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Mathematical Analysis and Modelling Research Group

About Us

This group develops theoretical techniques through analytical as well as topological and algebraic tools, and apply the ideas and results obtained in the theoretical investigation to problems arising from a wide range of areas, including various important models in differential geometry and natural sciences (biology, chemistry, ecology, physics and rural sciences). The concrete problems we currently work on include the Chern-Simons model in condensed matter physics, the logistic and related models in population biology, and image reconstruction for diffraction and phase contrast tomography. Important problems in these applied areas serve as a key source of the topics for investigation by the group.

Research Topics

  • Nonlinear partial differential equations
  • Reaction-diffusion equations
  • Mathematical biology
  • Nonlinear functional analysis
  • Complex analysis
  • CR-Geometry
  • Complex analytic geometry
  • Pseudoholomorphic curves
  • Coherent diffractive imaging
  • Phase-contrast tomography

Staff Researchers

Postgraduate Students

  • Linfeng Mei (PhD)
  • Lena Schmalz (PhD)
  • Ben Mclaughlin (PhD)

External Collaborators

  • Dr Florica Cirstea (Univ. of Sydney)
  • Prof Daomin Cao (Chinese Academy of Sci., China)
  • Prof Norman Dancer (Univ. of Sydney)
  • Prof Vladimir Ezhov (Univ. of South Australia)
  • Dr Timur Gureyev (CSIRO - Manufacturing and Infrastructure Technology, Melbourne)
  • Prof Zongming Guo (Henan Normal Univ., China)
  • Prof Sze-Bi Hsu (National Tsinghua Univ., Taiwan)
  • Prof Kang-Tae Kim (POSTECH, Rep. of Korea)
  • Prof Martin Kolar (Masaryk Univ., Czech Rep.)
  • Prof Xing Liang (Chinese Univ of Sci and Tech, China)
  • Prof Changshou Lin (National Taiwan Univ.)
  • Prof Zhigui Lin (Yangzhou Univ., China)
  • Dr David Paganin (Monash University)
  • Dr Gabriel Paternain (Cambridge Univ., UK)
  • Dr Rui Peng (Memorial Univ of Newfoundland, Canada)
  • Prof Vasilii Punegov (Russian Academy of Sci., Russia)
  • Prof Juncheng Wei (Chinese Univ. of Hong Kong)

Research Projects and Grants Since 2004

  1. Y. Du, E.N. Dancer and S. Yan, Transitions and Singular Behaviour in Nonlinear Partial Differential Equations, ARC discovery grant, 2010-2011, $306,000.
  2. Y. Du, Mathematical modelling of the spreading of invasive species, UNE PVCR grant, 2010, $30,000.
  3. K.M. Pavlov, High resolution deterministic coherent diffractive imaging, Australian Synchrotron (International Synchrotron Access Program), 2009, $9,800.
  4. Y. Du, S, Yan, G. Schmalz and A. Harris, Singularities in nonlinear and complex analysis, UNE IRL grant, 2009, $13,000.
  5. K.M. Pavlov, High contrast polychromatic biomedical imaging, UNE IRL, 2008, $12,000.
  6. K.M. Pavlov, Measuring the complex amplitude reflection coefficient of layer-structured crystals for use in analyzer-based phase contrast imaging, Australian Synchrotron Research Program Grant, 2008, $8,850.
  7. Y. Du, E.N. Dancer and S. Yan, Sharp Transitions in Partial Differential Equations and Related Problems, ARC discovery grant, 2007-2009, $240,000.
  8. Y. Du, A. Harris, G. Schmalz and S. Yan, Partial differential equations and complex geometry, UNE grant, 2007-2008, $11,000
  9. K.M. Pavlov, Measuring the complex amplitude reflection coefficient of analyser crystals for use in analyser-based contrast imaging, Australian Synchrotron Research Program Grant, 2007, $7,950.
  10. K.M. Pavlov, Direct and unambiguous coherent diffractive imaging using a rectangular aperture, Australian Synchrotron Research Program Grant, 2007, $8,250.
  11. D.E. Jesson, U. Denker, B.F. Usher, K.M. Pavlov and M.J. Morgan, Exploring the Dynamics of Nanostructure Self-Organisation during Compound Semiconductor Epitaxy, Australian Research Council Discovery Grant, 2005-2007, $187,518.
  12. Y. Du and E.N. Dancer, Free Boundary Problems in Partial Differential Equations and Related Topics, ARC discovery grant, 2003-2006, $158,000.
  13. V. Ejov, G. Schmalz and A Spiro, Normal forms and Chern-Moser Connection in the Study of Cauchy-Riemann Manifolds, ARC discovery grant, 2003-2006, $165,000.
  14. Y. Du, Sharp transitions in partial differential equations and related problems, UNE RPG, 2006, $14,500.
  15. Y. Du, Singular perturbation and transition layers in heterogeneous environment, Australian Academy of Science and JSPS (Japan), 2005, $7,600.
  16. Y. Du, Partial differential equations with unconventional boundary conditions and applications, UNE RPG, 2005, $13,000.
  17. K.M. Pavlov, Dynamic analyser-based and propagation based X-ray phase contrast imaging, Access to Major Research Facilities Program, Australian Synchrotron Research Program Grant, 2005, $7,450.
  18. K.M. Pavlov, Statistical Diffraction Reciprocal Space Mapping Study of the Epitaxial Growth of the Co Islands on Calcium Fluoride, Australian Synchrotron Research Program Grant, 2005, $6,680.
  19. K.M. Pavlov and D.M. Paganin, New quantitative methods in X-ray imaging using crystal optics, Australian Research Council Discovery Grant, 2005-07, $240,000.
  20. Y. Du, Partial differential equations with unconventional boundary conditions and applications, UNE RPG, 2004, $10,000.
  21. K.M. Pavlov, The foundations of DEI: Characterization of full crystal transfer functions, Access to Major Research Facilities Program, Australian Synchrotron Research Program Grant, 2004, $11,700.

