Research Interest

• Inverse scattering problem of two or multi-dimensional integrable systems
• Backlund transformation theory of two or multi-dimensional integrable systems
• Isomonodromy problem of self similar integrable systems.
  more...

Latest Research Work

The direct scattering problem for the perturbed Gr(1, 2)_≥0 Kadomtsev-Petviashvili II solitons  (Nonlinearity 33 (2020) 6729-6759)

D. Wu
The Kadomtsev-Petviashvili II equation

(-4u_x3+u_x1x1x1+6uu_x1)x1 + 3u_x2x2=0,

is a two-spatial dimensional integrable generalization of the Korteweg-de Vries (KdV) equation. It is an asymptotic model for dispersive systems in the weakly nonlinear, long wave regime, when the wavelengths in the transverse direction are much larger than in the direction of propagation. Regular Kadomtsev-Petviashvili II (KPII) solitons, which are non decaying and ray asymptotic at space infinity, have been investigated and classified successfully by the Grassmannian. We complete rigorous analysis for the direct scattering problem of perturbed Gr(1, 2)_≥0 KPII solitons by providing a λ-uniform estimate for the Green function and a Cauchy integral equation with controllable singularities.

The Cauchy problem for the Pavlov equation with large data  (Journal of Differential Equations (2017) v.263, no.3, pp. 1874-1906)

D. Wu
We prove a local solvability of the Cauchy problem for the Pavlov equation

v_xt+v_yy+v_xv_xy-v_yv_xx=0,

with large initial data by the inverse scattering method. The Pavlov equation arises in studies Einstein-Weyl geometries and dispersionless integrable models. Our theory yields a local solvability of Cauchy problems for a quasi-linear wave equation with a characteristic initial hypersurface.

The Cauchy Problem for the Pavlov equation  (Nonlinearity (2015), v. 28, no. 11, pp. 3709-3754)

P. G. Grinevich, P. M. Santini, and D. Wu

v_xt+v_yy+v_xv_xy-v_yv_xx=0

Abstract: 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.