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# Lee, Jyh-Hao

 Mr. Jyh-hao LEE obtained his Ph.D at Yale in 1983, then worked as associte research fellow since then. He was promoted as research fellow in 1995.Dr. Lee ever taught at the Institute of Applied Mathematics, National Tsing-Hua University. He taught  at the Department of Mathematics, National Taiwan University from  Fall 1988 to 2009  summer . 　　   Dr. Lee's major research interests are the inverse scattering transform and the associated evolution equations and wavelet analysis. He is also interested in the differential equations and classical analysis. Dr. Lee's research interest lies on P.D.E, integrable systems and soliton equations. He has worked on the analysis part for the AKNS-ZS system with quadratic, polynomial spectral parameters based on the set-up of Beals-Coifman.This involves the Riemann-Hilbert problem. He has been aware of  many  aspects related to Riemann-Hilbert problem. In the past decade, he has considered several integrable systems with "quantum potentials" (jointly with Oktay Pashaev). A novel integrable version of the Nonlinear Schrodinger Equation (NLS) equation namely , $$i \frac{\partial \psi}{\partial t} + \frac{\partial^2 \psi}{\partial x^2} + {\Lambda \over 4} |\psi|^2 \psi = s \frac{1}{|\psi|}\frac{\partial^2 |\psi|}{\partial x^2} \psi,$$ has been termed the resonant nonlinear Schrödinger equation (RNLS)(also called Nonlinear Schrödinger equation with quantum potential ). It can be regarded as a third version of the NLS, intermediate between the defocusing and focusing cases. The critical value $s = 1$ separates two distinct regions of behaviour. Thus, for $s < 1$ the model is reducible to the conventional NLS, (focusing for $\Lambda > 0$ and defocusing for $\Lambda < 0$). However, for $s > 1$ it is not reducible to the usual NLS, but rather to a reaction-diffusion system, which can also be transform into Broer-Kaup system.   Email : leejh AT math.sinica.edu.tw Phone:+886 2 2368-5999 ext. 735 Fax: +886 2 2368-9771

### Publications : :

 1 (with O.K. Pashaev) Relativistic DNLS and Kaup-Newell Hierarchy, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) , 2017(2017-07), no. 13, 058-071-(http://www.emis.de/journals/SIGMA/2017/058/) 2 NLS with Quantum Potential and the Related Reaction-Diffusion Systems, presented in 2015 Cross-Strait Joint Conference on Integrable Systems and Related Topics (and a colloq. talk in Quanzhou campus of HuaQiao Univ.) (2016-06) 3 NSC report(2014-2015)具量子位能之可積系統與 Riemann-Hilbert 問題(II) Integrable systems with quantum potentials and Riemann-Hilbert problem(II), NSC(MOST) report(2014-2015) (2016-01) 4 NLS with quantum potential and the related reaction-diffusion systems and Kaup-Broer systems, Prof. of 10th Taiwan-Philippines Symposium on Analysis, Mar. 30-April 3, 2014 (2015-04) 5 (with Oktay Pashaev) Resonant dispersive Benney and Broer-Kaup systems in 2+1 dimensions, Journal of Physics: Conference Series (Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013)) , 482(2014-03), 12026-0(http://iopscience.iop.org/1742-6596/482/1) 6 (with Oktay K. Pashaev) Resonat dispersive nonlinear Broer-Kaup type Benny system in 2+1 dimension, preprint (2013-06), 0-0 7 (with Oktay K. Pashaev) NLS with quantum potential, the related reaction-diffusion systems and Kaup-Broer system, Special Session 57: Nonlinear and Dispersive Partial Differential Equations,”The 9th AIMS Conference on Dynamical Systems, Differential Equations and Application,Orlando, Florida, USA, July1-5 (2012-07), 0-0 8 (with Oktay K. Pashaev) Chiral