Publications since 2007

Books

  1. Y. Du, Hitoshi Ishii and Wei-Yueh Lin (editors), Recent Progress in Nonlinear Partial Differential Equations and Viscosity Solutions, World Scientific Publishing, 372 pages. ISBN 981-283-473-7, 2009.
  2. Y. Du (author), Order structure and topological methods in nonlinear partial differential equations. Vol. 1, Maximum principles and applications, Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, x+190 pages. ISBN 981-256-624-4, 2006.

Papers

  1. F. Cirstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations, J. Functional Anal., 259, pp 174-202, 2010. (A*)
  2. Y. Du and Zhigui Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42(2010), 377-405. (A)
  3. Y. Du and Hiroshi Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. European Math. Soc., 12(2010), 279-312. (A*)
  4. J. Schmalz, G. Schmalz, T. Gureyev, and K. Pavlov, On the derivation of the Greens function for the Helmholtz equation using generalized functions, Am. J. Phys., (2010), Vol. 78, No. 2, pp. 181-186. (B)
  5. Y. Du and Yoshio Yamada, On the long-time limit of positive solutions to the degenerate logistic equation, Discrete and Continuous Dynamical Systems A, 25(2009), 123-132. (A)
  6. Y. Du, Rui Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Eqns., 246, pp. 3932-3956, 2009. (A*)
  7. Y. Du and Z.M. Guo, Positive solutions of an elliptic equation with negative exponent: stability and critical power, J. Diff. Eqns., 246, pp. 2387-2414, 2009. (A*)
  8. Y. Du, Change of environment in model ecosystems: effect of a protection zone in diffusive population models, Recent Progress in Nonlinear Partial Differential Equations and Viscosity Solutions, Y. Du, H. Ishii and W.Y. Lin (eds.), World Scientific Publishing, ISBN 978-981-283-473-7, 2009.
  9. D.J. Vine, D. M. Paganin, K. M. Pavlov, K. Uesugi, A. Takeuchi, Y. Suzuki, N. Yagi, T. Kampfe, E.-B. Kley and E. Forster, Deterministic Green’s function retrieval using hard X-rays, Phys. Rev. Lett., 102, 043901, 2009. (IF=6.944) (A*)
  10. Cao Daomin, Peng Shuangjie and Yan Shusen, Asymptotic behaviour of ground state solutions for the Henon equation, IMA J. Appl. Math., 74, no. 3, pp. 468-480, 2009. (A)
  11. Dancer Edward N., Hilhorst Danielle and Yan Shusen, Peak solutions for the Dirichlet problem of an elliptic system, Discrete Contin. Dyn. Syst. no. 3, pp. 731-761, 2009. (A)
  12. K.T.Kim, E.Poletsky and G.Schmalz, Functions holomorphic along holomorphic vector fields, Journal of Geometric Analysis, accepted 14/2/2009. (A)
  13. V. Ejov, M. Kolar and G. Schmalz, Degenerate hypersurfaces with a two-parametric family of automorphisms, Complex Variables and Elliptic Equations, accepted 20/1/2009. (B)
  14. Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasiliear problem modelling phytoplankton I: Existence, SIAM J. Math. Anal., 40, pp. 1419-1440, 2008. (A)
  15. Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasiliear problem modelling phytoplankton II: Limiting profile, SIAM J. Math. Anal., 40, pp. 1441-1470, 2008. (A)
  16. Y. Du, P.Y.H. Pang and M. Wang, Qualitative analysis of a predator-prey model with stage structure for the predator, SIAM J. Appl. Math., 69, pp. 596-620, 2008. (A*)
  17. Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Eqns., 244, pp. 61-86, 2008. (A*)
  18. Y. Du, The heterogeneous Allen-Cahn equation in a ball: solutions with layers and spikes, J. Diff. Eqns., 244, pp. 117-169, 2008. (A*)
