Derchyi Wu

The Cauchy problem for the Pavlov equation

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.

Twisted hierarchies associated with the generalized sine-Gordon equation

Twisted U- and twisted U/K-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted O(J,J)O(J)×O(J)-hierarchies are among the most important classes of twisted hierarchies. In this paper, we derive explicit interesting first and higher flows of twisted O(J,J)O(J)×O(J)-hierarchies, justify that the one-dimensional systems of twisted O(J,J)O(J)×O(J)-hierarchies for J = Iq, n − q(1 ⩽ q ⩽ n − 1), called the generalized sinh-Gordon equations, are the Gauss-Codazzi equations for n-dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo-Euclidean (2n − 1)-dimensional space R2q−12n−1 with index 2q − 1. Furthermore, a unified treatment of the inverse scattering theory for twisted O(J,J)O(J)×O(J)-hierarchies is provided.

The -approach to the dispersionless (2+1)-Harry Dym hierarchy

A -approach is adopted to study the dispersionless Harry Dym (dHD) hierarchy. Moreover, this formulism is applied to construct some explicit solutions of the dHD hierarchy.

Global Birkhoff factorization on loop groups of the ZS-AKNS flows

The global Birkhoff factorization on the loop groups of the SU(2)-and SU(1, 1)-ZS-AKNS flows is investigated using the inverse scattering method.

Harmonic Diffeomorphisms between Complete Surfaces

We prove an existing theorem of homotopic harmonic diffeomorphisms between complete surfaces of finite total curvature and nontrivial genus. This is a generalization of a theorem of Jost and Schoen [4].

The Cauchy problem of the Ward equation with mixed scattering data

We solve the Cauchy problem of the Ward equation with both continuous and discrete scattering data.

Gauge transformation and spectral decomposition for the Ishimori-II equations

We study the gauge transformation between spectral problems and their adjoints for the Ishimori-II (IS-II) and Davey–Stewartson-II (DS-II) equations. The commutativity between the gauge and adjoint transformations is proved. The commutativity is used for spectral decomposition and surface representation theorems for the Ishimori-II equations.

A Bridge Principle for Harmonic Maps

We prove a bridge principle for harmonic maps between generalmanifolds.

Isomonodromy deformations for the ZS-AKNS system with quadratic spectral variables

We solve the monodromy problem and prove the Painleve property for self-similar ZS-AKNS flows with a quadratic spectral variable in this report. In particular, we obtain meromorphic solutions for the Cauchy problem of the self-similar derivative nonlinear Schrodinger equation.

A Bridge Principle for Harmonic Diffeomorphisms between Surfaces

We show a bridge principle for harmonic diffeomorphisms between closed surfaces with higher genus.