resonant solitons in Chern–Simons theory and Broer–Kaup type new hydrodynamic systems, Chaos, Solitons and Fractals , 45(2012-08), 1041-1047(http://www.sciencedirect.com/science/article/pii/S0960077912001075) 9 (with Oktay Pashaev) Chiral Resonant Solitons in Chern-Simons Theory and Broer-Kaup Type New Hydrodynamic Systems, Chaos Solitons and Fractals (2012-04), 0-0 10 (with Oktay Pashev) The Reaction-Diffusion Systems Related to the Nonlinear Schrödinger Equation with Quantum Potential”, 7th IMACS conference on Nonlinear Evolution Equation and Wave Phenomena: Computation and Theory at Athens, Georgia, USA, April 4-7, 2011. (2011-12), 0-0 11 (with Oktay Pashaev) Chiral Resonant Solitons in Chern-Simons Theory and Broer-Kaup Type New Hydrodynamic Systems, (2010-12), 0-0 12 (with Oktay Pashaev) Solutions of the Reaction-Diffusion System Related to the Resonant Nonlinear Schrodinger Equation,, Shanghai International Symposium on Nonlinear Science and Application, June 29-July 4, 2010, Xuzhou & Shanghai, China. (2010-10), 0-0 13 (with Oktay Pashaev) Chiral Resonat Solitons in Broer-Kaup Type New Hydrodynamics System, proc. of 3rd International Conference on Nonlinear Science and Complexity,28-31 July, 2010, Ankara, Turkey (2010-10), 0-0 14 (with Dmitry Demskoi) On Non-Abelian Toda $A_2^{(1)}$ Model and Related Hierarchies (SCI:1.085)(Mathematical Physics), Journal of Mathematical Physics (2009-12), 0-0 15 (with Oktay Pashaev and R.R. Parwani) (with Oktay Pashaev and R.R. Parwani) Information measure and Chern-Simon Solutions, (2009-07-30), 0-0 16 (with Oktay Pashaev) transformation of Reaction-Diffusion system into Kaup-Broer hierarchy,, (2009-07-30), 0-0 17 (with Wen-Xiu Ma) A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos, Solitons and Fractals (2009-04), 0-0 18 (with Oktay K. Pashaev, Colin Rogers) Soliton resonances in a generalized nonlinear Schr\"{o}dinger equation, J. Phys. A :Math. Theor. , 41(2008), 452001-0 19 (with Oktay K. Pashaev) Chiral solitons in quantum potentials( presented in the workshop on Nonlinear Physics-Theory and Experiment V Gallipoli(Lecce),Italy, June 12-21,2008,), Theoretical and Mathematical Physics. , 160(2009-06-30), 987-995 20 Analytic properties of Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence of degree 1 in the potential, Ph.D Dissertation, Yale University (1983-05) 21 A Zakharov-Shabat inverse scattering problem and the associated evolution equations, Chinese J. of Mathematics , 12(1984), 223-233 22 Hamiltonian structure of soliton equations, Proc. of CCNAA-AIT Seminar on Differential Equations, Hsinchu, Taiwan (1985-06), 205-214 23 On the derivative nonlinear Schrodinger equation, N.S.C. Mathematics Center Symposium of Summer'85, July 1985., (1985-07) 24 Analytic properties of a Zakharov-Shabat inverse scattering problem(I), Chinese J. of Mathematics , 14(1986), 225-248 25 Global solvability of the derivative nonlinear Schrodinger equation, The Transactions of the American Mathematical Society , 314(1989), 107-118 26 A 2 X 2 Zakharov-Shabat system with a quadratic spectral parameter, Proc. of the 4th Workshop of Nonlinear Evolution Equations and Dynamical Systems, Balaruc Les Bains(Montpellier), France, June 11-25, 1987, World Scientific Publishing Co., 199-208., (1987) 27 Analytic properties of a Zakharov-Shabat inverse scattering problem(II), Chinese J. of Mathematics , 16(1988), 81-110 28 On the dissipative evolution equations associated with the Zakharov- Shabat system with a quadratic spectral parameter, The Transactions of the American Mathematical Society , 