  19. Y. Du, Z. Liu, A. Pistoia and Shusen Yan, Sign changing solutions with clustered layers near the origin for singularly perturbed semilinear elliptic problems on a ball, Methods and Applications of Analysis (special issue dedicated to N.S. Trudinger), 15, pp. 137-148, 2008.
  20. A. Harris and M. Kolar, A Remark on cohomology with supports in the complement of a cone singularity, RIMS Kokyuroku, 1610, pp. 32-37, 2008.
  21. A. Harris and G. Paternain, Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders, Ann. Global Anal and Geom., 34, pp. 115-134, 2008. (A)
  22. A. Harris and K. Wysocki, Branch-structure of J-holomorphic curves near periodic orbits of a contact manifold, Trans. Amer. Math. Soc., 360, pp. 2131-2152, 2008. (A*)
  23. M.J. Kitchen, K.M. Pavlov, S.B. Hooper, D.J. Vine, K.K.W. Siu, M.J. Wallace, M.L.L. Siew, N. Yagi, K. Uesugi and R.A. Lewis. Simultaneous acquisition of dual analyser-based phase contrast X-ray images for small animal imaging, Eur J. Radiol, 68S, S49-S53, 2008. (B)
  24. V. Beloshapka, V. Ejov and G. Schmalz, Holomorphic classification of 4-dimensional surfaces in C3, Izv. Ross. Akad. Nauk Ser. Mat., 72:3, pp. 3-18, 2008. (B)
  25. V. Ejov, G. Schmalz and A. Spiro, CR-manifolds of codimension two of parabolic type, Indiana University Mathematics Journal, 57:1, pp. 309-342, 2008. (A)
  26. C.-K. Han, J.-W. Oh and G. Schmalz, Symmetry algebra for multi-contact structures given by 2n vector fields on ?, Mathematische Annalen, 341:3, pp. 529-542, 2008. (A*)
  27. Cao Daomin, Noussair Ezzat S. and Yan Shusen, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Trans. Amer. Math. Soc., 360, no. 7, pp. 3813-3837, 2008. (A*)
  28. Dancer EN and Yan S, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, Journal of the London Mathematical Society, 78, pp. 639-662, 2008. (A)
  29. Dancer EN and Yan S, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods and Applications of Analysis, 15, no. 1, pp. 97-120, 2008. (B)
  30. W. X. Tang, D. E. Jesson, K. M. Pavlov, M. J. Morgan and B. F. Usher, Ga droplet morphology on GaAs (001) studied by Lloyd's Mirror photo-emission electron microscopy, J. Phys. - Condens. Matter, 21, 314022(2009). (A)
  31. F. Cirstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity, J. Functional Anal., 250, pp. 317-346, 2007. (A*)
  32. Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation, J. Diff. Eqns., 238, pp. 87-117, 2007. (A*)
  33. F. Cirstea and Y. Du, Large solutions of elliptic equations with a weakly superlinear nonlinearity, J. d'Analyse Math., 103, pp. 261-277, 2007. (A)
  34. Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359, pp. 4557-4593, 2007. (A*)
  35. Y. Du, Z.M. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem, Discrete and Continuous Dynamical Systems, 19, pp. 271-298, 2007. (A)
  36. D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Krausslich, O. Wehrhan, I. Uschmann, and E. Forster, Analyser-based phase contrast imaging and phase retrieval using a rotating anode X-ray source, Appl. Phys. Lett., 91, 254110, 2007. (IF=3.596) (A*)
  37. D. E. Jesson, K. M. Pavlov, M. J. Morgan and B. F. Usher, Imaging surface topography using Lloyd’s mirror in Photo-Emission Electron Microscopy, Phys. Rev. Lett., 99, 016103, 2007. (IF=6.944) (A*)
