On the multi-symplectic structure of Boussinesq-type equations I
Abstract.

The Boussinesq equations are known since the end of the XIX century. However, the proliferation of various Boussinesq-type systems started only in the second half of the XX century. Today they come under various flavours depending on the goals of the modeller. At the beginning of the XXI century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the Hamiltonian. In the present paper a family of Boussinesq-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known Boussinesq models, the identification of those systems with additional Hamiltonian structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full Euler equations is also discussed.



Key words and phrases: multi-symplectic structure; long dispersive wave; Boussinesq equations; surface waves

MSC [2010]: 76B15 (primary), 76B25 (secondary)
PACS [2010]: 47.35.Bb (primary), 47.35.Fg (secondary)

Key words and phrases:
multi-symplectic structure; long dispersive wave; Boussinesq equations; surface waves
Corresponding author

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Angel Durán
Universidad de Valladolid, Spain
Denys Dutykh
CNRS, Université Savoie Mont Blanc, France
Dimitrios Mitsotakis
Victoria University of Wellington, New Zealand

On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

arXiv.org / hal

Last modified: July 1, 2019

1. Introduction

The first Boussinesq-type equation was proposed by Joseph Boussinesq in 1877 [Boussinesq1877], who gave the name to different families of nonlinear wave equations proposed since then. This topic remained a sleeping beauty, [Ke2015], during almost one hundred years. The modern era was opened by Howell Peregrine in his theoretical and numerical investigations of long wave transformations on uniform slopes [Peregrine1967]. A proliferation of various systems started, serving various purposes of the near-shore hydrodynamics. Some reviews of this topic can be found in [Madsen1999, DMII, Brocchini2013].

At the beginning of the XXI century an effort to classify such systems, at least for even bottoms, was undertaken. In this way, the following four-parameter family

(1.1)
(1.2)

denoted here as the family, was formulated and analyzed by Bona et al., [BCS, Bona2004]. In (1.1), (1.2), and are real-valued functions defined for and , while the coefficients are defined as

(1.3)
(1.4)

where , and . The Systems (1.1), (1.2) are proposed for modelling the two-way propagation of one-dimensional, small amplitude, irrotational long surface waves in a channel of constant depth, see Figure 1. The variables and represent, respectively, the position along the channel and time, while and are proportional to the free surface excursion and to the horizontal velocity of the fluid at at a non-dimensional height , respectively. A review of the existing mathematical theory for the Systems (1.1), (1.2) which includes a list of systems of the family of especial interest, well-posedness results of the corresponding Initial – Value Problem (IVP) and periodic IVP as well as existence and stability results of solitary-wave solutions, can be seen in [DMII]. A symmetric111Here, the symmetry is understood in the sense of FriedrichsLax [Friedrichs1971]. variant of the system of the general form

(1.5)
(1.6)

with as in (1.3), (1.4), was proposed by Bona et al. in [BCL]. These systems retain the same order of approximation to the Euler equations as (1.1), (1.2), reduce to a symmetric hyperbolic system when the dispersive third-order derivative terms are omitted, preserve the norm

(1.7)

and, concerning existence and uniqueness of solutions of the corresponding IVP, are locally well-posed. Higher-order Boussinesq-type systems are proposed in e.g. [BCS] and also by Daripa in [Daripa2006]. Although we focus here on surface wave theory, it may be worth mentioning the derivation of Boussinesq systems for internal waves in, e.g. [BLS2008].

Figure 1. Sketch of the fluid domain. We note that it is always possible to choose dimensionless variables such that and . That is why these constants are absent in governing Equations (1.5), (1.6) (such a change of dependent and independent variables is given in [DMII, Equation (3.13)]). The brackets denote the spatial averaging operator, which fix the still water level in our problem.