316(1989), 327-336 29 Inverse problem for a N X N system with a quadratic parameter, in " Nonlinear Analysis " edited by F.C. Liu and T.P. Liu (Proc. of 1989 Conference on Nonlinear Analysis, Academia Sinica, Taipei, June 19-24, 1989 ), 1991 , World Scientific, 105-114., (1989), 105-114 30 A N X N Zakharov-Shabat system with a quadratic spectral parameter, in "Nonlinear Evolution Equations and Dynamical Systems " edited by S. Carillo and O. Ragnisco (Proc. of the 5th Workshop on Nonlinear Eqolution equations and Dynamical Systems, Kolymbari, Crete, Greece, July 2-16, 1989), 1990, Research Reports in Physics, Springer-Verlag, 73-76., (1989/1990) 31 Inverse scattering transform and the D-BAR method , NSC 1989-90 Report, Taipei., (1989) 32 A N X N Zakharov-Shabat system with a polynomial spectral parameter, Proc. of 7th Workshop on nonlinear evolution equations and dynamical systems, Gallipoli, Lecce, Italy, June 19-29, 1991, (Nonlinear Evolution Equations and Dynamical Systems), ed. by M. Boiti et al, World Scientific, 148-152., (1991) 33 N X N Zakharov-Shabat system of the form df/dx=(z2 - 1/z2)Jf + (zQ+P+R/z)f, presented at the III Potsdam-V Kiev International Workshop on Nonlinear Processes in Physics, Clarkson University, Potsdam, New York, U.S.A. , Aug.(1991), 1-11 34 A N X N Zakharov-Shabat system with a polynomial spectral parameter and the associated evolution equations, , June 5-9, 1992., the Proc. of 21th International Conf. on the Differential Geometry Methods on the Theoretical Physic, Tianjin (1992-06) 35 Inverse scattering problem and the associated evolution equations, Proc. of the 2nd Nonlinear Partial Differential Equation Workshop, Hsin-Chu, Chiao-Tung Univ., Jan. 6-7, 1993., (1993) 36 (with Chao-Ting Lin) Solitons of the derivative nonlinear Schrodinger equation and the Massive Thirring model, preprint 37 Solvability of the derivative nonlinear Schrodinger equation and the massive Thirring model, Theoretical and Mathematical Physics , 99(1994), 322-328 38 (with Boris G. Konopelchenko) Harry Dym equation on the complex plane: Inverse scattering transform and exact solutions, Theoretical and Mathematical Physics , 99(1994), 337-344 39 (with B.G. Konopelchenko) Inverse Spectral transform for the Harry Dym equation on the complex plane, Physica D , 81(1995), 32-43 40 (with Oktay K. Pashaev) Integrable hierarchy from topological gauge theory, International J. of Modern Physics A , 12(1997), 213-218 41 (With Oktay K. Pashaev) Integrable gauge theory and integrable models , preprint, 1997., 42 (with Oktay K. Pashaev) Moving frames hierarchy for BF theory, J. of Mathematical Physics , 39(1998), 102-123 43 (with Oktay K. Pashaev) Resonance NLS solitons as black holes in Madelung fluid, Preprint, hep-th/9810139 v.2, Oct. 1998., , 2(1998) 44 (with Oktay K. Pashaev) Dissipative and topological solitons resonance interacions in RD and Ads Heisenberg models, Proc. of the 7th Workshop on Differential Equations and its Application, Jan. 14-16, 1999, Hui-Sun Forest Park, Nantou., (1999-01) 45 (with Oktay K. Pashaev and Chi-Kun Lin) Bilinear form and equivalence relations for DNLS, Proc. of the Workshop on Nonlinearity, Integrability and All That-20 year after NEEDS'79, Gallipoli, Lecce, Italy, July 1-10, 1999, p. 175-181, World Scientific, 2000., (2000) 46 (with Oktay K. Pashaev) Integrability and quantum potential-form dissipatons to exponentially localized Chern-Simon solitons, Proc. n Nonlinearity, Intrgrability and All That-20 year after NEED'79, Gallipoli, Lecce, Italy, July 1-10, 1999, p. 479-503, World Scientific, 2000., (2000) 47 (with Oktay K. Pashaev) Chern-Simons solitins in quantum potential, Chaos, Solitons & Fractals , 11(2000), 2193-2202 48 (with Yen-Ching Lee) Exact solutions of the derivative nonlinear Schro dinger equation and the derivation Reaction-Diffusion system, Proc. of 8th Workshop on Differential Equations Kaohsiung, Sun Yat-sen University , Jan.