The Bi-Hamiltonian Structure of the Perturbation Equations of KdV Hierarchy

The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions. Perturbation theory is the study of the effects of small disturbances. Its basic idea is to find approximate solutions to a concrete problem by exploiting the presence of a small dimesionless parameter. There have been a lot of works to investigate the perturbated soliton equations (see for example [1] [2] and references therein). Tamizhmani and Lakshmanan have considered a perturbation effect of the unperturbated KdV equation and they have given rise to infinitely many Lie-Bäcklund symmetries and a Hamiltonian structure for the resulting equations in Ref. [3]. However they haven’t obtained the corresponding bi-Hamiltonian formulation. In this letter, we would like to discuss the perturbation equations of the unperturbated whole KdV hierarchy, i.e. the effect of the disturbance around solutions of the following original KdV hierarchy utn = Φ (u)ux, Φ(u) = ∂ 2 + 2ux∂ −1 + 4u, ∂ = d dx , n ≥ 1. (1) The first equation is exactly the usual KdV equation ut1 = uxxx + 6uux, (2) ∗On leave of absence from Institute of Mathematics, Fudan University, Shanghai 200433, China 1 which describes the unidirectional propagation of long waves of small amplitude and has a broad of applications in a number of physical contexts such as hydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics (for details see, for example, [4] [5]). The operator Φ(u) is a common hereditary recursion operator for the whole KdV hierarchy. We shall show that all of the resulting perturbation equations possess a bi-Hamiltonian structure and thus they constitute a typical integrable soliton hierarchy. We shall also point out a mistake on the Hamiltonian formulation in Ref. [3]. Finally, a more general bi-Hamiltonian integrable soliton hierarchy is established and some further discussion is presented. We first consider the case of KdV equation (2) and then consider the case of higher order KdV equations. Let us make a perturbation expansion û = N ∑ i=0 ηiε , ηi = ηi(x, t1, t2, · · ·), N ≥ 1, (3) and call the equation ût1 = Φ(û)ûx + o(ε ) = ûxxx + 6ûûx + o(ε ), (4) the N -th order perturbation equation of KdV equation (2). It is easy to find that Φ(û) = N ∑ i=0 Φ(ηi)ε , Φ(ηi) = δi0∂ 2 + 2ηix∂ −1 + 4ηi, 0 ≤ i ≤ N, where δi0 is the Kronecker’s symbol. Therefore the N -th order perturbation equation (4) may be rewritten as ût1 ≡ ( N ∑