## Abstract

Many well-known partial differential equations can be written as multisymplectic systems. Such systems have a structural conservation law from which scalar conservation laws can be derived. These conservation laws arise as differential consequences of a 1-form ‘quasi-conservation law’, which is related to Noether's theorem. This paper develops the above framework and uses it to introduce a multisymplectic structure for differential-difference equations. The shallow water equations and the Ablowitz–Ladik equations are used to illustrate the general theory. It is found that conservation of potential vorticity is a differential consequence of two conservation laws; this surprising result and its implications are discussed.

## 1. Introduction

Hamiltonian systems of ordinary differential equations (ODEs) are endowed with a rich geometrical structure that is associated with the symplectic 2-form. This structure is masked in the Lagrangian formulation, which is nonlocal. Some partial differential equations (PDEs) have one or more Hamiltonian formulations (Olver 1993), but these, like their Lagrangian counterparts, are non-local.

Bridges (1997*a*,*b*) observed that many PDEs can be written as *multisymplectic* systems, which are local and have a distinct symplectic structure associated with each independent variable. The multisymplectic formulation is useful for the study of nonlinear waves (Bridges 1997*a*), stability and bifurcation analysis (Bridges & Derks 2001), and the development of numerical methods (Bridges & Reich 2001).

For any Hamiltonian ODE, the symplectic structure is preserved on trajectories. Similarly, every multisymplectic system has a conservation law that involves each of the symplectic structures. We shall refer to this as the *structural conservation law*, because it is associated with the multisymplectic structure rather than the specific PDE. Other conservation laws have been obtained for some systems by using a restricted version of Noether's theorem (Bridges 1997*a*).

The current paper shows how the general (unrestricted) form of Noether's theorem is linked to the structural conservation law for multisymplectic systems. This knowledge provides a rationale for generalizing multisymplectic structures to include differential-difference equations. For brevity, we restrict attention to differential-difference equations with only two independent variables. (Multisymplectic difference equations and their applications to geometric integration will be treated in a more extensive paper.) As examples, we consider the shallow water equations and the Ablowitz–Ladik equation, which each have a multisymplectic formulation with associated conservation laws.

One observation which comes from the analysis has an application in meteorology: conservation of potential vorticity is a differential consequence of two more fundamental conservation laws. This surprising result and its possible implications for numerical weather prediction are discussed briefly.

## 2. The structural conservation law and its consequences

It is helpful to begin by considering some well-known features of the following simple Hamiltonian system of ODEs:(2.1)

The dot denotes the derivative with respect to the independent variable, *t*. The Hamiltonian system (2.1) has the symplectic 2-form

Here we use the standard notation for Cartan's exterior calculus; in particular, d is the exterior derivative, which maps *r*-forms to (*r*+1)-forms and satisfies the identity d^{2}=0. Consequently, the symplectic 2-form is closed, i.e.

Let *D*_{t} denote the total derivative with respect to *t*, which treats the dependent variables and their derivatives as functions of the independent variable:

The exterior derivative d commutes with *D*_{t}; hence, on solutions of (2.1),(2.2)

In other words, the symplectic 2-form is conserved, irrespective of the function *H*(*q*, *p*). This exemplifies the idea of a ‘structural conservation law’.

The symplectic 2-form and its time-derivative are locally exact, and

The Poincaré lemma implies that, locally, there exists a function *g*(*q*, *p*) satisfying(2.3)

We call such a relationship a ‘1-form quasi-conservation law’. Explicitly,

Pulling this relationship back to the base space (replacing each 1-form d*f*(*q*, *p*) by *D*_{t}{*f*(*q*, *p*)}d*t*) yieldson solutions of (2.1), this amounts to(2.4)

Therefore, *D*_{t}(*H*)=0 on the trajectories in phase space. In the following, these simple observations for the Hamiltonian system (2.1) are extended to multisymplectic systems, with useful consequences.

The general form for a multisymplectic system of PDEs with dependent variables *z*^{i} and independent variables is(2.5)

We use the Einstein summation convention in (2.5) and hereafter; total differentiation with respect to is denoted by the subscript *α* after a comma. The following constraints are usually imposed on the functions .