It is well-known that only a sub-family of systems (1.1), (1.2) with possesses a co-symplectic Hamiltonian structure

(1.8)

on a suitable functional space for , where the co-symplectic structure is given by the nonlocal matrix operator

(1.9)

(the symbol denotes composition) and is the Hamiltonian functional given by

with , standing for the variational (i.e. Fréchet) derivatives of with respect to \texteta and , respectively. (This Hamiltonian structure is lost in the symmetric version (1.5), (1.6).) Note that the co-symplectic operator does not depend explicitly neither on time , nor on space , nor on the solution . In this sense the symplectic structure is quite rigid for Partial Differential Equations (PDEs). For each IVP of (1.8), the Hamiltonian functional is conserved in time and its value, determined by the corresponding initial conditions, can be considered as a generalized ‘energy’ of the state of the system represented by the solution, [Christov2001]. In addition to , the impulse functional

is preserved by smooth, decaying enough at infinity solutions.

The present paper delves into the properties of Boussinesq-type systems focusing on the multi-symplectic (MS) structure. It is the first part of a study devoted by the authors to this question and deals with the derivation of a MS family of equations of Boussinesq type, the comparison with other existing Boussinesq systems and the description of some mathematical properties, with special emphasis on the existence of traveling-wave solutions. The second part of the study, devoted to the construction of multi-symplectic schemes for the approximation to the systems, will be the subject of a forthcoming paper.

The multi-symplectic theory generalizes the classical Hamiltonian formulations, [Basdevant2007], to the case of PDEs such that the space and time variables are treated on the equal footing [Bridges1997] (see also [Leimkuhler2004, Chapter 12]). Multi-symplectic formulations are also gaining popularity for both mathematical investigations, [Marsden1998], along with numerical structure preserving modeling, [Bridges2001, Dutykh2013a].

The history of multi-symplectic formulations can be traced back to V. Volterra (1890) who generalized Hamiltonian equations for variational problems involving several variables, [Volterra1890, Volterra1890a]. Later these ideas were developed in 1930’s, [DeDonder1930, Weyl1935, Lepage1936]. Finally, in 1970’s this theory was geometrized by several mathematical physicists, [Goldschmidt1973, Kijowki1974, Krupka1975, Krupka1975a] similarly to the evolution of symplectic geometry from the ideas of J.-L. Lagrange, [Lagrange1853, Souriau1997]. In our study we will be inspired by modern works on multi-symplectic PDEs, [Bridges1997, Marsden1998]. Recently this theory has found many applications to the development of structure-preserving integrators with different strategies, [Bridges2001, Moore2003a, Chen2011, Dutykh2013a].

Let us also briefly review the main known equations arising in the modeling of long waves with MS formulation. The KdV equation is among the multi-symplectic veterans, [Zhao2000, Bridges2001], along with the NLS equation, [Chen2002]. Among one-way propagation models, the multi-symplectic structure of the Benjamin–Bona–Mahony (BBM) and generalized222The generalization consisted in taking higher order nonlinearities, i.e. , . BBM equations was highlighted in [Sun2004]. Some interesting numerical results for the BBM equation were presented in [Li2013]. The ‘good’ scalar Boussinesq equation was studied in [Huang2003]. The multi-symplectic structure of the celebrated SerreGreenNaghdi (SGN) equations was unveiled in [Chhay2016]. Finally, the multi-symplectic structure of two-layer SGN system modeling the propagation of long fully nonlinear internal waves was highlighted recently in [Clamond2016a]. We also remind that the full water wave problem is multi-symplectic, [Bridges1997].

The main highlights of the present paper revolve around the derivation of a family of Boussinesq systems with multi-symplectic structure. The procedure for the derivation starts from a system of form with a general homogeneous quadratic polynomial as nonlinear term and determines a combination of parameters allowing the multi-symplectic formulation. Additional advantages of the technique implemented are the identification of the MS structure in Boussinesq systems of the family (1.5), (1.6) and the way how to modify the Equations (1.1), (1.2) to admit the MS property.

