Publications

Monographs:

• Chaos in Partial Differential Equations.  International Press, (2004). 

• Lie-B\"acklund-Darboux Transformations.  International Press, Boston, USA and Higher Education Press, Beijing, China  Surveys of Modern Mathematics, Vol.8, (2014). (with A. Yurov) 

• Invariant Manifolds and Fibrations for Perturbed  Nonlinear Schr\"odinger Equations.  Springer-Verlag, Applied Mathematical Sciences, vol.128, (1997).  (with S. Wiggins) 

Articles:

• Superfast amplification and superfast nonlinear saturation of perturbations as a mechanism of turbulence (with Richard Ho, Arjun Berera, Z. C. Feng)  J. Fluid Mech., vol. 904, A27, 1-13, (2020) 

• Linear hydrodynamic stability  Notices of the AMS, vol.65, no.10, 1255-1259, (2018) 

• Infinite norm of the derivative of the solution operator of Euler equations  Physics of Fluids, vol.33, 035120, (2021) 

• Quasi-optimal limiting resource leads to coexistence - toward a resolution of the paradox of plankton, (with Frank Feng), Submitted, (2024) 

• A mathematical model of arm races, (with Hong Yang), Submitted, (2024) 

• On the relation between wind speed and maximum or mean water wave height, (with Balkissoon, S.; Lupo, A.R.;Walsh, S.; McGuire, L.) Atmosphere 2024, 15, 948. 

• Nowhere-differentiability of the solution map of 2D Euler equations on bounded spatial domain  (with Hasan Inci)  Dynamics of PDE, Vol.16, No.4, 383-392, (2019) 

• Short term unpredictability of high Reynolds number turbulence — rough dependence on initial data  (with Frank Feng)  Electronic J. of Diff. Eq., 2020, No. 104, pp. 1–21 (2020) 

• Determining chaotic characteristics and forecasting tall tower  wind speeds in Missouri using empirical dynamical modeling (EDM)  (with Sarah Balkissoon, Neil Fox, Anthony Lupo, Sue Ellen Haupt, Patrick Market, Samuel Walsh)  Renewable Energy, 170, 1292-1307,(2021) 

• The PAT model of population dynamics. (with Frank Feng) Ann. Appl. Math. vol. 38, no.2, 223-239 (2022)

• Enrichment paradox and applications  (with Frank Feng)  EJDE, 2023, no.17, 1-10, (2023) 

• Dependence upon initial conditions  Commun Nonlinear Sci Numer Simulat, 70, 74–79 (2019) 

• Rough dependence upon initial data exemplified by explicit solutions and the effect of viscosity  Nonlinearity, 30, 1097-1108, (2017) 

• The distinction of turbulence from chaos --- rough dependence on initial data  Electronic Journal of Differential Equations, Vol. 2014, No. 104, pp. 1-8, (2014) 

• Toward the description of short term unpredictability for fully developed turbulence  (How to describe turbulence in view of the short term unpredictability,  this is the ongoing project with Frank Feng, January 15, 2014) 

• On the so-called rogue waves in the nonlinear Schr\"odinger equation  Electron. J. Diff. Equ., Vol. 2016 (2016), No. 101, pp. 1-10. 

• A mathematical model of demand-supply dynamics with collectability and saturation factors  (with Hong Yang)  Intl. J. Bifurcation and Chaos, Vol. 27, No. 1 (2017), 1750016. 

• Sensitive versus rough dependence under initial conditions in atmospheric flow regimes  (with A. Lupo, Z. Feng, N. Fox, J. Rabinowitz, M. Simpson)  Atmosphere 7, 157; doi:10.3390/atmos7120157Submitted, (2016) 

• Major open problems in chaos theory and nonlinear dynamics  Dynamics of PDE, Vol.10, No.4, 379-392, (2013) 

• A Resolution of the Sommerfeld Paradox  (with Z. Lin)  SIAM J. Math. Anal., Vol.43, No.4, 1923-1954, (2011) 

• A Resolution of the Sommerfeld Paradox: Numerical Implementation  (with Y. Lan)  Intl. J. Non-Linear Mechanics, vol.51, 1-9, (2013) 

• On the Arrow of Time  (with H. Yang)  Discrete and Continuous Dynamical Systems, S, vol.7, No. 6, 1287-1303 (2014) 

• A Resolution of the Paradox of Enrichment  (with Frank Feng)  Intl. J. Bifurcation and Chaos, Vol. 25, No. 6, 1550094, (2015) 

• Symbolic dynamics and chaos in plane Couette flow  Dynamics of PDE, vol.14, no.1, 79-85, (2017) 

• On the Paradox of Pesticides  (with Yipeng Yang)  Commun. Nonlinear Sci. Numer. Simulat., vol. 29, 179-187, (2015) 

• The Effect of Synchronous Firing on the Clustering Dynamics of Social Amoebae  (with Yipeng Yang)  Complexity, vol.20, no.1, 16-26, (2014) 

• Recurrence in 2D Inviscid Channel Flow  Applied Mathematics Letters, vol.25, 2188-2192, (2012) 

• Stability Criteria and Turbulence Paradox Problem For Type II 3D Shears  J. Phys. A: Math. Theor. 45, 175501, (2012) 

• Chaos Phenotypes in Fluids  Discrete and Continuous Dynamical Systems, vol.26, no.4, 1383-1398, (2010) 

• Exact and Asymptotic Conditions on Traveling Wave Solutions  of the Navier-Stokes Equations  (with D. Viswanath)  Physics of Fluids, vol.21, 101703, (2009) 

• A Markov Chain Approximation of a Segment Description of Chaos.  (with Alexander Labovsky)  Dynamics of PDE, vol.7, no.1, 65-76, (2010) 

• Stability Criteria of 3D Inviscid Shears.  Quart. Appl. Math., vol. LXIX, no.2, 379-387, (2011) 

• Existence of Chaos in Weakly Quasilinear Systems.  Communications on Pure and Applied Analysis, vol.10, no.5, 1331-1344, (2011) 

• On the Chaotic Flux Dynamics in a Long Josephson Junction.  (with Frank Feng)  Int. J. Bifurcation and Chaos, vol.21, no.1, 137-153, (2011) 

• 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.  Dynamics of PDE, vol.8, no.1, 69-77, (2011) 

• 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 Articles:

• 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) 

• (Review Article) 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. 2nd Chinese National Turbulence and Hydrodynamic Stability Conference, Wuhan, China (in Chinese, 1986)