Reduction in soliton hierarchies, elliptic Lie algebras of Holod and Krichever-Novikov-type equations

Abstract In the present paper we develop one of the main Lie-algebraic approaches to the theory of soliton equations based on the special infinite-dimensional Lie algebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> (Flaschka et al 1983 Physica D 9 300–323; Newell 1985 Solitons Math. Phys. (Society for Industrial and Applied Mathematics); Kulish and Reiman 1983 Zap. Nauchn. Sem. LOMI 123 67–76; Holod 1982 Preprint ITP-82-144; Holod 1984 DAN Ukrainian SSR A 5 5–8; Holod 1987 Theor. Math. Phys. 70 18–29; Holod 1984 Preprint ITP-84-87P). We argue that the restriction of the integrable hierarchy onto the nilpotent coadjoint orbits in the dual space of the underlying finite-dimensional Lie algebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> </mml:math> leads to the additional reduction in the considered integrable hierarchy. We illustrate this phenomenon by the examples of the elliptic Lie algebra of Holod (1984 DAN Ukrainian SSR A 5 5–8; 1987 Theor. Math. Phys. 70 18–29) and its even subalgebra (Holod 1984 Preprint ITP-84-87P; 1987 DAN USSR 292 1087–91). Using them and the nilpotent coadjoint orbits of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> we derive Krichever-Novikov equation (Krichever and Novikov 1980 Russ. Math. Surv. 35 53–80) and its ‘twisted’ analogue—one component evolutionary PDE present in the list of Mikhailov (1987 Uspehi Mat. Nauk 42 3–53). For the twisted Krichever-Novikov hierarchy we also derive the hierarchy of non-evolutionary ‘negative’ flows and explicitly write the first negative flow equation. The last equation seems to be new.