## Abstract

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler–Poincaré equations defined on the Virasoro–Bott group, by using the inverse map (also called ‘back-to-labels’ map). This family contains as special cases the well-known Korteweg–de Vries, Camassa–Holm and Hunter–Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.

## 1. Introduction

The family of equations
*a*, *α*, *β* are real non-negative parameters, was introduced in 1 as the geodesic flow dynamics associated with a variety of right-invariant metrics on the Virasoro–Bott group (see also 2; 3). Various well-known completely integrable soliton equations are special cases of (1.1). For example, when *α*=1 and *β*=0, then equation (1.1) specializes to the Korteweg–de Vries (KdV) equation 4; 5

whereas for *α*=*β*=1 one obtains the Camassa–Holm (CH) equation 6; 7; 8
*α*=0 and *β*=1 one finds the Hunter–Saxton (HS) equation 9; 10

The aim of this paper is to derive a new canonical variational principle for the family of equations (1.1), and thereby determine its new multisymplectic formulation. By doing so, we obtain unified variational and multisymplectic characterizations of all three of the well-known integrable soliton equations, KdV, CH and HS, which are subcases of the general family of equations in (1.1).

Variational principles have proved extremely useful in the study of nonlinear evolution partial differential equations (PDEs). For instance, they often provide physical insights into the problem being considered; facilitate discovery of conserved quantities by relating them to symmetries via Noether’s theorem; allow one to determine approximate solutions to PDEs by minimizing the action functional over a class of test functions (see, e.g., 11); and provide a way to construct a class of numerical methods called variational integrators (see 12; 13). A canonical variational principle for the KdV equation expressed in terms of the velocity potential was first proposed by Whitham 14; see also 11; 5; 15; 16. In fact, there is an infinite family of such Lagrangians, as shown by Nutku 17. Two canonical variational principles for the dispersionless CH equation (*a*=0) were introduced in 18; 19. Two variational structures are also known for the HS equation with *a*=0 (see 20; 9; 10).

Multisymplectic structures of Hamiltonian PDEs were first considered by Bridges 21 as a natural generalization of the symplectic structure of Hamiltonian ODEs. Among other applications, the multisymplectic formalism is useful for, e.g., the stability analysis of water waves (see 21; 22) and construction of a class of numerical methods known as multisymplectic integrators (see 23; 12). It has been observed in the literature that, as for symplectic integrators for Hamiltonian ODEs, multisymplectic integrators demonstrate superior performance in capturing long-time dynamics of PDEs (see 24). To the best of our knowledge, only one multisymplectic formulation of the KdV equation has been considered so far (see 22; 25). Four different multisymplectic formulations are known for the dispersionless CH equation (see 26; 18; 19). Two multisymplectic structures for the HS equation with *a*=0 were described in 27.

### (a) Main content

The main content of the remainder of this paper is as follows.

In §2, we review the Euler–Poincaré theory on the Virasoro–Bott group and then construct a new canonical variational principle in terms of the inverse map. The main result of this section is theorem 2.2, which provides the Clebsch variational principle for Euler–Poincaré equations on the dual of the Virasoro–Bott algebra. The variational equations yield the new Clebsch momentum map

In §3, we use the Clebsch representation based on the inverse flow map to derive the multisymplectic form formula associated with our variational principle. We then deduce a new multisymplectic formulation of the family of equations (1.1). The main result of this section is theorem 3.1, which derives the multisymplectic formulation based on the inverse flow map.

Section 4 contains the summary of the present work and a discussion of several new directions for research that it reveals.

## 2. The inverse map and Clebsch representation

Equation (1.1) was first introduced in the Lie–Poisson context (see 1; 2; 3). In this section, we take the Lagrangian point of view, instead, and formulate (1.1) as an Euler–Poincaré equation on the Virasoro–Bott group. For this, we construct a canonical variational principle that will later allow us to determine a multisymplectic formulation of (1.1).

