, ,

(School of Science, Dalian Ocean University, Dalian 116023, China)

0 Introduction

Shallow water equations are widely applicable models in mathematical physics. Whitham-Broer-Kaup(WBK) system for the dispersive long wave in the shallow water

(1)

In this paper, a complete basis of the invariantized Maurer-Cartan forms of Whitham-Broer-Kaup system is obtained by the method of equivariant moving frames. The invariantized Maurer-Cartan forms play an important role in finding the complete system of differential invariants for the nonlinear partial differential equations. Sophus Lie used his infinitesimal method based on the infinitesimal determining system obtained by linearizing the determining system, and Cartans method using intricate recursive prolongation of exterior differential systems are either limited in scope or impractical from the standpoint of applications. Along this line of research in the last decade, many other methods were developed[8-11]. Their algorithms were successfully applied to certain types of Lie symmetry pseudo-groups of differential equations. A major drawback, however, is that their methods were based on ad hoc series expansions and became significantly more complicated, requiring more case-by-case analyses, if they worked at all, when it came to intransitive pseudo-group actions. More recently, the invariant contact forms on the diffeomorphism jet bundle were interpreted as the Maurer-Cartan forms of the Lie pseudo-group[12]. As a result, a very efficient method for constructing the structure equations of the Maurer-Cartan forms was discovered. This method bypasses the troublesome process of integrating either the determining system or its linearization, or the complicated Cartan prolongation process. Moreover, the algorithm directly applies to completely general Lie pseudo-group actions, whether finite or infinite dimensional, transitive or intransitive, and can be easily implemented in computer algebra systems. This efficient method is based on a new, equivariant formulation of Cartan’s method of moving frames that was initiated in[13-14]and developed[17-21]. The goal of this paper is to use the constructive computational algorithms[12-14]to determine a complete basis of the invariantized Maurer-Cartan forms of WBK system.

The paper is organized as follows. In Section 1, the preliminaries about the algorithms we use are presented. In Section 2, a complete basis of the invariantized Maurer-Cartan forms of WBK system is solved. Section 3 includes the conclusion and discussion about the further research.

1 Preliminaries

In this part, we will generally show the theoretical preliminaries, which are from[12-16]. Firstly we consider the point symmetry group of a system of differential equations

Δσ(x,un)=0,σ=1,2,…,k,

Let

denote thenthorder prolongation of the vector field toJn(M,p). A vector fieldvis an infinitesimal symmetry of the system of differential equations if and only if it satisfies the infinitesimal symmetry conditions.

Let

(2)

denote the completion of the system ofinfinitesimaldeterminingequations, which includes the original determining equations along with all equations obtained by repeated differentiation.

Definition1 Annthordermovingframefor a pseudo-groupGacting onp-dimensional submanifoldsN?Mis a locallyG-equivariant sectionρ(n):Jn(M,p)→H(n).

Local coordinates on H(n). Have the form(x,u(n),λ(n)), where (x,u(n)) arejet coordinates onJn(M,p) while the fiber coordinatesλ(n)represent the pseudo-group parameters of order≤n.

Theorem1 Alocallyequivariantmovingframeexists in a neighborhood of a jet(x,u(n))∈Jn(M,p) if and only ifGacts locally freely at(x,u(n)).

A practical wayto construct a moving frameρ(n)is through the normalization procedure based on the choice of a cross-section to theG-orbits. Once a moving frame is fixed, invariantizing thenthorder jet coordinates (x,u(n)) leads to thenormalizeddifferentialinvariants

(3)

whereιis the inducedinvariantizationprocess.

A basis for theinvariantdifferentialoperatorsD1,…,Dpcan be obtained by invariantizing the total differential operatorsD1,…,Dp. Thecontact-invarianthorizontalcoframeωi=ι(dxi), wherei=1,2,…,p, can be obtained by invariantizing the horizontal coordinate coframe.

Theorem2 The restricted Maurer-Cartan forms satisfy the lifted determining equations

which are obtained by applying the following replacement rules:

xiXi,uαUα,,

for alli,α,A, to the infinitesimal determining equations (2).

Definition2 Given a moving frameρ(n):Jn(M,p)→H(n), we define theinvariantizedMaurer-Cartanformsto be the horizontal components of the pull-backs

(4)

Theorem3 The invariantized Maurer-Cartan forms satisfy the invariantized determining equations

(5)

Extending the invariantization process, we set

to be the corresponding invariantized Maurer-Cartan forms (4).

Theorem4 The recurrence formulas for the normalized differential invariants (3) are

(6)

2 The invariantized Maurer-Cartan forms for Whitham-Broer-Kaup system

For Whitham-Broer-Kaup system (1)

the underlying total space isM=4with coordinates (t,x,u,v). A vector field

is an infinitesimal symmetry of the WBK system if and only if its coefficients satisfy the infinitesimal symmetry determining equations

(7)

along with all their differential consequences.

Let

denote the corresponding normalized differential invariants and

denote theinvariantized Maurer-Cartan forms. The complete system of linear dependencies among them is derived from (7) as

(8)

and so on. Therefore a basis of the invariantized Maure-Cartan forms can be obtained from (8) as followsμ,γ,μT,γT.

(9)

Then the recurrence formulas are directly obtained from (9) and (6) as follows

(10)

The normalizations are chosen as

(11)

Substituting (11) into equations (10) yields the basis of the invariantized Maurer-Cartan forms,

(12)

for the basic invariant forms. The higher order invariantized Maurer-Cartan forms can berecursively deduced from them.

3 Conclusion

The equivariant moving frame method has proved to be a very powerful tool in determining a basis of the invariantized Maurer-Cartan forms. In this paper, only using the infinitesimal determining equations and choice of cross-section normalization completely has yielded a basis of the invariantized Maurer-Cartan forms of Whitham-Broer-Kaup system. These results can be used to find the complete system of differential invariants for Whitham-Broer-Kaup system. Further more, how to solve the original equation via the invariantized Maurer-Cartan forms and differential invariants, which is interesting and meaningful, deserves our further research.

Acknowledgement

This work is supported by General Scientific Research Project of Liaoning Province(L2014279),Natural Science Foundation of Liaoning Province(20170540103), Foundation of Dalian Ocean University(HDYJ201409), National Natural Science Foundation of China(11501076).

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