Lagrangian Approach to Dispersionless KdV Hierarchy

Equation (1.1), often called the Riemann equation, serves as a prototypical nonlinear partial differential equation for the realization of many phenomena exhibited by hyperbolic systems [2]. This might be one of the reasons why, during the last decade, a number of works [3] was envisaged to study the properties of dispersionless KdV and other related equations with special emphasis on their Lax representation and Hamiltonian structure.

The complete integrability of the KdV equation yields the existence of an infinite family of conserved functions or Hamiltonian densities H n ’s that are in involution. All H n ’s that generate flows which commute with the KdV flow give rise to the KdV hierarchy. The equations of the hierarchy can be constructed using [4] u t = Λ n u x (x, t), n = 0, 1, 2, . . . (1.2) with the recursion operator

In the dispersionless limit the recursion operator becomes Λ = u + 1 2 u x ∂ -1

x .

(1.3)

According to (1.2), the pseudo-differential operator Λ in (1.3) defines a dispersionless KdV hierarchy. The first few members of the hierarchy are given by n = 0 :

n = 1 :

)

)

n = 4 : u t = 315 128 u 4 u x .

(1.4e)

Thus the equations in the dispersionless hierarchy can be written in the general form

where the values of A n should be computed using (1.3) in (1.2). We can also generate A 1 , A 2 , A 3 etc recursively using

The Hamiltonian structure of the dispersionless KdV hierarchy is often studied by taking recourse to the use of Lax operators expressed in the semi-classical limit [5]. In this work we shall follow a different viewpoint to derive Hamiltonian structure of the equations in (1.5). We shall construct an expression for the Lagrangian density and use the time-honoured method of classical mechanics to rederive and reexamine the corresponding canonical formulation. A single evolution equation is never the Euler-Lagrange equation of a variational problem. One common trick to put a single evolution equation into a variational form is to replace u by a potential function u = -w x . In terms of w, (1.5) will become an Euler-Lagrange equation. We can, however, couple a nonlinear evolution equation with an associated one and derive the action principle. This allows one to write the Lagrangian density in terms of the original field variables rather than the w’s, often called the Casimir potential. In Section 2 we adapt both these approaches to obtain the Lagrangian and Hamiltonian densities of the Riemann type equations. In Section 3 we study the bi-Hamiltonian structure [6]. One of the added advantage of the Lagrangian description is that it allows one to establish, via Noether’s theorem, the relationship between variational symmetries and associated conservation laws. The concept of variational symmetry results from the application of group methods in the calculus of variations. Here one deals with the symmetry group of an action functional A[u] = Ω 0 L x, u (n) dx with L, the so-called Lagrangian density of the field u(x). The groups considered will be local groups of transformations acting on an open subset M ⊂ Ω 0 × U ⊂ X × U . The symbols X and U denote the space of independent and dependent variables respectively. We devote Section 4 to study this classical problem. Finally, in Section 5 we make some concluding remarks.

For u = -w x (1.5) becomes

The Fréchet derivative of the right side of (2.1) is self-adjoint. Thus we can use the homotopy formula [7] to obtain the Lagrangian density in the form

In writing (2.2) we have subtracted a gauge term which is harmless at the classical level. The subscript n of L merely indicates that it is the Lagrangian density for the nth member of the dispersionless KdV hierarchy. The corresponding canonical Hamiltonian densities obtained by the use of Legendre map are given by

Equation (1.5) can be written in the form

There exists a prolongation of (1.5) or (2.4) into another equation

with the variational derivative

such that the coupled system of equations follows from the action principle [8] δ L c dxdt = 0.

The Lagrangian density for the coupled equations in (2.4) and (2.6) is given by

(2.7)

For the system represented by (1.5) and (2.7) we have

The result in (2.7) could also be obtained using the method of Kaup and Malomed [9]. Referring back to the supersymmetric KdV equation [10] we identify v as a fermionic variable associated with the bosonic equation in (1.5). It is of interest to note that the supersymmetric system is complete in the sense of variational principle while neither of the partners is. The Hamiltonian density obtained from the Lagrangian in (2.8) is given by

(2.9)

It remains an interesting curiosity to demonstrate that the results in (2.3) and (2.9) represent the conserved densities of the dispersionless KdV and supersymmetric KdV flows. We demonstrate this by examinning the appropriate

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