Pavel M. Lushnikov

Distinguished Professor of Mathematics

 


Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131 USA
Phone: +1 505 277-2104
E-mail:  
 
Resume
Media and online video presentations
Recent meetings organization and upcoming workshops
Teaching
  • Math 412/512: Fall 2026, Introduction to Ordinary Differential Equations
  • Math 579: Fall 2026, Nonlinear Partial Differential Equations with Applications to Fluid Dynamics, Quantum Fluids, Optics, Plasmas and Biology
    • Research interests
    include a wide range of topics in applied mathematics, nonlinear waves and theoretical physics. Among them are laser fusion and laser-plasma interaction; dynamics of fluids with free surface, Kelvin-Helmholtz instability and nonlinear interactions of surface waves; theory of the wave collapse, singularity formation and its application to plasma physics, hydrodynamics, biology and nonlinear optics; bacterial aggregation, chemotaxis, cell-cell interactions. collapse of bacterial colonies, stochastic Potts model of biological cell; pattern formation in photorefractive crystals and other nonlinear optical media; high-bit-rate optical communication; dispersion-managed optical fiber systems; soliton propagation in optical systems; high performance parallel simulations of optical fiber systems; Bose-Einstein condensation of ultracold dipolar gases.
    Publications
    Articles in Refereed Journals
    1. D.A. Silantyev, P.M. Lushnikov, M. Siegel, and D.M. Ambrose, Exact Periodic Solutions of the Generalized Constantin–Lax–Majda Equation With Dissipation, Studies in Applied Mathematics, v. 155, e70115 (2025). (DOI:10.1111/sapm.70115) (arXiv:2411.01891 PDF)
    2. D.M. Ambrose, P.M. Lushnikov, M. Siegel, and D.A. Silantyev, Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains, Nonlinearity, v. 37, 025004 (2024). (DOI: 10.1088/1361-6544/ad140c) (arXiv:2207.07548 PDF)
    3. B. Deconinck, S.A. Dyachenko, P.M. Lushnikov, A. Semenova, The dominant instability of near-extreme Stokes waves. The Proceedings of the National Academy of Sciences (PNAS), v. 120 (32) e2308935120 (2023). (DOI: 10.1073/pnas.2308935120) (arXiv:2211.05473 PDF)
    4. A.O. Korotkevich, P.M. Lushnikov, A. Semenova, S.A. Dyachenko, Superharmonic Instability of Stokes Waves, Studies in Applied Mathematics, v.150, 119 (2022). (DOI: 10.1111/sapm.12535) (arXiv:2206.00725 PDF)
    5. P.M. Lushnikov, D.A. Silantyev and M. Siegel, Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation, Journal of Nonlinear Science, v. 31, 82 (2021). (DOI: 10.1007/s00332-021-09737-x) (arXiv:2010.01201 PDF)
    6. A.I. Dyachenko, S.A. Dyachenko, P.M. Lushnikov and V.E. Zakharov, Short branch cut approximation in two-dimensional hydrodynamics with free surface, Proc. Roy. Soc. A, v. 477, 20200811 (2021). ( PDF) DOI: 10.1098/rspa.2020.0811
    7. P.M. Lushnikov and V.E. Zakharov, Poles and branch cuts in free surface hydrodynamics, Water Waves, v. 3, 251-266 (2021). ( PDF). DOI: 10.1007/s42286-020-00040-y
    8. A. Semenova, S.A. Dyachenko, A.O. Korotkevich, P.M. Lushnikov, Comparison of Split-Step and Hamiltonian Integration Methods for Simulation of the Nonlinear Schrödinger Equation, Journal of Computational Physics, v. 427, 110061 (2021). ( PDF). DOI: 10.1016/j.jcp.2020.110061
    9. I. Kolokolov, V. Lebedev, P.M. Lushnikov, Statistical properties of a laser beam propagating in a turbulent medium, Physical Review E, v. 101, 042137 (2020). ( PDF) DOI:10.1103/PhysRevE.101.042137
    10. P.M. Lushnikov and N.M. Zubarev, Explosive development of the Kelvin-Helmholtz quantum instability on the Helium II free surface, JETP, v. 129, 651-658 (2019). (arXiv:1809.09584 PDF) DOI: doi.org/10.1134/S1063776119100157
    11. A.I. Dyachenko, S.A. Dyachenko, P.M. Lushnikov and V.E. Zakharov, Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Constants of Motion, Journal of Fluid Mechanics, v. 874, 891-925 (2019). (arXiv:1809.09584 PDF) DOI:10.1017/jfm.2019.448