(2000), 8-9 49 (with Oktay K. Pashaev) Black holes and solitons of quantized dispersionless NLS and DNLS, to appear in ANZIAMJ. of Applied Maths. (Continues of J. of the Australian Mathematical Society , Ser. B: Applied Math.), 50 (with Oktay K. Pashaev) Self- dual vortices in Chern-Simons hydrodynamics , to appear in THeor. Math. Phys. (proc. of the Workshop NEEDS 2000 at Turkey), THeor. Math. Phys. (2000) 51 (with Yen-Ching Lee, Chien-Chih Lin) Exact solutions of DNLS and the derivative Reaction-Diffusion System, to appear in J. of Nonlinear Math. Physics, Supplement 1 to Vol.9 (2002), 87-97 Proc. of the Special Session on Integrable Systems of the First Joint Meeting of AMS-HKMS , Dec. 13-16, 2000, Hong Kong., (2000/2002) 52 (with Oktay K. Pashaev) Soliton Resonances, Black Holes and Madelung Fluid, to appear in J. of Nonlinear Mathematical Physics, Feb. 2001, V.8, Supplement(Proceedings of NEEDS'99) ,230-234., (2001) 53 (with Oktay K. Pashaev) Self-Dual Chern-Simons Solitons and Quantum Potential, to appear in J. of Nonlinear Mathematical Physics, 2000, V.8, Supplement(Proceedings of NEEDS'99), 225-229., (2000) 54 (with Chi-Kun Lin) The Behaviour of Solitons of NLS Equations of Derivative Type in the Semiclassical Limits, Chaos, Solitons and Fractals , 13(2002), 1475-1492 55 (with Oktay K. Pashaev) Resonance solitons as black holes in Madelung fluid, Modern Physics Letters A , 17(2002), no. 24, 1601-1619 56 (with Yen-Ching Lee, Chien-Chih Lin) Some exact solutions of DNLS and derivative Reaction-Diffusion systems via Hirota method, Proc. of 10th Workshop on Differential Equations, Chang-Hua Univ. of Education, Nov. 2-4, 2001., (2001-11) 57 (with Oktay K. Pashaev) Bilinear representation for the modified nonlinear Schrodinger equation and their quantum potential deformation, presented in the Workshop on Nonlinear Physics: Theory and Experiment. II , Gallipoli (Lecce), Italy, June 27-July 6, 2002., proc. of Nonlinear Physics: Theory and Experiment,Gallipoli,World Sci.,2003 (2003), 79-82 58 (with Chi-Kun Lin, Oktay K. Pashaev) Shoch waves,Chiral solitons and semiclassical limit of one-dimensional Anyons, Chaos, Solitons and Fractals , 19(2004), 109-128 59 (with Oktay K. Pashaev), Solitons resonances for MKP-II, presented in the Workshop on Nonlinear Physics: Theory and Experiments.II, Gallipolli(Lecce),Italy,June-July,2004, Theoret. and Mat. Fiz. , 144(2005), 995-1003 60 (with Meltem L. Y. Francisco! Oktay K. Pashaev) Four Soliton Resonances for KP-II and MKP-II.(Special Issue('s) IMACS 2005 International conference entitled, Mathematics and Computers in Simulation , 74(2007), no. 4-5, 323-332 61 (with Oktay K. Pashaev) Solitons of the Resonant Nonlinear Schrüinger Equation with Nontrivial Boundary Conditions and Hirota Bilinear Method(presented in the Workshop on Nonlinear Physics. Theory and Experiment IV, Gallipoli (Lecce), Italy - June 22 - July 1, 2006.), Teoret. Mat. Fiz. , 152(2007), 133-146 62 (with Oktay K. Pashaev, C. Rogers and W. K. Schief) The resonant nonlinear Schrodinger equation in cold plasma physics. Application of Bäklund-Darboux transformations and superposition principles, (SCI:1.037) (Physics, Fluids & Plasma), J. Plasma Physics , 73(2007), no. part 2, 257-272