The symplectic 2-forms satisfy for each

*α*..

Note that (2.5) is a generalization of the Hamiltonian ODE system (2.1). The consequence of the first of the above constraints is that, at least locally, there exist functions such that

Therefore,and so the second constraint is automatically satisfied. In practice, it is usually reasonably easy to construct the functions by inspection; if necessary, they can be constructed systematically with the aid of a homotopy operator (Olver 1993).

Bridges (1997*b*) derived the structural conservation law(2.6)

By following the argument that was used above for Hamiltonian ODEs, we derive the 1-*form quasi-conservation law*:(2.7)which holds on solutions of the multisymplectic system (2.5).

A set of conservation laws can be found by pulling (2.6) back to the base space (Bridges *et al*. 2005). To do this, replace each d*z*^{i} by and use the anti-symmetry of the wedge product. If there are *n* independent variables this yields conservation laws:(2.8)

Just as for the Hamiltonian ODE, we can also pull the quasi-conservation law back to the space of independent variables, obtaining *n* conservation laws:(2.9)where is the Kronecker delta. The first set of conservation laws (2.8) can be obtained from (2.9) by cross-differentiation, and therefore it is reasonable to regard (2.9) as the more fundamental set. However, in some instances the interpretation of a 2-form law may be more obvious than that of the 1-form laws. To illustrate this, we use the conservation laws (2.8) and (2.9) for the shallow water equations.

The shallow water equations are written in the Lagrangian (particle) description. Let represent the position and velocity of the fluid particles, which are functions of *t* and the Lagrangian ‘label-space’ coordinates ** m**=(

*m*

^{1},

*m*

^{2}). (Note: we use

*t*rather than

*m*

^{3}to denote time in this example.) The system rotates with constant angular velocity

*f*/2. The internal energy of the fluid is

*e*(

*τ*), where

*τ*is the reciprocal of the fluid depth. Writing

*τ*in terms of new dependent variables , we obtain

To write the equations of motion in multisymplectic form, we introduce four further dependent variables

The shallow water equations can then be written as the following multisymplectic system (Bridges *et al*. 2005):

This system is of the form (2.5), withand

The only nonzero components with *i*<*j* are

By substituting the results into (2.8) and (2.9), we obtain conservation laws. In the following, the *potential vorticity* for the shallow water system is

The d*m*^{1}∧d*m*^{2} component of the 2-form conservation law (2.6) is(2.10)

In other words, potential vorticity is conserved by fluid particles. The d*t* component of the (pulled back) 1-form quasi-conservation law describes conservation of energy:(2.11)

The remaining components of the structural conservation law are merely label-space derivatives of the energy conservation law (2.11), so they are of no interest. However, the remaining components of the 1-form quasi-conservation law are interesting; they are(2.12)

(2.13)

Conservation of potential vorticity is a differential consequence of (2.12) and (2.13). At present, it is generally agreed that numerical schemes should respect the conservation of potential vorticity. However, we have just seen that fluid particles retain potential vorticity because they are bound by the conservation laws (2.12) and (2.13). Furthermore, potential vorticity is not the only information contained in these conservation laws. This raises an intriguing question: would schemes that preserve (2.12) and (2.13) be more accurate than those which preserve potential vorticity? A major advantage of using constant potential vorticity is that singularities such as fronts can be dealt with. It remains to be seen whether the conservation laws (2.12) and (2.13) are as useful as (2.10); however, they are more fundamental (mathematically speaking) than the conservation of potential vorticity.

As always, calculations that use differential forms on a Riemannian manifold can also be done with tensor analysis. Viúdez (2001) used tensor analysis to derive the conservation law for potential vorticity in its local form (2.10), relating this to Beltrami's material vorticity (see Casey & Naghdi 1991). Similarly, the laws (2.12) and (2.13) can be derived using tensor analysis; they are label-space components of the Lagrangian description of the momentum equation. The chief advantages of using differential forms are simplicity and transparency of structure.