### (a) Euler–Poincaré equation on the Virasoro–Bott group

Let *S*^{1}) be the diffeomorphism group of *S*^{1}. The tangent bundles can be identified as *S*^{1}. In particular, the Lie algebra of *S*^{1} is *S*^{1}) is *B*(*ψ*_{1},*ψ*_{2}) is given by
*L*^{2} inner product

Suppose a Lagrangian system is defined on *u*,*a*)=*L*(id,0,*u*,*a*) and the reduced variational principle
*δ*(*u*,*a*)=(*d*/*dt*)(*v*,*b*)−[(*u*,*a*),(*v*,*b*)], where (*v*(*t*),*b*(*t*)) vanish at the end points, and the time derivative of a Virasoro algebra-valued function of time is understood as (*d*/*dt*)(*v*(*t*),*b*(*t*))=((∂*v*/∂*t*)(⋅,*t*),(*db*/*dt*)(*t*)). This variational principle leads to the *Euler–Poincaré equation*:

### Theorem 2.1

*Let the reduced Lagrangian be defined as
**where α,β≥0. Then the corresponding Euler–Poincaré equations take the form
*

### Proof.

The case *α*=1 and *β*=0 is shown in 16. The case *α*,*β*≥0 is a straightforward generalization. ▪

The first equation in (2.9) is equivalent to (1.1), because the second equation in (2.9) implies *a*=const.

### (b) Reconstruction equations and the inverse map

A solution (*u*(*t*),*a*(*t*)) of (2.7) describes the evolution of the (right-invariant) Lagrangian system in the Virasoro algebra, denoted by *u*(*t*),*a*(*t*)), i.e. in short-hand notation *R* denotes right translation on the Virasoro–Bott group and *TR* its tangent lift (see 29; 16). By using (2.1) and (2.2), we obtain the reconstruction equations

In the context of fluid dynamics, a time-dependent diffeomorphism *ψ*(*t*)∈Diff(*S*^{1}) maps a given reference configuration to the fluid domain at each instant of time, i.e. *ψ*(*X*,*t*) represents the position at time *t* of the fluid particle labelled by *X*. On the other hand, the inverse map *l*(*t*)=*ψ*^{−1}(*t*) maps from the current configuration of the fluid to the reference configuration, i.e. *l*(*x*,*t*) is the label of the fluid particle occupying the position *x* at time *t*. The Eulerian velocity field *u*(*x*,*t*) gives the velocity of the fluid particle that occupies the position *x* at time *t*, i.e. *u*(*t*),*a*(*t*)) of (2.7), one can easily solve (2.14) for *l*(*x*,*t*) and *θ*(*t*).

### (c) Clebsch variational principle

#### (i) General reduced Lagrangian

As discussed in §2a, equation (1.1) has an underlying variational structure. However, the Euler–Poincaré variational principle (2.6) imposes constraints on the variations of the functions *u* and *a*, which may be inconvenient in some applications, for instance, when one is interested in deriving variational integrators, or determining the underlying multisymplectic structure, as is our goal in this work. One can circumvent this issue by considering an augmented action functional which includes the reconstruction equations as constraints. This idea was formalized in the context of variational Lie group integrators in back-to-back papers in 30; 31. The idea of using the inverse map *l*(*x*,*t*) (also called ‘back-to-labels’ map) and the advection condition (2.12) appeared in 32; 33, and was later used in 18 to construct multisymplectic formulations of a class of fluid dynamics equations. We extend these ideas to systems defined on the Virasoro–Bott group.

The Clebsch variational principle (also sometimes called the Hamilton–Pontryagin principle) enforces stationarity of the action *π*=*π*(*x*,*t*) and λ=λ(*t*) are Lagrange multipliers, and consider the variational principle
*δu*, *δa*, *δπ*, *δ*λ, and vanishing endpoint variations *δl* and *δθ*, i.e. *δl*(*x*,*t*_{1})=*δl*(*x*,*t*_{2})=*δθ*(*t*_{1})=*δθ*(*t*_{2})=0. The resulting variational equations are

### Remark

Equation (2.18) is the Clebsch momentum map

We will now show that the dynamics generated by the system (2.17) is equivalent to the dynamics generated by the Euler–Poincaré equation (2.7).

