An integrable hierarchy including the AKNS hierarchy and its strict version

G. F. Helminck

Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 3, pp. 444-458

Résumé

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space $\mathfrak{g}$ of $gl_2$ into a Lie subalgebras $\mathfrak{g}_{\ge0}$ and $\mathfrak{g}_{<0}$ of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra $C$ in the whole loop space $\mathfrak{s}$ of $sl_2$ and represent it as $C=C_{\ge0}\oplus C_{<0}$. We deform the Lie subalgebras $C_{\ge0}$ and $C_{<0}$ by the respective groups corresponding to $\mathfrak{g}_{<0}$ and $\mathfrak{g}_{\ge0}$. Further, we require that the evolution equations of the deformed generators of $C_{\ge0}$ and $C_{<0}$ have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.

Détail

Mots-clés : AKNS equation, compatible Lax equations, wave matrix
Keywords: AKNS hierarchy, strict version, zero curvature form, linearization, oscillating matrix, loop group, loop algebra.

Bibliographie

[1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems”, Studies Appl. Math., 53:4 (1974), 249–315 | DOI | MR | Zbl

[2] H. Flaschka, A. C. Newell, T. Ratiu, “Kac–Moody Lie algebras and soliton equations. II. Lax equations associated with $A_1^{(1)}$”, Physica D, 9:3, 300–323 | DOI | MR

[3] G. F. Helminck, “The strict AKNS hierarchy: its structure and solutions”, Adv. Math. Phys., 2016 (2016), 3649205, 10, 10 pp. | DOI | MR

[4] G. F. Khelmink, A. F. Khelmink, E. A. Panasenko, “Integriruemye deformatsii v algebre psevdodifferentsialnykh operatorov s tochki zreniya algebraicheskoi teorii Li”, TMF, 174:1 (2013), 154–176 | DOI | DOI | MR | Zbl

[5] E. Date, M. Kashiwara, M. Jimbo, T. Miwa, “Transformation groups for soliton equations”, Nonlinear Integrable Systems – Classical Theory and Quantum Theory (Kyoto, May 13–16, 1981), eds. M. Jimbo, T. Miwa, World Sci., Singapore, 1983, 39–119 | MR

[6] G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publ. Math. IHES, 61 (1985), 5–65 | MR | Zbl

[7] A. Pressley, G. Segal, Loop Groups, Oxford Univ. Press, New York, 1986 | MR

[8] R. S. Hamilton, “The inverse function theorem of Nash and Moser”, Bull. Amer. Math. Soc. (N. S.), 7:1 (1982), 65–222 | DOI | MR | Zbl