## 3. Conservation laws from symmetries

In a seminal paper on multisymplectic systems, Bridges (1997*a*) demonstrated that Noether's theorem could be applied to the multisymplectic system (2.5), subject to some constraints on the symmetry group. Most notably, these constraints included the requirement that *H*(** z**) should be invariant under the group action.

In fact, Noether's theorem applies in full generality to any multisymplectic system (2.5), without the need for additional constraints. To see this, all that is needed is a variational formulation. It is straightforward to check that (2.5) is the Euler–Lagrange equation for the variational problemwhere(3.1)

(Here *z*^{(1)} denotes the set of first derivatives of ** z**.) Note that the right-hand side of the quasi-conservation law (2.7) is the exterior derivative of the Lagrangian,

*L*. By applying Noether's theorem to this Lagrangian, we can use variational symmetries to derive conservation laws for multisymplectic systems.

Let [** m**,

**] denote the set of independent and dependent variables together with derivatives of**

*z***of all orders. The partial differential operatorgenerates variational symmetries of the Lagrangian (3.1) iffor some functions . Equivalently,**

*z**XL*is in the kernel of the Euler–Lagrange operator (see Olver 1993). Note that

*X*acts only on the dependent variables and their derivatives; every non-trivial symmetry can be written in this form without loss of generality. Noether's theorem states that the variational symmetry generator

*X*yields the conservation law(3.2)

This result can also be proved directly by taking the interior product of the generalized vector field *X* with the 1-form quasi-conservation law (2.7).

As a particular application of Noether's theorem, note that every multisymplectic system (2.5) is invariant under the group of translations in each independent variable . The corresponding symmetry generators are

Then and so Noether's theorem yieldswhich are the conservation laws (2.9) that were obtained by pulling the quasi-conservation law back to the base space. Therefore, the structural conservation law can be regarded as a differential consequence of the conservation laws corresponding to translational invariance in the independent variables.

## 4. Multisymplectic differential-difference equations

So far, we have established that every multisymplectic system of PDEs is a Euler–Lagrange equation, where the Lagrangian *L* is first-order and has linear derivatives. Furthermore, the interior product of any variational symmetry generator *X* with the 1-form quasi-conservation law gives the conservation law that corresponds to *X* according to Noether's theorem. These facts may be used to introduce a multisymplectic structure for differential-difference equations, and to obtain conservation laws. For brevity, we shall adopt the approach of considering only the simplest class of problems, because the generalizations are obvious but notationally cumbersome. Therefore, attention is restricted to equations whose dependent variables *z*_{i} are functions of two variables only, namely, and .

The forward shift operator *S*: *n*→*n*+1 acts on functions of *n* as follows:and therefore

Henceforth, we omit the argument *n* except where it is needed for clarity. The forward difference and backward difference operators are, respectively, Δ^{+}=*S*−id and Δ^{−}=*S*^{−1}Δ^{+}=id−*S*^{−1}, where `id' is the identity map. Unlike *D*_{t}, these difference operators are not derivations; however, they satisfy the useful identity

The adjoint of the forward shift operator is the backward shift: *S**=*S*^{−1}, becauseprovided that these series converge. Consequently (Δ^{+})*=−Δ^{−}.

In the following, we use the notation *z*_{+}^{j} and *z*_{−}^{j} to denote Δ^{+}*z*^{j} and Δ^{−}*z*^{j}, respectively. Consider the variational problem(4.1)

The simplest type of first-order Lagrangian that is linear in derivatives and differences is of the form(4.2)where *L*_{j}(** z**),

*R*

_{j}(

**), and**

*z**H*(

**) are given functions. The Euler–Lagrange equation obtained by varying**

*z**z*

^{i}is

When this is written out explicitly and simplified, it amounts to the following *multisymplectic differential-difference equation*:(4.3)where

By analogy with the continuous case, the 1-form quasi-conservation law is derived by taking the exterior derivative of *L* and using (4.3) to obtain(4.4)

Taking the exterior derivative of (4.4) yields the structural conservation law(4.5)

