Jen-Hsu Chang

On the Miura Map between the dispersionless KP and dispersionless modified KP hierarchies

The Miura map between the dispersionless KP and dispersionless modified KP hierarchies is investigated. It is shown that the Miura map is canonical with respect to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki and Takebe, the twistor construction of solution structure for the dispersionless modified KP hierarchy is given.

On the waterbag model of the dispersionless KP hierarchy (II)

We construct the bi-Hamiltonian structure of the waterbag model of dKP and establish the third-order Hamiltonian operator associated with the waterbag model. Also, the symmetries and conserved densities of the rational type are discussed.

On the waterbag model of dispersionless KP hierarchy

We investigate the bi-Hamiltonian structure of the waterbag model of dispersionless K (dKP) for the two-component case. One can establish the third- and first-order Hamiltonian operators associated with the waterbag model. Also, the dispersive corrections are discussed.

Poisson algebras associated with constrained dispersionless modified Kadomtsev–Petviashvili hierarchies

We investigate the bi-Hamiltonian structures associated with constrained dispersionless modified Kadomtsev–Petviashvili (KP) hierarchies which are constructed from truncations of the Lax operator of the dmKP hierarchy. After transforming their second Hamiltonian structures to those of the Gelfand–Dickey-type, we obtain the Poisson algebras of the coefficient functions of the truncated Lax operators. Then we study the conformal property and free-field realizations of these Poisson algebras. Some examples are worked out explicitly to illustrate the obtained results.

Hydrodynamic reductions of the dispersionless Harry Dym hierarchy

We investigate the reductions of the dispersionless Harry Dym hierarchy to systems of finitely many partial differential equations. These equations must satisfy the compatibility condition and they are diagonalizable and semi-Hamiltonian. By imposing a further constraint, the compatibility is reduced to a system of algebraic equations, whose solutions are described.

Rational approximate symmetries of KdV equation

We construct one-parameter deformation of the Dorfman Hamiltonian operator for the Riemann hierarchy using the quasi-Miura transformation from topological field theory. In this way, one can get the rational approximate symmetries of the KdV equation and then investigate its bi-Hamiltonian structure.

Asymptotic Analysis of Multilump Solutions of the Kadomtsev–Petviashvili-I Equation

We construct lump solutions of the Kadomtsev–Petviashvili-I equation using Grammian determinants in the spirit of the works by Ohta and Yang. We show that the peak locations depend on the real roots of the Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. We also prove that if the time goes to −∞, then all the peak locations are on a vertical line, while if the time goes to ∞, then they are all on a horizontal line, i.e., a π/2 rotation is observed after interaction.

Solutions of the real dispersionless Veselov-Novikov hierarchy

We investigate the dispersionless Veselov-Novikov (dVN) equation in the framework of the dispersionless two-component BKP hierarchy. We consider symmetry constraints for the real dVN system and show that the conserved densities are related to Faber polynomials and can be solved recursively. In addition, we use the Faber polynomials to find hodograph solutions of the dVN hierarchy.

Dressing operator approach to the toroidal model of higher-dimensional dispersionless KP hierarchy

Based upon the dressing operator approach we investigate the twistor theoretical construction, additional symmetries, hodograph solutions, and Miura transformation for the toroidal model of higher-dimensional dispersionless KP hierarchy introduced by Takasaki.