Zero Curvature Formulations of Dual Hierarchies

Zero curvature formulations are given for the “dual hierarchies” of standard soliton equation hierarchies, recently introduced by Olver and Rosenau, including the physically interesting Fuchssteiner-Fokas-CamassaHolm hierarchy. PACS Numbers: 03.40.Kf, 47.20.Ky, 52.35.Sb Through the years since the discovery of the notion of “integrability” in PDEs, quite a number of integrable PDEs have been discovered, most of which remain obscure for lack of any physical significance. For nearly fifteen years now, the equation ut + 2κux − uxxt + 3uux = 2uxuxx + uuxxx, (1) derived by Fuchssteiner and Fokas [1] has enjoyed such obscurity; but in a recent paper of Camassa and Holm [2], this equation was rediscovered, and looks likely to be of some importance. Like Fuchssteiner and Fokas, Camassa and Holm showed that, for κ = 0, (1) has bihamiltonian structure: if we write m = u− uxx, then (1) takes the form mt = − J1 δH2 δm = − J2 δH1 δm , (2) where J1 = ∂ − ∂, J2 = ∂m+m∂ (3) are two compatible hamiltonian operators, and H2 = 1 2 ∫ ∞ −∞ (u + uux)dx, H1 = 1 2 ∫ ∞ −∞ (u + ux)dx. (4) The novelty of Camassa and Holm’s work was that they gave a physical derivation of (1). Furthermore, for κ = 0, they found solutions to (1) which they named “peakons” (travelling wave solutions with a corner at their peak); these take the simple form u = c exp(−|x− ct|). (5) More generally they showed that u = N ∑ i=1 pi(t) exp(−|x− qi(t)|) (6) gives an N -peakon solution, provided {pi(t), qi(t)} solves hamilton’s equations for the hamiltonian HA = 1 2 N ∑ i,j=1 pipj exp(−|qi − qj |). (7) Camassa and Holm proved this hamiltonian system is integrable, and gave its solution for N=2. For κ 6= 0, solutions of (1) have been investigated numerically in [3]. A little prior to Camassa and Holm’s work, Rosenau and Hyman [4] made the remarkable observation that a large class of nonlinear PDEs with nonlinear dispersion terms 1 exhibited “compacton” solutions, viz. solitons with compact spatial support. Rosenau [5] further showed that this phenomenon can also occur in integrable PDEs; in particular, if we replace (x, t) in the Fuchssteiner-Fokas-Camassa-Holm equation (1) by (ix, it), we find the equation ut + 2κux + uxxt + 3uux + 2uxuxx + uuxxx = 0, (8) and this admits, for κ = 0, the compacton solution u = c cos(x− ct) |x− ct| ≤ π 2 . (9) (The compacton solutions of (8) are actually unstable; but it serves to illustrate that compactons can occur in the framework of integrability; in addition it seems further equations in its hierarchy have acceptable properties. I thank Philip Rosenau for information on this point.) In the wake of this work, two apparently widely applicable constructions of integrable PDEs with nonlinear dispersion terms have been given. The first, due to Rosenau [6], consists of applying Lagrange transformations to soliton-bearing integrable PDEs, such as the KdV and MKdV equations. The philosophy here is that the standard solitons in such equations, despite being of infinite spatial extent, carry finite mass and/or momentum, and hence must be of compact support when measured in mass and/or momentum units. The second construction, due to Olver and Rosenau [7] (again a rediscovery of Fuchssteiner and Fokas’ earlier work [1]; the reader should also see the modern work [8] of Fokas), starts from the observation that the two hamiltonian operators J1, J2 given in (3) look like recombinations of terms from the two standard hamiltonian operators of the KdV equation (see, for example, [9]). In fact it turns out that if a bihamiltonian integrable hierarchy has one hamiltonian operator which is a constant coefficient differential operator, and another hamiltonian operator which is a linear combination of a constant coefficient differential operator and another operator which scales homogeneously with non-zero degree when the fields are rescaled, then by recombining terms from these hamiltonian operators one can construct a new hierarchy. In [7] this procedure is followed to construct dual hierarchies of the KdV, MKdV, Broer-Kaup-Kupershmidt and Ito hierarchies (the NLS hierarchy is also dualized by a variant of the general procedure). The aim of this paper is to provide yet another method of constructing dual hierarchies, reproducing the results of [7]. This time the initial observation is the similarity of the linear system associated with the CamassaHolm equation (the linear system is given in equation (6) of [2]), and the linear system associated with the KdV equation. We will see that zero curvature formulations of dual hierarchies can be obtained by a simple modification of the well-known zero curvature formulations of the standard soliton equation hierarchies. 2 The original purpose of this work was twofold. First, for standard soliton equation hierarchies, the zero curvature formulation is a springboard for revealing many other properties of the hierarchies. In particular, in the zero curvature formulation one sees a natural group action on the space of solutions (the group of “dressing transformations”), which, when it can be made explicit, gives rise to a host of solutions of the hierarchies (for a compact overview of how the group of dressing transformations gives rise to the tau-function formalism for the MKdV equation, see [10]). Alas, while the zero curvature formulations of dual hierarchies are only slight variations of those for standard hierarchies, this slight variation complicates the explicit realization of dressing transformations, and we have been unable, as of yet, to compute explicit dressing transformations and generate solutions this way. The second hope in undertaking this work was that, while Olver and Rosenau’s construction [7] cannot be extended to, for example, the Boussinesq (SL(3) KdV) equation (one hamiltonian operator is a constant coefficient differential operator, as required, but the other is the sum of a constant coefficient differential operator and another term that does not scale homogeneously under any rescaling of the fields), it was hoped that the zero curvature formulations would suggest an extension. Extensive experiments in this direction — which will not be reported here — have so far yielded only negative results. It seems quite possible that dual hierarchies can only be constructed for a handful of soliton equation hierarchies, and not for all the various infinite chains of hierarchies, like the SL(N) KdV hierarchies [11], that exist. The content of this note is therefore limited to presenting zero curvature formulations of the existing dual hierarchies. It is to be hoped that these will be of use in further studies of these hierarchies, and in finding solutions. We will see some minor immediate benefits of our labor; in particular, we will see that the dual Broer-Kaup-Kupershmidt hierarchy and the dual Ito hierarchy are equivalent, and we clarify a little further the structure of the dual NLS hierarchy. Also, of course, the zero curvature forms we will present, can be used to derive “standard” Lax pairs for the dual hierarchies, via a simple procedure we will illustrate. Zero Curvature Formulations. The notion of a zero curvature formulation for a soliton equation dates back to the work [12] (and other works in the Soviet literature). In [12] it was observed that several equations of physical interest could be written in the form ∂tA = ∂xB + [B,A], (10) where A,B are functions of x, t valued in the Lie algebra of the SL(2) loop group, that is, A,B are traceless, 2 × 2 matrix valued functions of x, t, λ. Equation (10) reduces to 3 the desired soliton equation by specifying a very particular dependence on the “spectral parameter” λ. In greater generality, the majority of (if not all) soliton equation hierarchies can be written in the form ∂trA = ∂xBr + [Br, A], (11) where A, {Br} (r runs over an appropriate index set) are functions of x, {tr}, λ, valued in some matrix Lie algebra, with a certain specified λ dependence. The classic example is the KdV hierarchy, for which r ∈ {1, 3, 5, ...} and A = (