  38. S.B. Hooper, M.J. Kitchen, M.J. Wallace, N. Yagi, K. Uesugi, M.J. Morgan, C. Hall, K.K.W. Siu, I.M. Williams, M. Siew, S.C. Irvine, K. Pavlov and R.A. Lewis, Imaging lung aeration and lung liquid clearance at birth, FASEB (The Journal of the Federation of American Societies for Experimental Biology), 21, pp. 3329-3337, 2007. (IF=6.791) (A)
  39. S.G. Podorov, K.M. Pavlov and D.M. Paganin, A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging, Optics Express, 15(16), pp. 9954-9962, 2007. (IF=3.709) (A*)
  40. M.J. Kitchen, K.M. Pavlov, K.K.W. Siu, R.H. Menk, G. Tromba and R.A. Lewis, Analyser-based phase-contrast image reconstruction using geometrical optics, Phys.Med. Biol., 52, pp. 4171-4187, 2007. (IF=2.528) (A)
  41. K.M. Pavlov, D.M. Paganin, D.J. Vine and L. Kirste, Wide angle X-ray dynamical diffraction by deformed crystals: recurrence relations, Phys. Status Solidi A, 204(8), pp. 2613-2619, 2007. (B)
  42. D.J. Vine, D.M. Paganin, K.M. Pavlov and S.G. Podorov, Unambiguous reconstruction of the complex amplitude reflection coefficient of a laterally homogeneous crystal using analyser-based phase-contrast imaging, J. Appl. Cryst., 40, pp. 650-657, 2007. (IF=3.629) (A)
  43. T.E. Gureyev, Ya.I. Nesterets, K.M. Pavlov and S.W. Wilkins, Computed tomography with linear shift-invariant optical systems, J. Opt. Soc. Am. A., 24(8), pp. 2230-2241, 2007. (A*)
  44. V. I. Punegov, A. I. Maksimov, S. I. Kolosov and K. M. Pavlov, Calculating X-Ray diffraction from multilayer lateral crystal structures with arbitrary shapes and composition profiles, Technical Physics Letters, 33(2), pp. 125-127, 2007. (C)
  45. V. Ezhov and G. Schmalz, Elliptic CR-manifolds and shear invariant ODE with additional symmetries, Arkiv Matematik, 45:2, pp. 253-268, 2007. (B)
  46. V. Beloshapka, V. Ejov and G. Schmalz, Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russ. J. Math. Phys., 14:2, pp. 121-133, 2007.
  47. Dancer E.N. and Yan S., Remarks on the existence of many solutions of certain nonlinear elliptic equations, Bolletino U.M.I., 8, no. 10 (B), pp. 1013-1023, 2007. (B)
  48. Li Gongbao, Peng Shuangjie and Yan Shusen, A new type of solutions for a singularly perturbed elliptic Neumann problem, Rev. Mat. Iberoam, 23, no. 3, pp. 1039-1066, 2007.
  49. Wei Juncheng and Yan Shusen, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl., (9) 88, no. 4, pp. 350-378, 2007. (A*)
  50. Wei Juncheng and Yan Shusen, New solutions for nonlinear Schrodinger equations with critical nonlinearity, J. Differential Equations, 237, no. 2, pp. 446-472, 2007. (A*)
  51. Dancer Edward Norman, Ren Xiaofeng and Yan Shusen, On multiple radial solutions of a singularly perturbed nonlinear elliptic system, SIAM J. Math. Anal., 38, no. 6, pp. 2005-2041, 2007. (A)
  52. Wei Juncheng and Yan Shusen, Lazer-McKenna conjecture: the critical case, J. Funct. Anal., 244, no. 2, pp. 639-667, 2007. (A*)
  53. Li Gongbao, Yan Shusen and Yang Jianfu, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations, 28, no. 4, pp. 471-508, 2007. (A*)
  54. Dancer E. N. and Yan Shusen, A new type of concentration solutions for a singularly perturbed elliptic problem, Trans. Amer. Math. Soc., 359, no. 4, pp. 1765-1790, 2007. (A*)

Events

Contact us

    Professor Yihong Du
    School of Science and Technology
    University of New England, Armidale, NSW 2351
    Email: ydu@turing.une.edu.au
    Phone: 02 6773 3066 (or +61 2 6773 3066 overseas)
    Homepage: http://turing.une.edu.au/staff/~ydu