Niann-Chern Lee

Conformal covariantization of Moyal-Lax operators

A covariant approach to the conformal property associated with Moyal-Lax operators is given. By identifying the conformal covariance with the second Gelfand-Dickey flow, we covariantize Moyal-Lax operators to construct the primary fields of one-parameter deformation of classical W-algebras.

Polytropic gas dynamics revisited

We investigate the Hamiltonian as well as Lax formulations of the polytropic gas dynamics which can be characterized by the polytropic exponent γ=1+h. We show that when h=1∕m with ∣m∣∊N⩾2, the resonant phenomena occur in bi-Hamiltonian formulation and logarithmic-type conserved charges emerge naturally. We provide a logarithmic Lax representation for conserved charges and the whole hierarchy flows, which coincides with the bi-Hamiltonian formulation constructed from a two-dimensional Frobenius manifold associated with the polytropic gas dynamics.

A note on reductions of the dispersionless Toda hierarchy

We investigate the algebraic structure of the dispersionless Hirota equations for the m + 1-variable reductions of the dispersionless Toda hierarchy which contains the special case for m = 1 considered by Kodama and Pierce. We demonstrate the m = 2 case to illustrate the obtained result.

Operator approach to solutions of the constrained BKP hierarchy

The operator formalism to the vector k-constrained BKP hierarchy is presented. We solve the Hirota bilinear equations of the vector k-constrained BKP hierarchy via the method of neutral free fermion. In particular, by choosing suitable group element of O(?), we construct rational and soliton solutions of the vector k-constrained BKP hierarchy.

Kernel formula approach to the universal Whitham hierarchy

Based on the kernel formula proposed by Carroll and Kodama, we derive the dispersionless Hirota equations of the universal Whitham hierarchy. We also verify the associativity equations in this hierarchy from the dispersionless Hirota equations and give a realization of the associative algebra with the structure constants expressed in terms of residue formulas.

On dispersionless Hirota equations of the dispersionless Dym hierarchy

We show that, under twistor construction, the only finite-dimensional reduction of the dispersionless Dym hierarchy which can be characterized by a free energy is the two-reduction model with Lax operator of the form λ = up + u0 + p−1. The primary variables u and u0 can be expressed in terms of second derivatives of the associated free energy and the hierarchy flows can be written as a set of dispersionless Hirota equations.