Research Interests

Publications

• MONOGRAPHS:
• .
• Chaos in Partial Differential Equations.
  International Press, (2004).
• Lie-B\"acklund-Darboux Transformations.
  (with A. Yurov), (2009).
• Invariant Manifolds and Fibrations for Perturbed
  Nonlinear Schr\"odinger Equations.
  Springer-Verlag, Applied Mathematical Sciences, vol.128, (1997).
  (with S. Wiggins)
• .
• ARTICLES:
• .
• A Resolution of the Sommerfeld Paradox
  (with Z. Lin)
  Submitted, (2009)
• Sommerfeld Paradox - A Novel Numerical Study
  (with Y. Lan, Z. Lin)
  Still in progress, (2009)
• Chaos Phenotypes in Fluids
  in press, (2009)
• Exact and Asymptotic Conditions on Traveling Wave Solutions
  of the Navier-Stokes Equations
  (with D. Viswanath)
  Physics of Fluids, vol.21, 101703, (2009)
• Stability Criteria of 3D Inviscid Shears.
  Submitted, (2009)
• Existence of Chaos in Weakly Quasilinear Systems.
  in press, (2009)
• On the Chaotic Flux Dynamics in a Long Josephson Junction.
  (with Frank Feng)
  Submitted, (2009)
• The Poincar\'e Recurrence Problem of Inviscid Incompressible Fluids
  Asian J. Math, vol.13, no.1, 7-14, (2009)
• A Recurrence Theorem on the Solutions to the 2D Euler Equation
  Asian J. Math, vol.13, no.1, 1-6, (2009)
• Issues of Chaos and Recurrence in Infinite Dimensions.
  Submitted, (2009)
• Simple Explicit Formulae For Finite Time Blow Up Solutions
  To The Complex KdV Equation
  Chaos, Solitons and Fractals, vol.39, 369-372, (2009)
• Chaotic Spin Dynamics of a Long Nanomagnet Driven by a Current
  (with Y. Lan)
  Nonlinearity, vol.21, 2801-2823, (2008)
• Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence
  Proceedings of the 4th International Congress of Chinese Mathematicians,
  (Hangzhou, December 17-22, 2007), vol. III, 110-122,
  Higher Education Press, Beijing, China
• On the Dynamics of Navier-Stokes and Euler Equations
  (with Y. Lan)
  J. Stat. Phys., vol. 132, 35-76, (2008)
• Tubes in Dynamical Systems.
  Chaos, Solitons and Fractals, vol.36, 1215-1224, (2008)
• Segment Description of Turbulence
  Dynamics of PDE, vol.4, no.3, 283-291, (2007)
• Chaotic Dynamics of Spin Valve Oscillators
  (with Z. Yang, S. Zhang)
  Physical Review Letters, 99, 134101, (2007)
• Strange Tori of the Derivative Nonlinear Schr\"odinger Equation.
  Letters in Mathematical Physics, vol.80, no.1, 83-99, (2007)
• On the True Nature of Turbulence.
  The Mathematical Intelligencer, vol.29, no.1, 45-48, (2007)
• Chaos in Wave Interactions.
  International J. Bifur. Chaos, vol.17, no.1, 85-98, (2007)
• Arnold Diffusion of the Discrete Nonlinear Schr\"odinger Equation.
  Dynamics of PDE, vol.3, no.3, 235-258, (2006)
• Nanoscale Spontaneous Patterning.
  (with F. Feng)
  J. Stat. Phys., vol.123, no.4, 741-751, (2006)
• Phase Transition as a Mechanism of DNA Opening for
  Replication and Transcription.
  (with D. Retzloff)
  Math. Biosci., vol. 203, no.1, 137-147, (2006)
• Synchronization, Modification and Chaos
  Induced by Spin-Polarized Currents
  (with Z. Li, S. Zhang)
  Phys. Rev. B., vol.74, 054417, (2006)
• Chaos and Shadowing Around a Heteroclinically Tubular Cycle
  With an Application to Sine-Gordon Equation.