After the derivation, it is expected to discuss some properties of the new systems. In this sense, the present paper is focused on the identification of those equations of the family with additional symplectic structure, the well-posedness of the corresponding IVP and the existence of different types of solitary-wave solutions. For the last two points, the form of the nonlinearity enables us to make use of the existing literature on them. The discussion of consistency of the Euler equations with the new system is here initiated from the MS structure of Equations (1.5), (1.6). Considered as part of this discussion are also some comparisons of the solitary-wave speed-amplitude relations for the MS systems with systems (1.1), (1.2) and the Euler system. These comparisons suggest the existence of new MS systems, derived in this paper, with good performance in this sense. Nevertheless, a complete analysis of the question deserves a future research. The description of the models is completed with the construction and development of multi-symplectic schemes of approximation, being the subject of a second, forthcoming part.

The paper is organized according to the following structure. After a brief reminder of MS theory for PDEs, Section 2 is devoted to the procedure of derivation of the MS structure. The main steps are first described to obtain the MS formulations of the KdV–BBM equation and the symmetric System (1.5), (1.6). The technique is then extended to construct a family of MS Boussinesq-type systems and to explain how to modify (1.1), (1.2) in such a way that the resulting systems are multi-symplectic. Some properties of the new equations concerning well-posedness and existence of solitary-wave solutions are studied in Section 3. The main conclusions and perspectives are outlined in Section 4.

2. Multi-symplectic structure

We recall first some basic facts about multi-symplectic geometry and PDEs. A PDE333Note that there is no assumption that the IVP for this PDE is well-posed. To give an example, the multi-symplectic setting may include some elliptic PDEs in space-time which are ill-posed in the sense of Hadamard. (or a system of PDEs) is said to be multi-symplectic in one space dimension if it can be written in the following canonical form:

(2.1)

for some , where , and are some real, skew-symmetric matrices, the dot denotes the matrix-vector product in and is the classical gradient operator in . The given function is assumed to be a smooth function of its variables .

The formulation (2.1) represents a direct generalization of Hamiltonian PDEs where the space and time are treated on equal footing. On the other hand, the main drawback of formulation (2.1) is that it is not intrinsic, i.e. not coordinate-free on the base manifold . However, it is sufficient for our purposes, since in wave propagation we would like to keep the distinction between space and time coordinates.

All equations with the multi-symplectic form (2.1), have also a certain number of relevant properties. One is the existence of conservation laws. Note first that Equations (2.1) satisfy the multi-symplectic conservation law:

(2.2)

with

(2.3)

with being the standard exterior product of differential forms, [Spivak1971]. The form \textomega defines a symplectic structure on (where denotes the rank of the matrix ), which is associated with the time direction and the form \textkappa defines a symplectic structure , which is associated with the space direction, [Bridges2001]. From definitions of forms (2.3), we can see that the multi-symplectic approach relies on a local concept of symplecticity, since symplectic forms may vary with the solution in space and in time similarly to a Lagrangian tracer in a fluid flow. This observation explains why the class of multi-symplectic PDEs is more general. On the other hand, when designing a geometric integrator, the principal requirement is that the discretization conserves exactly the symplecticity (2.2).

Remark 1.

In classical field theories sometimes the following form is called a multi-symplectic form:

In [Bridges2001] it was proposed to call a meta-symplectic form and we follow this terminology. Throughout our manuscript the term multi-symplectic form refers to the system of PDEs (2.1) or to the couple of differential forms depending on the context.

Additionally, other conservation laws can be derived when the function does not depend explicitly on or . Defining the generalized energy and generalized momentum densities as

and corresponding fluxes

(the symbol denotes the standard scalar product in ) then the generalized energy and momentum are conserved:

(2.4)

Note that one of the advantages of the multi-symplectic structure is that a geometric interpretation of the conservation laws can be derived from the application of Noether theory. Note also that, in general, it is not possible to conserve energy and momentum (2.4) with uniform discretizations additionally to multi-symplecticity (2.2).

Finally, multi-symplectic PDEs (2.1) automatically possess also the Lagrangian variational structure. Indeed, it is not difficult to check that Equation (2.1) is the EulerLagrange equation to the following Lagrangian functional:

with the Lagrangian density defined as

Here , denote arbitrary instances of time and, similarly, , are some locations in space . From the last definition, observe that the function plays the rôle of the generalized potential energy. The Hamilton principle states that the physical trajectory corresponds to stationary values of the action functional, i.e. .