By construction, the 2-form enclosed in braces in (4.5) is closed, because it is exact. The difference term in the structural conservation law would not have been easy to find by direct extrapolation from the continuous multisymplectic system because it is not anti-symmetric in (*i*, *j*). By construction, it gives a continuous multisymplectic system in the limit as the lines of constant *n* coalesce. Most importantly, because it comes from a variational problem, Noether's theorem (for differential-difference equations) is retained. The differential operatorgenerates variational symmetries of (4.1) iffor some functions *B*^{t}[*t*, *n*, ** z**] and

*B*

^{n}[

*t*,

*n*,

**]. Here, [**

*z**t*,

*n*,

**] denotes dependence on**

*z**t*,

*n*,

**, and shifts and derivatives of**

*z***. Taking the interior product of**

*z**X*and the 1-form quasi-conservation law yields (by Noether's theorem) the conservation law(4.6)

Every multisymplectic system (4.3) is invariant under translations in *t*, which are variational symmetries, so Noether's theorem gives the conservation law(4.7)

Unlike the continuous case, there is no local conservation law corresponding to invariance under translations in the second independent variable, because such translations cannot be made infinitesimally. (For information on nonlocal symmetries of difference equations, see Levi & Winternitz (1993, 1996).)

In many continuous multisymplectic systems (including the shallow water equations) all coefficients are constants, so each is linear. If the functions *R*_{j}(** z**) are linear then (4.3) simplifies as follows. Suppose thatwhere each

*M*

_{ij}is constant. Then the multisymplectic system (4.3) reduces to(4.8)

The structural conservation law for the simplified system (4.8) is(4.9)and the conservation law (4.7) becomes(4.10)

In a pioneering paper, Maeda (1980) introduced a symplectic structure for ordinary difference equations. The discrete part of the simplified multisymplectic system (4.8) is a generalization of Maeda's structure (which corresponds to particular choices of *M*_{ij}).

As an application of the multisymplectic differential-difference structure, consider the Ablowitz–Ladik equation,(4.11)which is integrable (Ablowitz & Ladik 1976). In terms of the variablesthe Ablowitz–Ladik equation amounts to the multisymplectic system (4.8), wherethe only non-zero coefficients *K*_{ij}(** z**) and

*M*

_{ij}are

Having identified the multisymplectic formulation of the Ablowitz–Ladik equation, we can immediately write down the conservation law (4.10). In terms of the original complex variable *u*, (4.10) amounts to(4.12)

The conservation law (4.12) is equivalent to

This is the local formulation of a global law given in Ablowitz & Herbst (1990). Other conservation laws can be obtained from (4.6) provided that *X* generates variational symmetries. Some densities of conservation laws and higher symmetries of the Ablowitz–Ladik equation have recently been computed by Göktaş & Hereman (1998, 1999). The Ablowitz–Ladik equation is one of a class of integrable differential-difference equations whose Hamiltonians contain logarithmic terms. Adler *et al*. (2000) includes a general approach to such systems. For further discussion of local symmetries of general differential-difference equations, see Quispel *et al*. (1992).

## 5. Conclusions

For any multisymplectic system of PDEs, the 2-forms are closed and satisfy the structural conservation law . From this starting-point, we have shown that the structural conservation law is a differential consequence of a quasi-conservation law that is connected with Noether's theorem. Furthermore, we have also seen that every multisymplectic system of PDEs can be written as the Euler–Lagrange equations for a Lagrangian that is first-order and linear in the derivatives. By reversing the line of argument, we have constructed a multisymplectic structure for a discrete independent variable that shares almost all of the important features of the continuous multisymplectic structure. The generalization to multiple discrete variables is obvious: simply add these variables to the Lagrangian while retaining the property that the Lagrangian is first-order and linear in all differences and derivatives.

For brevity, most of this paper has been devoted to developing the general theory of multisymplectic systems. Nevertheless, for each of the applications that we have considered, the multisymplectic formulation gives new information. Application of the discrete multisymplectic structure to numerical integration will be considered in a separate paper. It remains to be seen whether this structure is as useful as its continuous counterpart for stability analysis.

## Acknowledgements

I thank Mark Ablowitz, Tom Bridges and Ian Roulstone for helpful discussions.

## Footnotes

- Received August 6, 2004.
- Accepted December 1, 2004.

- © 2005 The Royal Society