  Studies in Applied Mathematics, vol.116, 145-171, (2006)
• Zero Dispersion and Viscosity Limits of Invariant Manifolds
  for Focusing Nonlinear Schr\"odinger Equations.
  J. Math. Anal. Appl., vol.315, 642-655, (2006)
• Invariant Manifolds and Their Zero-Viscosity Limits
  for Navier-Stokes Equations.
  Dynamics of PDE, vol.2, no.2, 159-186, (2005)
• Ergodic Isospectral Theory of the Lax Pairs of Euler
  Equations with Harmonic Analysis Flavor.
  Proc. Amer. Math. Soc., vol.133, 2681-2687, (2005)
• On Quasi-Periodic Boundary Condition Problem.
  J. Math. Phys., vol.46, no.1, 013503, (2005)
• Melnikov Analysis for Singularly Perturbed DSII Equation
  Studies in Applied Math., vol.114, 285-306, (2005)
• Persistent Homoclinic Orbits for Nonlinear Schr\"odinger Equation
  Under Singular Perturbation.
  Dynamics of PDE, vol.1, no.1, 87-123, (2004)
• Existence of Chaos for Nonlinear Schr\"odinger Equation
  Under Singular Perturbation.
  Dynamics of PDE, vol.1, no.2, 225-237, (2004)
• Chaos in Miles' Equations.
  Chaos, Solitons and Fractals, vol.22, no.4, 965-974, (2004)
• Isospectral Theory of Euler Equations.
  (with R. Shvidkoy),
  J. Math. Anal. Appl., vol.292, no.1, 311-315, (2004)
• Euler Equations of Inviscid Fluids.
  Advances in Mathematics Research, Vol. 6, ed. by Gabriel Oyibo,
  Nova Scientific Publishers, (2004)
• The Spectrum of a linearized 2D Euler operator.
  (with Y. Latushkin and M. Stanislavova),
  Studies in Applied Mathematics, vol.112, 259-270, (2004)
• Global Regularity for the Viscous Boussinesq Equations.
  Math. Methods in the Appl. Sci., vol.27, no.3, 363-369, (2004)
• Homoclinic Tubes and Chaos in Perturbed Sine-Gordon Equation.
  Chaos, Solitons and Fractals, vol.20, no.4, 791-798, (2004)
• Chaos and Shadowing Lemma for Autonomous Systems
  of Infinite Dimensions
  J. Dyn. Diff. Eq., vol.15, no.4, 699-730, (2003) [Most Viewed Article]
• Chaos and Shadowing Around a Homoclinic Tube.
  Abstract and Applied Analysis, vol.2003, no.16, 923-931, (2003)
• Lax Pairs and Darboux Transformations for Euler Equations.
  (with A. Yurov),
  Studies in Appl. Math., vol.111, 101-113, (2003)
• Chaos in PDEs and Lax Pairs of Euler Equations.
  Acta Appl. Math., vol.77, 181-214, (2003)
• Homoclinic Tubes in Discrete Nonlinear Schroedinger
  Equation Under Hamiltonian Perturbations.
  Nonlinear Dynamics, vol.31, No.4, 393-434, (2003)
• On 2D Euler Equations: Part II. Lax Pairs and Homoclinic Structures.
  Commun. on Appl. Nonlinear Analysis, vol.10, no.1, 1-43, (2003).
• Singularly Perturbed Vector and Scalar Nonlinear Schroedinger
  Equations with Persistent Homoclinic Orbits.
  Studies in Applied Mathematics, 109: 19-38, (2002)
• Existence of Chaos in Evolution Equations.
  Mathematical and Computer Modelling, vol.36, issues 11-13,
  1211-1219, (2002)
• Chaos in Partial Differential Equations.
  Contemporary Mathematics, 301: 93-115, (2002)
• Integrable Structures for 2D Euler Equations of Incompressible Inviscid Fluids.
