omega}_Q(f_1,f_2)& = \int (f_1^\alpha {\rm pr}{\textbf{X}}_{f_3}Q'(f_2)_\alpha - f_2^\alpha {\rm pr}{\textbf{X}}_{f_3}Q'(f_1)_\alpha)\, d^nx\\[5pt]&= \int (f_1^\alpha Q''(f_3,f_2)_\alpha - f_2^\alpha Q''(f_3,f_1)_\alpha)\, d^nx .\end{aligned}\end{equation}

Then, in the cyclic sum ${\rm pr}{\textbf{X}}_{f_3}{\boldsymbol\omega}_Q(f_1,f_2) + {\rm pr}{\textbf{X}}_{f_2}{\boldsymbol\omega}_Q(f_3,f_1) + {\rm pr}{\textbf{X}}_{f_1}{\boldsymbol\omega}_Q(f_2,f_3)$ , all terms cancel pairwise, due to the symmetry of Q ^′′ in its two arguments. Hence, the condition (6.24) is satisfied.

The proof can be straightforwardly generalised (using the methods in [Reference Olver23]) to show that

(6.26) \begin{equation}{\rm pr}{\textbf{X}}_{P_3}{\boldsymbol\omega}_Q(P_1,P_2) +\textrm{ cyclic } =0\end{equation}

holds for all symmetries $P_1^\alpha\partial_{u^\alpha}$ , $P_2^\alpha\partial_{u^\alpha}$ , $P_3^\alpha\partial_{u^\alpha}$ .

The formal inverse of the Noether operator (6.22) 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

(6.27) \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, $\delta{F}/\delta u^\alpha = E_{u^\alpha}(f)$ .

An interesting general question for future work is to determine under what conditions on ${\mathcal{J}}^{-1}$ or $Q_\alpha$ will a given evolution equation possessing a Hamiltonian formulation.

Running example: The non-gradient adjoint-symmetry $Q_3=2u_x + p x u_{xx} - 3p t (u_x{}^p u_{xx} + u_{xxxx})$ of the p-gKdV equation (2.2) yields the Noether operator (cf (4.17)) ${\mathcal{J}} = Q'_{\!\!3} - Q'_{\!\!3}{}^* = 2(p-2) D_x$ from the relation (6.16). Note that the inverse operator acting on $Q_3$ yields a multiple of the scaling symmetry $P_4=(p-2)u -3p t u_t -xp u_x$ . The associated symplectic 2-form on ${\rm span}(P_1,P_2,P_3,P_4)$ is explicitly given by

(6.28) \begin{equation}{\boldsymbol\omega}_{Q_3}\!\left(\sum_{i=1}^{4} a_i P_i, \sum_{j=1}^{4} b_j P_j\right) = 2 \sum_{i,j=1}^{4} a_i b_j \int P_i D_x P_j \, dx .\end{equation}

Its components are shown in Table 4. Note that, by skew-symmetry, the omitted entries in the lower left will be the negative of the entries in the upper right. Also observe that the non-zero entries are precisely the conserved integrals for momentum (2.19) and energy (2.20). The scaled Noether operator $D_x$ is in fact the inverse of the well-known Hamiltonian operator $D_x^{-1}$ for which the p-gKdV equation (2.2) has the Hamiltonian structure

(6.29) \begin{equation}u_t = -\frac{1}{p+1} u_x{}^{p+1} - u_{xxx} = - D_x^{-1}(\delta H/\delta u),\quad\;H = \int \left(\frac{1}{2}u_{xx}{}^2 - \frac{1}{(p+1)(p+2)} u_x{}^{p+2} \right)\,dx ,\end{equation}

where H is the energy integral.

Table 4. p-gKdV symplectic 2-form

Further examples of the symplectic structure in Theorem 6.2 will be given in a subsequent paper.