    12. A.I. Dyachenko, P.M. Lushnikov and V.E. Zakharov, Non-Canonical Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with Free Surface, Journal of Fluid Mechanics, v. 869, pp. 526-552 (2019). (arXiv:1711.02841 PDF) DOI: dx.doi.org/10.1017/jfm.2019.219
    13. P.M. Lushnikov and N.M. Zubarev, Exact solutions for nonlinear development of Kelvin-Helmholtz instability for counterflow of superfluid and normal components of Helium II. Phys. Rev. Lett., v. 120, 204504 (2018). ( PDF) DOI:10.1103/PhysRevLett.120.204504
    14. P.M. Lushnikov and N. Vladimirova, Toward defeating diffraction and randomness for laser beam propagation in turbulent atmosphere. JETP Lett., v. 108, issue 9, p. 571-576 (2018). (arXiv:1711.02841 PDF) DOI: 10.1134/S0021364018210026
    15. L.G. Wright, Z.M. Ziegler, P.M. Lushnikov, Z. Zhu, M.A. Eftekhar, D.N. Christodoulides, F.W. Wise, Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial and Outlook. IEEE Journal of Selected Topics in Quantum Electronics, v. 24, 5100516 (2018). (arXiv:1708.05324 PDF) DOI: 10.1109/JSTQE.2017.2779749
    16. P.M. Lushnikov, S.A. Dyachenko and D.A. Silantyev, New conformal mapping for adaptive resolving of the complex singularities of Stokes wave. Proc. Roy. Soc. A, v. 473, 20170198 (2017). (arXiv:1703.06343 PDF) DOI: dx.doi.org/10.1098/rspa.2017.0198
    17. D.A. Silantyev, P.M. Lushnikov and H.A. Rose, Langmuir wave filamentation in the kinetic regime. II. Weak and strong pumping of nonlinear electron plasma waves as the route to filamentation. Phys. of Plasmas, v. 24, 042105 (2017). (PDF) DOI: 10.1063/1.4979290
    18. D.A. Silantyev, P.M. Lushnikov and H.A. Rose, Langmuir wave filamentation in the kinetic regime. I. Filamentation instability of Bernstein-Greene-Kruskal modes in multidimensional Vlasov simulations. Phys. of Plasmas, v. 24, 042104 (2017). (PDF) DOI: 10.1063/1.4979289
    19. P.M. Lushnikov, Structure and location of branch point singularities for Stokes waves on deep water.
      Journal of Fluid Mechanics, v. 800, pp. 557-594 (2016). (PDF) DOI: 10.1017/jfm.2016.405
    20. S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich, Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pade approximation.
      Studies in Applied Mathematics, v. 137, pp. 419-472 (2016). DOI: 10.1111/sapm.12128 (arXiv:1507.02784 PDF)
    21. P.M. Lushnikov and N. Vladimirova, Nonlinear combining of multiple laser beams in Kerr medium.
      Optics Express, v. 23, pp. 31120-31125 (2015). (PDF)
    22. A.O. Korotkevich, P.M. Lushnikov, and H.A. Rose, Beyond the random phase approximation: Stimulated Brillouin backscatter for finite laser coherence times.
      Phys. of Plasmas, v. 22, 012107 (2015). (PDF) DOI: 10.1063/1.4906057
    23. P.M. Lushnikov and N. Vladimirova, Nonlinear combining of laser beams.
      Optics Letters, v.39, 3429-3432 (2014). (PDF) DOI: 10.1364/OL.39.003429
    24. P.M. Lushnikov, H.A. Rose, D.A. Silantyev, and N. Vladimirova, Vlasov multi-dimensional model dispersion relation.
      Phys. of Plasmas, v. 21, 072103 (2014). (PDF) DOI: 10.1063/1.4886122
    25. S.I. Dejak, D. Egli, P.M. Lushnikov, and I.M. Sigal. On blowup dynamics in the Keller-Segel model of chemotaxis.
      St. Petersburg Mathematical Journal, v. 25,547-574 (2014). (PDF)
    26. S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich. The complex singularity of a Stokes wave.
      JETP Letters, v. 98, 675-679 (2014). (PDF) DOI: 10.1134/S0021364013240077
    27. S.A. Dyachenko, P.M. Lushnikov and N. Vladimirova. Logarithmic scaling of the collapse in the critical Keller-Segel equation.
      Nonlinearity, v. 26, 3011-3041 (2013). (PDF) DOI: 10.1088/0951-7715/26/11/3011
    28. P.M. Lushnikov, S.A. Dyachenko and N. Vladimirova. Beyond leading-order logarithmic scaling in the catastrophic self-focusing of a laser beam in Kerr media.