### Theorem 2.2

*Suppose the functions u(x,t), a(t), l(x,t), θ(t), π(x,t) and λ(t) satisfy the Euler–Lagrange equations (2.17). Then the functions u(x,t) and a(t) satisfy the Euler–Poincaré equation (2.7).*

### Proof.

Let (*w*,*c*) be an arbitrary element of the Virasoro algebra *A* contains the functions *u*, *l*, *π*, λ, and their spatial derivatives. On the other hand, we have
*B* contains the functions *u*, *l*, *π*, λ, and their spatial derivatives. After rather tedious, albeit straightforward algebraic manipulations we find that *A*+*B*=0. Therefore, we have that, for all

#### (ii) Separable reduced Lagrangian

The variational principle (2.16) simplifies significantly when one considers separable Lagrangians of the form
*a*=const. Treating *a* as a constant, we can eliminate the variables *θ* and λ from the action functional (2.15). Consider the action functional
*δS*=0 with respect to arbitrary variations *δu*, *δπ*, and vanishing end-point variations *δl*, yields the variational equations

### Remark

The action functional (2.25) provides a new variational formulation for equation (1.1) when the Lagrangian (2.8) is considered. For *a*=0 this action functional reduces to the action functional for the dispersionless CH equation (*α*=*β*=1) introduced in 18 and one of the action functionals for the HS equation (*α*=0 and *β*=1) described in 10. For *α*=1 and *β*=*a*=0 we also obtain a variational principle for the inviscid Burgers’ equation.

## 3. Inverse map multisymplectic formulation

The action functional and variational principle introduced in §2c(ii) allow the identification and analysis of a new multisymplectic formulation of the family of equations (1.1). Multisymplectic geometry provides a covariant formalism for the study of field theories in which time and space are treated on equal footing. Multisymplectic formalism is useful for, e.g., the stability analysis of water waves (see 21; 22) or construction of structure-preserving numerical algorithms (see 23; 12). The multisymplectic form formula was first proved by Marsden *et al.* 12 and provides an intrinsic and covariant description of the conservation of symplecticity law, first introduced by Bridges 21 in the context of multisymplectic Hamiltonian PDEs. In §3a, we review the multisymplectic geometry formalism and derive the multisymplectic form formula associated with (2.25). We further make a connection with Bridges’ approach to multisymplecticity in §3b and determine a multisymplectic Hamiltonian form of the Euler–Lagrange equations (2.26).

### (a) Multisymplectic form formula and conservation of symplecticity

The multisymplectic form formula is the multisymplectic counterpart of the fact that in finite-dimensional mechanics, the flow of a mechanical system consists of symplectic maps. It was first proved for first-order field theories in 12, and later generalized to second-order field theories in 19. As the field theory described by the action functional (2.25) with the Lagrangian (2.8) is second order, we follow the theory developed in 19. For the convenience of the reader, below we briefly review multisymplectic geometry and jet bundle formalism necessary for our discussion.

Let *x*^{μ})=(*x*^{1},*x*^{0}), where *x*^{1}≡*x* is the spatial coordinate and *x*^{0}≡*t* is time. Define the configuration fibre bundle *τ*_{XY}:*Y* →*X* as *y*^{A})=(*y*^{1},*y*^{2},*y*^{3}) with *y*^{1}≡*l*, *y*^{2}≡*u*, and *y*^{3}≡*π*. Physical fields are sections of the configuration bundle, that is, continuous maps *ϕ*:*X*→*Y* such that *τ*_{XY}°*ϕ*=id_{X}. In the coordinates (*x*^{μ},*y*^{A}), a field *ϕ* is represented as *ϕ*(*x*,*t*)=(*x*^{μ},*ϕ*^{A}(*x*^{μ}))=(*x*,*t*,*l*(*x*,*t*),*u*(*x*,*t*),*π*(*x*,*t*)).

