Abstract
The new (2+1)-(γn,σk)-Harry Dym hierarchy and(γn,σk)-mKP hierarchy with two new time seriesγnandσk, which consist ofγn-flow,σk-flow, and mixedγnandσkevolution equations of eigenfunctions, are proposed. Gauge transformations and reciprocal transformations between(γn,σk)-KP hierarchy,(γn,σk)-mKP hierarchy, and (2+1)-(γn,σk)-Harry Dym hierarchy are studied. Their soliton solutions are presented.
Highlights
Generalizations of soliton hierarchies are important topics since last century
We first constructed the (2+1)-(γn, σk)-Harry Dym hierarchy ((2+1)-(γn, σk)-HDH), which consists of γn-flow, σk-flow, and one mixed γn and σk evolution equation of eigenfunctions
It is natural to think whether these transformations can be generalized to the-KP hierarchy (KPH),mKPH, and (2+1)-(γn, σk)-HDH
Summary
Generalizations of soliton hierarchies are important topics since last century. In 2008, KdV6 equation is studied as a nonholonomic deformation of KdV equation. The (γn, σk)-KPH consists of γn-flow, σk-flow, and one mixed γn and σk evolution equation of eigenfunctions. We first constructed the (2+1)-(γn, σk)-Harry Dym hierarchy ((2+1)-(γn, σk)-HDH), which consists of γn-flow, σk-flow, and one mixed γn and σk evolution equation of eigenfunctions. Based on these transformations and the solutions of the (γn, σk)-KPH [15], the solutions of (γn, σk)mKPH and (2+1)-(γn, σk)-HDH are obtained, respectively. Lσk = [Bk + βk∑qi∂−1ri∂2, L] , i=1 (11b) αn (qi,σk − Bk (qi)) − βk (qi,γn − Bn (qi)) = 0, αn (ri,σk + ∂−2B∗k ∂2 (ri)) − βk (ri,γn + ∂−2B∗n ∂2 (ri))
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