      Physical Review A, v. 88, 013845 (2013). (PDF) DOI: 10.1103/PhysRevA.88.013845
    29. S.I. Dejak, P.M. Lushnikov, Y.N. Ovchinnikov, and I.M. Sigal. On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis.
      Physica D, v. 241, 1245-1254 (2012). (PDF) DOI: 10.1016/j.physd.2012.04.003
    30. P.M. Lushnikov, P. Sulc, and K.S. Turitsyn, Non-Gaussianity in single-particle tracking: Use of kurtosis to learn the characteristics of a cage-type potential.
      Physical Review E, v. 85, 051905 (2012) (PDF). DOI: 10.1103/PhysRevE.85.051905
    31. R. Gejji, P.M. Lushnikov, and M. Alber, Macroscopic model of self-propelled bacteria swarming with regular reversals.
      Physical Review E, v. 85, 021903 (2012). (PDF). DOI: 10.1103/PhysRevE.85.021903
    32. Y. Chung and P.M. Lushnikov, Strong Collapse Turbulence in Quintic Nonlinear Schrodinger Equation.
      Physical Review E, v. 84, 036602 (2011) (PDF). DOI: 10.1103/PhysRevE.84.036602
    33. A.O. Korotkevich, and P.M. Lushnikov, Proof of concept implementation of the massively parallel algorithm for simulation of dispersion-managed WDM optical fiber systems.
      Optics Letters, v. 36, 1851-1853 (2011). (PDF).
    34. P.M. Lushnikov and N. Vladimirova. Non-Gaussian Statistics of Multiple Filamentation.
      Optics Letters v. 35, 1965-1967 (2010). (PDF)
    35. P.M. Lushnikov. Collapse and stable self-trapping for Bose-Einstein condensates with 1/r^b type attractive interatomic interaction potential.
      Physical Review A, v. 82, 023615 (2010). (PDF)
    36. P.M. Lushnikov. Critical chemotactic collapse.
      Physics Letters A, v. 374, 1678-1685 (2010). (PDF)
    37. P.M. Lushnikov, N. Chen, and M. Alber. Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact.
      Physical Review E, v. 78, p. 061904 (2008). (PDF)
    38. M. Alber, N. Chen, P.M. Lushnikov, and S.A. Newman. Continuous macroscopic limit of a discrete stochastic model for interaction of living cells.
      Physical Review Letters, v. 99, p. 168102 (2007). (PDF)
    39. I. Gabitov, R. Indik, P.M. Lushnikov, L. Mollenauer, and M. Shkarayev. Twin Families of Bisolitons in Dispersion Managed Systems.
      Optics Letters, v. 32, pp. 605-607 (2007). (PDF)
    40. P.M. Lushnikov and H.A.Rose. How much laser power can propagate through fusion plasma?
      Plasma Physics and Controlled Fusion, v. 48, pp. 1501-1513 (2006). (PDF)
    41. M. Alber, N. Chen, T. Glimm and P.M. Lushnikov. Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description.
      Phys. Rev. E, v.73, p.051901 (2006). (PDF)
    42. P.M. Lushnikov, and V.E. Zakharov. On optimal Canonical Variables in the Theory of Ideal Fluid with Free Surface. Physica D, v.203, pp.9-29 (2005). (PDF)
    43. P.M. Lushnikov. Diffusion of optical pulse in dispersion-shifted randomly birefringent optical fibers. Optics Communications, v. 245, pp. 187-192 (2005). (PDF)
    44. P.M. Lushnikov and H.A.Rose. Instability versus equilibrium propagation of laser beam in plasma.
      Physical Review Letters, v.92 (#25), p. 255003 (2004). (PDF)
