Soliton equations related to the affine Kac-Moody algebra D 4 (1)

Abstract

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D 4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data. IntroductionAfter the discovery of the inverse scattering method (ISM) in 1967 by the Princeton group of Gardner, Greene, Kruskal and Miura [1] the study of soliton equations and the related integrable nonlinear evolution equations (NLEEs) became one of the most active branches of nonlinear science. In this pioneering paper, the Korteveg-de Vries (KdV) equation was exactly integrated using the ISM. One year later, Peter Lax presented a method, which would later be instrumental in the search of new integrable equations different from the KdV[2]. This started numerous attempts to extend the range of application of the ISM to other equations.Three years had passed, when Gardner started the canonical approach to the NLEEs. He was the first to realize that the KdV equation can be written in a Hamiltonian form [3]. Soon after that, Zakharov and Faddeev proved that KdV equation is a completely 1 integrable Hamiltonian system [4] and Wadati solved the modified Korteweg-de Vries (mKdV) equation [5].This explosion of interest in the soliton science led to the discovery of a completely new world of unknown before integrable equations together with methods and techniques for finding their exact solutions. We should mention some of those methods and techniques, starting with the fundamental analytic solution (FAS), which was introduced by Shabat [6]. Within this approach one can show that the ISM is equivalent to a multiplicative Reimman-Hilbert problem (RHP), which is the basis for the dressing method invented by Zakharov and Shabat [7,8]. Another important result was derived by Ablowitz, Kaup, Newell and Segur [9]. They showed that the ISM can be interpreted as a generalized Fourier transform. This approach is based on the Wronskian relations, which are basic tools for the analysis of the mapping between the potential and the scattering data. They allow the construction of the "squared solutions" of the Lax operator, which form a complete set of functions and play the role of the generalized exponentials (see [10,11,12] and the references therein). Another powerful technique we should mention here is that of the so called recursion operators. This method has in principal a long history but the application to the NLEEs started in [13] (see also [14] and [15]) where with the help of the recursion operator the KdV equation was written in a compact form. Within the study of the Nonlinear Schrödinger (NS) equation, considered in the rapidly decreasing case and imposing the requirement that the squared Jost solutions of th...

Recursion operators and the hierarchies of MKdV equations related to $D_4^{(1)}$, $D_4^{(2)}$ and $D_4^{(3)}$ Kac-Moody algebras

“…The latter can be constructed starting from simple Lie algebra g graded by its Coxeter automorphism C, for details see Section 2 below. This present paper is an extension of our previous results reported in [22,23]. Its main purpose is to present the modified Korteweg-de-Vries (mKdV) equations related to Kac-Moody algebras of type D (k) 4 , k = 1, 2, 3.…”

“…From (22) it is easy to check that C 8 2 = 1 1 and its eigenvalues are ω 1 2 , ω 3 2 , ω 5 2 , ω 7 2 , where ω 2 = exp(2πi/8); compare with the Coxeter number and the exponents for D…”

Soliton equations related to the affine Kac-Moody algebra D^(1)_4

“…A natural extension of these results is to study the soliton equations associated with the twisted affine Kac-Moody algebras. The corresponding integrable hierarchies related to the twisted affine Kac-Moody algebras A are under investigations and have been presented elsewere [68,69,70].…”

Riemann-Hilbert problem, integrability and reductions

“…Here ξ k (x, t, λ) is an analytic function in the sector kπ/h ≤ arg λ ≤ (k + 1)π/h and k − 1 is evaluated modulo h. Among the relevant new NLEE with Z h and D h symmetries related to such RHP we note the 2-dimensional Toda field theories [48], mKdV equations related to the Kac-Moody algebras [63,47,65,53,15,64], see also [17,31,22,19,23,30,24,25] and others.…”