2.1. MS structure of the KdV–BBM equation

As a necessary step towards understanding the way how the multi-symplectic structure of the type systems will be derived later, we first study the case of the KdV–BBM equation

(2.5)

where , are some real coefficients. The variable can be related both to the velocity or to the free surface elevation. The KdV–BBM Equation (2.5) arises in the modeling of unidirectional water wave propagation [Dutykh2010e] and in the study of non-integrable solitonic gases [Dutykh2014d]. It was also used to study the well-posedness of the KdV equation in [Bona1975a].) The multi-symplectic structure of (2.5) has never been reported before to the best of our knowledge. We provide below an example of such structure in .

At the first step, we rewrite Equation (2.5) in a conservative form:

(2.6)

In order to lower the order of the equation, the following variables are introduced:

The new variable is a generalized potential for the field . This potential appears, for example, in Lagrangian variational formulations of both KdV and BBM equations. As the final conceptual step, we rewrite Equation (2.6) in the following slightly unusual way:

The last equation suggests the introduction of an extra variable, which has the meaning of a flux:

Now, we have all elements to present the desired multi-symplectic structure. The vector including the field along with ‘conjugate momenta’ is defined as

Note then that (2.5) is equivalent to the system of equations:

(2.7)
(2.8)
(2.9)
(2.10)
(2.11)

We now consider the vector field given by the right hand side of (2.7) – (2.11):

Since the Jacobian of is symmetric for all , then Poincaré lemma implies that is conservative and thus for some potential . Therefore, System (2.7) – (2.11) can be written in the canonical matrix-vector form (2.1) with the following skew-symmetric matrices:

A potential energy for the KdV–BBM Equation (2.5) is

This completes the definition of the multi-symplectic structure for (2.5). We conjecture that another multi-symplectic structure for this equation in , with is not possible.

2.2. MS structure of the symmetric Boussinesq system

The previous steps can be adapted to obtain the MS structure of some systems of the family (1.5), (1.6). Similarly to Section 2.1, these can first be written in the conservative form

(2.12)
(2.13)

and we introduce the additional variables:

(2.14)
(2.15)

where are generalized potentials and , are space and time gradients of dynamic variables \texteta and correspondingly. Using these variables, we can rewrite Equations (2.12), (2.13) as

Thus, defining two additional fluxes:

(2.16)
(2.17)

then the vector with auxiliary variables can now be defined:

(2.18)

System (2.12), (2.13) is given below in the expanded form:

Now, if is the vector field whose components are given by the right hand side:

then we observe that the Jacobian of is symmetric for all if and only if . Under this condition, the last system of scalar equations can be recast into the canonical matrix-vector form (2.1), if we introduce the following skew-symmetric matrices:

(2.19)
(2.20)

and where a potential energy for the symmetric sub-family reads:

This completes the presentation of the multi-symplectic structure of Equations (2.12), (2.13). It is noted that only a sub-class of symmetric Systems (1.5), (1.6) (with ) possesses a multi-symplectic structure.

2.3. A MS family of Boussinesq-type systems

The construction presented in the previous sections can be conformed to derive a new family of Boussinesq-type systems with MS structure. We start from a system of equations of the general form

(2.21)
(2.22)

of Boussinesq type. The parameters are as in (1.3), (1.4) and the nonlinearities and are homogeneous, quadratic polynomials

with real coefficients , which can be chosen on modeling or geometric structure bases. System of the form (2.21), (2.22) have been used for modelling nonlinear waves in different situations. Several examples are given below:

  1. The System (1.1), (1.2) trivially corresponds to the choice

  2. The symmetric version (1.5), (1.6) is obtained from (2.21), (2.22) with

  3. Some examples of (2.21), (2.22) also appear by choosing specific values of instead of the parameters of the nonlinearities. This is the case of the system of KdV type considered by Bona et al. in [BonaChK]:

    (2.23)