## Wu, Derchyi

Associate Research Fellow

Research Fields: Integrable Systems,

Email : wudc@math.sinica.edu.tw

Extension : 636

**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_{x}v_{xy}-v_{y}v_{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_{x}v_{xy}-v_{y}v_{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.

- B. S., National Tsing Hua University, Taiwan 1985
- Ph. D., Yale University, USA 1990

- Assistant Research Fellow, Institute of Mathematics, Academia Sinica, (1990-2007)
- Associate Research Fellow, Institute of Mathematics, Academia Sinica, (2008-Present)
- Teaching Assistant, National Tsing Hua University, Taiwan, (1984-1985)

- (with Derchyi Wu) "Stability of KdV solitons",
*arXiv:2401.15819*, 2024, https://arxiv.org/abs/2401.15819 - "Stability of Kadomtsev-Petviashvili multi-line solitons",
*arXiv:2205.07432*, 1-68, 2022 - "The direct scattering problem for perturbed Kadomtsev-Petviashvili multi line solitons",
*Journal of Mathematical Physics*, 62 (9), 91513-91513, 2021-09 - "The direct scattering problem for the perturbed Gr(1,2)≥0 Kadomtsev-Petviashvili solitons",
*Nonlinearity*, 33 (2020), 6729-6759, 2020-12 - "The direct problem for the perturbed Kadomtsev-Petviashvili II one line solitons",
*arXiv:1807.01420*, 1-34, 2018, Link - "The Cauchy problem for the Pavlov equation with large data",
*Journal of Differential Equations*, 263 (3), 1874-1906, 2017-08 , Link - (with P. G. Grinevich, P. M. Santini) "The Cauchy problem for the Pavlov equation",
*Nonlinearity, arXiv:1310.5834[nlin.SI]*, 28 (11), 3709-3754, 2015, Link - "A twisted integrable hierarchy with $\mathbb D_2$ symmetry",
*Journal of Mathematical Physics*, 53, 103708-103730, 2012, Link - (with Hui Ma) "Twisted hierarchies associated with the generalized sine-Gordon equation",
*Journal of Mathematical Physics*, 52 (9), 1-33, 2011-04 , Link - "The Cauchy problem of the Ward equation",
*Journal of Functional Analysis*, 256 (1), 215-257, 2009-01 , Link - "The Cauchy problem of the Ward equation with mixed scattering data",
*Journal of Mathematical Physics*, 49 (11), 113507-113520, 2008-11 , Link - (with Ming-Hsien Tu, Yu-Tung Chen and Jen Hsu Chang) "The debar-approach to the dispersionless (2+1)-Harry Dym hierarchy",
*Journal of Physics A-Mathmatical and General*, 38 (27), 6167-6181, 2005, Link - " Isomonodromy deformations for the ZS-AKNS system with quadratic spectral variables",
*JOURNAL OF MATHEMATICAL PHYSICS*, 44 (10), 4640-4651, 2003, Link - (with D. Pelinovsky ) "Gauge transformation and spectral decomposition for the Ishimori-II equations",
*JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL*, 36 (20), 5557-5574, 2003, Link - "Global factorization theorems for the ZS-AKNS System",
*ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS*, 77, 1-13, 2002 - "Global Birkhoff factorization on loop groups of the ZS-AKNS flows ",
*Inverse Problems*, 18 (1), 95-109, 2002, Link - (with Y. Lee, A.N. Wang) "A bridge for principle harmonic diffeomorphism between surfaces ",
*Ann. Global Analysis and Geometry*, 21 (2), 141-149, 2002, Link - (with Y. Lee and A.N. Wang) "A bridge principle for harmonic maps ",
*Ann. Global Analysis and Geometry*, 18 (2), 107-127, 2000, Link - "Harmonic diffeomphisms between complete surfaces",
*Ann. Global Analysis and Geometry*, 15 (2), 133-139, 1997, Link - (with Y. Lee and A.N. Wang) "On a theorem of Rado",
*Bull. London Math. Soc.*, 28 (4), 398-400, 1996, Link - (with Y. Lee) "The closed embedded minimal surface in an almost positively curved three manifold ",
*Ann. Global Analysis and Geometry*, 13 (3), 231-237, 1995, Link - "A Davey-Stewartson related scattering problem with non small potentials",
*J. Diff. Equations*, 111 (1), 27-57, 1994, Link - "A 2 X 2 scattering problem and its related nonlinear evolutions",
*Dissertation, Yale University*, 1990