An explicit reciprocal transformation between a 2-component generalization of the CamassaHolm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented.
Mathematics Subject Classifications(2000). 35Q53, 37K35
We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP 1 topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy.
We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W -algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov -Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.
We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEsUnder certain genericity assumptions it is proved that any bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so-called quasiMiura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi-Hamiltonian structure of hydrodynamic type (the quasi-triviality theorem). We also describe, following [35], the invariants of such bi-Hamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.
We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie
algebra of infinitesimal symmetries isomorphic to the half of the Virasoro
algebra. The generators $L_m$, $m\geq -1$ of the Lie algebra act by linear
differential operators onto the tau function of the hierarchy. We also prove
that the tau function of a generic solution to the extended Toda hierarchy is
annihilated by a combination of the Virasoro operators and the flows of the
hierarchy. As an application we show that the validity of the Virasoro
constraints for the $CP^1$ Gromov-Witten invariants and their descendents
implies that their generating function is the logarithm of a particular tau
function of the extended Toda hierarchy.Comment: A remark at the end of Section 5 is added; more detailed explanations
in Appendix; references adde
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