For a *k*th order field theory, the evolution of the field takes place on the *k*th jet bundle *J*^{k}*Y* . The first jet bundle *J*^{1}*Y* is the affine bundle over *Y* with the fibres *y*∈*Y* _{(x,t)}, where the linear maps *ϑ* represent the tangent mappings *T*_{(x,t)}*ϕ* for local sections *ϕ* such that *ϕ*(*x*,*t*)=*y*. The local coordinates (*x*^{μ},*y*^{A}) on *Y* induce the coordinates *J*^{1}*Y* . Intuitively, the first jet bundle consists of the configuration bundle *Y* , and of the first partial derivatives of the field variables with respect to the independent variables. We can think of *J*^{1}*Y* as a fibre bundle over *X*. Given a section *ϕ*:*X*→*Y* , we can define its first jet prolongation
*J*^{1}*Y* over *X*. For higher-order field theories we consider higher-order jet bundles, defined iteratively by *J*^{k+1}*Y* =*J*^{1}(*J*^{k}*Y*). We denote the local coordinates on *J*^{2}*Y* by *j*^{2}*ϕ*:*X*→*J*^{2}*Y* is given in coordinates by *j*^{2}*ϕ*(*x*^{μ})=(*x*^{μ},*ϕ*^{A},∂*ϕ*^{A}/∂*x*^{μ},∂^{2}*ϕ*^{A}/∂*x*^{μ}∂*x*^{ν}). Let *J*^{3}*Y* . The third jet prolongation *j*^{3}*ϕ* is defined similar to *j*^{1}*ϕ* and *j*^{2}*ϕ*. For more information about the geometry of jet bundles see 34; 35.

In the jet bundle formalism introduced above, the action functional (2.25) with the reduced Lagrangian (2.8) can be written as
*d*^{2}*x*=*dx*∧*dt*, and the Lagrangian density *ϕ*(*x*,*t*) that extremize *S*, that is,
*V* on *Y* . This leads to the Euler–Lagrange equations

For a second-order field theory, the multisymplectic structure is defined on *J*^{3}*Y* (see 19). Given the Lagrangian density *J*^{3}*Y* , in local coordinates given by
*dx*_{0}=−*dx* and *dx*_{1}=*dt*, and the *formal* partial derivative in the direction *x*^{ν} of a function *ϕ* satisfying (3.6) or (3.7). For a given *V* on *Y* such that *V* . The multisymplectic form formula for second-order field theories (see 19) states that if *V* and *W* in *j*^{3}*ϕ*)* denotes the pull back by the mapping *j*^{3}*ϕ*, and *j*^{3}*V* is the third jet prolongation of *V* , that is, the vector field on *J*^{3}*Y* whose flow is the third jet prolongation of the flow *V* , i.e.
*V* , *W*, in the local coordinates (*x*^{μ},*y*^{A}) represented by (*V* ^{μ}(*x*^{μ},*y*^{A}),*V* ^{A}(*x*^{μ},*y*^{A})) and (*W*^{μ}(*x*^{μ},*y*^{A}),*W*^{A}(*x*^{μ},*y*^{A})), respectively. Let us work out the form of the formula (3.11) for *τ*_{XY}-vertical first variations, i.e. *V* ^{μ}(*x*^{μ},*y*^{A})=*W*^{μ}(*x*^{μ},*y*^{A})=0. Denote the components of *j*^{3}*V* as *j*^{3}*W*. The multisymplectic form formula then becomes
*j*^{3}*ϕ*(*x*,*t*). By applying Stokes’ theorem and using the fact that

### (b) Multisymplectic Hamiltonian partial differential equation formulation

Bridges 21 introduced the notion of multisymplecticity by generalizing the notion of Hamiltonian systems to PDEs. A multisymplectic structure *ω* and *κ*, where pre-symplectic means that the 2-forms are closed, but not necessarily non-degenerate. A multisymplectic Hamiltonian system is a PDE of the form
*x* and *t*, *M*(*z*), *K*(*z*) are *n*×*n* antisymmetric matrices defined by