  Proc. of Inst. of Math. of NAS of Ukraine, vol.43, Part 1, 332-338, (2002)
• A Lax Pair for 2D Euler Equation.
  J. Math. Phys., vol.42, No.8, 3552-3553, (2001).
• On 2D Euler Equations: Part I. On the Energy-Casimir Stabilities
  and the Spectra for Linearized 2D Euler Equations.
  Journal of Mathematical Physics, vol.41, no.2, pp.728-758, (2000).
• ----- [ pde chaos series, part 1. ] -----
  Morse and Melnikov Functions for NLS Pde's.
  Comm. Math. Phys., vol.162, 175-214, (1994).
  (with D. McLaughlin)
• ----- [ pde chaos series, part 2. ] -----
  Persistent Homoclinic Orbits for Perturbed
  Nonlinear Schr\"odinger Equation.
  Comm. Pure and Appl. Math., XLIX: 1175-1255, (1996).
  (with D. McLaughlin, J. Shatah, and S. Wiggins)
• ----- [ pde chaos series, part 3. ] -----
  Smale Horseshoes and Symbolic Dynamics
  in Perturbed Nonlinear Schr\"odinger Equations.
  Journal of Nonlinear Sciences, vol.9, 363-415, (1999).
• ----- [ lattice chaos series, part 1. ] -----
  B\"acklund Transformations and Homoclinic Structures for
  the Integrable Discretization of the NLS Equation.
  Physics Letters A, vol.163, 181-187, (1992).
• ----- [ lattice chaos series, part 2. ] -----
  Homoclinic Orbits and Chaos in Discretized
  Perturbed NLS Systems: Part I. Homoclinic Orbits.
  Journal of Nonlinear Sciences, vol.7, 211-269, (1997).
  (with D. McLaughlin)
• ----- [ lattice chaos series, part 3. ] -----
  Homoclinic Orbits and Chaos in Discretized
  Perturbed NLS Systems: Part II. Symbolic Dynamics.
  Journal of Nonlinear Sciences, vol.7, 315-370, (1997).
  (with S. Wiggins)
• ----- [ high-D pde chaos series, part 1. ] -----
  Homoclinic Orbits and B\"acklund Transformations
  for the Doubly Periodic Davey-Stewartson Equation.
  Nonlinear Processes in Physics,
  Proceedings of III Potsdam - V Kiev
  International Workshop at Clarkson University,
  Potsdam, NY. pp.122-125, (1993).
  (with D. McLaughlin)
• ----- [ high-D pde chaos series, part 2. ] -----
  B\"acklund-Darboux Transformations and Melnikov Analysis
  for Davey-Stewartson II Equations.
  Journal of Nonlinear Sciences, Vol.10, 103-131, (2000).
• ----- [ pde Hamiltonian perturbation paper. ] -----
  Homoclinic Tubes in Nonlinear Schr\"odinger Equation.
  Under Hamiltonian Perturbations.
  Progress of Theoretical Physics, Vol.101, No.3(4), 559-577, (1999).
• CHAOTIC DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS AND LATTICES.
  Ph.D. Dissertation, Princeton University, Department of Mathematics, (1993).
• .
• Undergraduate Papers:
• .
• Auto-B\"acklund Transformations for Some
  Evolution Equations.
  Advances in Applied Mathematics and Mechanics in China,
  vol.4, pp.91-106, Internat. Acad. Publ., Beijing, (1992).
• New Methods and Developments of the B\"acklund Transformations.
  Advances in Mechanics (in Chinese), vol.21, No.4, 1991, 470-481.
• On the Equivalence Problem of the Navier-Stokes Equations.
  Mechanics and Engineering (in Chinese), Vol.11, No.5, 1989, 39-41.
• B\"acklund Transformations for the Equation
  \Laplacian u = f(u).
  Applied Mathematics and Mechanics, vol.10, No.2, 139-143, 1989.
• Singular perturbation analysis on the capillary instability of liquid jet.
  Proc. 1st Chinese National Turbulence Conference (in Chinese, 1986).