We will use the multisymplectic form formula (3.14) to deduce the multisymplectic Hamiltonian PDE form (3.17) of the Euler–Lagrange equations (2.26). We note that, for *a*>0, the vector components that appear in (3.15) only correspond to the seven coordinate directions *y*^{1}, *y*^{2}, *y*^{3}, *J*^{3}*Y* . We will therefore consider *l*,*u*,*π*,Δ,*Θ*,*Ξ*,*Π*). Define the projection map
*M*(*z*) and *K*(*z*) can be read off from (3.15) as
*H* can be read off from the *dx*∧*dt* term in (3.10) as

### Theorem 3.1

*Suppose a>0. Then the Euler–Lagrange equations (2.26) with the Lagrangian (2.8) are equivalent to the multisymplectic Hamiltonian system (3.17) with the matrices (3.20) and the Hamiltonian (3.21). That is, if ϕ(x,t)=(x,t,l(x,t),u(x,t),π(x,t)) is a solution of (2.26), then* *is a solution of (3.17). Conversely, if z(x,t) is a solution of (3.17), then ϕ(x,t)= (x,t,z*_{1}*(x,t),z*_{2}*(x,t),z*_{3}*(x,t))=(x,t,l(x,t),u(x,t),π(x,t)) is a solution of (2.26).*

### Proof.

Substituting (3.20) and (3.21) in (3.17) yields the system of equations
*l*_{x} and equation (3.22f) implies *Θ*=*u*_{x}. Then, equations (3.22e) and (3.22d) imply *Ξ*=*l*_{xx} and *Π*=*u*_{xx}, respectively. By substituting these identities in the remaining equations (3.22a)–(3.22c), we obtain a system equivalent to (2.26), which completes the proof. ▪

Bridges 21 showed that the conservation of symplecticity law
*z*(*x*,*t*) of (3.17), where *z*(*x*,*t*). This is an equivalent statement of (3.16), because if *W* and *V* are first variations for (3.7), then

### Remark

Equations (3.17), (3.20) and (3.21) provide a new multisymplectic formulation for the family of equations (1.1) with *a*>0. For *a*=0 several special cases can be obtained. If *β*>0, then equations (3.22d), (3.22f) and (3.22g) become trivial, and it is enough to consider the variables *z*=(*l*,*u*,*π*,*Θ*). The matrices *M* and *K* then take the form
*H*(*z*)=(*α*/2)*u*^{2}−(*β*/2)*Θ*^{2}. For *α*=*β*=1 this reproduces the multisymplectic structure for the dispersionless CH equation found in 18, and for *α*=0, *β*=1 we obtain a new multisymplectic formulation of the HS equation with *a*=0. If in addition *β*=0, then equation (3.22e) also becomes trivial, and a further simplification is possible: we consider the variables *z*=(*l*,*u*,*π*) with the matrices
*H*(*z*)=(*α*/2)*u*^{2}. This final simplification provides a multisymplectic formulation for the inviscid Burgers’ equation.

## 4. Summary, open problems and opportunities for future work

In this paper, we have introduced a new type of Clebsch representation that extends the momentum map formulation for fluid dynamics introduced in Holm and colleagues 32; 33 based on the inverse flow map to the case when the group action governing Lagrangian fluid paths includes the Bott 2-cocycle in equation (2.2). Physically, this means that linear dispersion with third-order spatial derivatives can be included, as required for investigating the multisymplectic structures of the KdV, CH and HS equations. Moreover, the multisymplectic form formula was shown to persist and was derived explicitly for this important class of equations, by using our new type of Clebsch representation, identified in equation (2.18) as the momentum map associated with particle relabelling with group actions which include the Bott 2-cocycle. In addition, symplecticity was found to be conserved in this new class of flows. Consequently, new types of structure-preserving numerics for soliton equations with linear dispersion can now be developed.

Multisymplectic integrators are methods that preserve a discrete version of the symplectic conservation law (3.23). There is numerical evidence that these schemes locally conserve energy and momentum remarkably well (see, e.g., 37; 38; 23; 26; 27; 39; 40; 41; 25), which is a much stronger property than merely global conservation over the whole spatial domain (see 42). Variational integrators are based on discrete variational principles, which provide a natural framework for the discretization of Lagrangian systems (see, e.g. 43; 12; 13; 44; 45; 46; 47). A discrete action functional can be obtained by discretizing the functional (2.25) on a space–time mesh. A variational numerical scheme is then derived by extremizing the discrete action with respect to the discrete set of the values of the fields *l*, *u* and *π*. Variational integrators satisfy a discrete version of the multisymplectic form formula (3.12), and are therefore multisymplectic. Moreover, in the presence of a symmetry, they satisfy a discrete version of Noether’s theorem, as a consequence of which many of the conservation laws of the continuous system persist. These directions will be explored in future work. They are beyond the scope of the present derivation and formulation.

Furthermore, the new Clebsch momentum map with the Bott 2-cocycle in equation (2.18) represents an opportunity to extend the approach in 48 of using Clebsch variational principles for introducing noise into continuum mechanics. The new Clebsch momentum map (2.18) will enable us to investigate a new type of interplay among nonlinearity and noise that also includes stochastic linear dispersion. This interplay introduces a class of dynamical problems addressing ‘wobbling’ solitons governed by stochastic PDEs with stochastic mass/label transport. Consider a stochastic deformation of the Euler–Poincaré equation (2.7) such that the coadjoint action ad* is taken with respect to the perturbed Virasoro algebra element *u*,*a*), where *ξ*(*x*) and element *a*, respectively. Because of the definition of the coadjoint action (2.5), the perturbation of *a* does not have any effect on the equation and can be omitted. The stochastic Euler–Poincaré equation will therefore take the form
*W*(*t*) is the standard Wiener process, d denotes the stochastic differential with respect to the time variable *t* and ° denotes Stratonovich time integration. For the Lagrangian (2.8), we obtain a stochastic deformation of the family of equations (1.1) as
*α*=*β*=1 and *a*=0 in the equation above), electromagnetic field equations and various fluid dynamics equations (see 49; 50; 51; 52; 48; 53; 54). This approach retains many properties of the unperturbed equations, such as the peaked soliton solutions of the CH equation, and the Kelvin circulation theorem for fluid dynamics. Moreover, for certain functional forms of *ξ*(*x*), the introduction of this type of coadjoint transport noise can preserve the deterministic isospectral problem, while introducing stochasticity into the evolution equations for the corresponding eigenfunctions. This stochastic process preserves certain aspects of the inverse scattering methods for determining the soliton solutions of stochastic PDEs, as discussed in 51. The results presented in 50; 51; 54 suggest that, for smooth initial conditions and for a proper class of the correlation functions *ξ*(*x*), the solutions of (4.2) are likely to retain their spatial regularity. For instance, in the case of spatially uniform noise, with *ξ*(*x*)=*γ*=const, if *u*(*x*,*t*) is a solution of (1.1), then *u*(*x*−*γW*(*t*),*t*) is a solution of (4.2), which can be easily verified by a direct substitution.

Under this regularity hypothesis, solutions of equation (4.2) are seen to be critical points of the action functional
*u* in the reconstruction equations (2.14) is replaced with *F*(*x*,*t*) and *G*(*x*,*t*) have been defined in (3.15), and the function

Finally, it is worth pointing out that equation (4.2) can be put into Lie–Poisson Hamiltonian form as coadjoint motion,
*m*=*αu*−*βu*_{xx} is the momentum map, *J*=∂_{x}*m*+*m*∂_{x}+*a*∂_{xxx} is the Lie–Poisson Hamiltonian operator, and the Hamiltonians are *δh*/*δm*=*u* and *α*=1 and *β*=0, expressed here as a member of the class of stochastic Hamiltonian PDEs in (4.6), has also appeared in 55, as

## Data accessibility

This paper has no data.

## Authors' contributions

Our contributions were equally balanced in a true collaboration.

## Competing interests

We declare we have no competing interests.

## Funding

During this work, the authors were partially supported by the European Research Council Advanced Grant 267382 FCCA and the UK EPSRC Grant EP/N023781/1 held by D.H.

## Acknowledgements

We are very grateful to Alexis Arnaudon and Nader Ganaba for many useful comments, references and stimulating discussions during the present work.

- Received January 29, 2018.
- Accepted April 9, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.