    45. P.M. Lushnikov. Exactly Integrable Dynamics of Interface between Ideal Fluid and Light Viscous Fluid.
      Physics Letters A, v.329 (#1-2), pp.49-54 (2004). (PDF)
    46. P.M. Lushnikov. Oscillating tails of dispersion-managed soliton. JOSA B, v.21, 1913 (2004). (PDF)
    47. P.M. Lushnikov. Collapse of Bose-Einstein condensate with dipole-dipole interactions. Phys. Rev. A,  v.66 (#5), p. 051601(R) (2002).  (PDF)
    48. P.M. Lushnikov. Fully parallel algorithm for simulating wavelength-division-multiplexed optical fiber systems. Optics Letters, v.27 (#11), pp.939-941 (2002). (PDF)
    49. M. Chertkov, I. Gabitov, P. Lushnikov, J. Moeser, Z. Toroczkai. Pinning method of pulse confinement in optical fiber with random dispersion. J. of the Optical Society of America B, v.19 (#11), pp. 2538-2550 (2002). (PDF)
    50. I.R. Gabitov and P. M. Lushnikov.   Nonlinearity management in dispersion managed system. Optics Letters, v.27 (#2), pp.113-115 (2002).  (PostScript) (PDF)
    51. P.M. Lushnikov.   Dispersion-managed soliton in a strong dispersion map limit. Optics Letters,  v.26 (#20), pp. 1535-1537 (2001).  (PostScript) (PDF)
    52. P.M. Lushnikov, and M. Saffman. Collapse in a forced three dimensional nonlinear Schrodinger equation. Phys. Rev. E, v. 62 (#4), pp. 5793-5796 (2000). (PostScript) (PDF)
    53. P.M. Lushnikov. Dispersion-managed soliton in optical fibers with zero average dispersion. Optics Letters, v. 25 (#16), pp. 1144-1146 (2000).  (PostScript) (PDF)
    54. P.M. Lushnikov. On the boundary of the dispersion-managed soliton existence. JETP Letters, v. 72 (#3), pp. 111-114 (2000).  (PostScript) (PDF)
    55. P.M. Lushnikov, and A.V. Mamaev. Spontaneous hexagon formation in photorefractive crystal with a single pump wave. Optics Letters, v. 24 (#21), pp. 1511-1513 (1999). (PostScript) (PDF)
    56. P.M. Lushnikov. Light propagation in photorefractive crystals: from rings to hexagons. Nature (Priroda Magazine of the Russian Academy of Science, in Russian), v.999 (#11), p. 29 (1998).
    57. P.M. Lushnikov, P. Lodahl, and M. Saffman. Transverse modulational instability of counterpropagating quasi-phase-matched beams in a quadratically nonlinear medium.Optics Letters , v. 23(#21) pp. 1650-1652 (1998).  (PostScript) (PDF)
    58. P.M. Lushnikov.  Two mechanisms of surface wave generation: Kelvin-Helmholtz and Miles instabilities Izvestiya,  Atmospheric and Oceanic Physics, v. 34, 370-377 (1998).   (GZippedPostScript) (PDF)
    59. P.M. Lushnikov. Hexagonal optical structures in photorefractive crystals with a feedback mirror JETP,  v. 86(#3), 614-627 (1998).  (PostScript) (PDF)
    60. P.M. Lushnikov. Dynamic criterion for collapse. JETP Letters, v. 62 (#5), pp. 461-467 (1995).  (GZiippedPostScript) (PDF)
    61. E.A. Kuznetsov, and P.M. Lushnikov. Nonlinear theory of the excitation of waves by a  wind  due to the  Kelvin-Helmholtz instability. JETP v. 81, 332-340 (1995).  (GZippedPostScript) (PDF)
       
    Book Chapters
    1. M. Alber, N. Chen, T. Glimm, and P.M. Lushnikov. Two-dimensional Multiscale Model of Cell Motion in a Chemotactic Field, pp. 53-76,
      In Single-Cell-Based Models in Biology and Medicine, Series: Mathematics and Biosciences in Interaction.
      Eds. A.R.A. Anderson, M.A.J. Chaplain, K.A. Rejniak. Birkhauser Verlag Basel/Switzerland (2007). (Link).
    Other publications
    1. P.M. Lushnikov, Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface.
      arXiv:1509.03393(2016)
    2. S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich, Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pade approximation.
      arXiv:1507.02784 (2015)
    3. A.O. Korotkevich, and P.M. Lushnikov. Nonlinear Waves and Singularities in Optics, Hydrodynamics and Plasmas.
      AIP Conf. Proc., v. 1389, p. 684-685 (2011). (PDF)
    4. S.A. Dyachenko, P.M. Lushnikov and N. Vladimirova. Logarithmic-type Scaling of the Collapse of Keller-Segel Equation.
      AIP Conf. Proc., v. 1389, p. 709-712 (2011). (PDF)
    5. A.O. Korotkevich, P.M. Lushnikov and H.A.Rose. Collective stimulated Brillouin scatter.
      arXiv:1105.2094 (2011).
    6. Y. Chung, P.M. Lushnikov, and N. Vladimirova. Collapse Turbulence in Nonlinear Schrodinger Equation.
      AIP Conf. Proc., v. 1168, p. 1235-1238 (2009). (PDF)
    7. I.R. Gabitov, and P.M. Lushnikov. Symposium: Nonlinear Waves and Singularities in Optics, Hydrodynamics and Plasmas.
      AIP Conf. Proc., v. 1168, p. 1217-1218 (2009). (PDF)
    8. J.L. Kline, D.S. Montgomery, H.A. Rose, S.R. Goldman, D.H. Froula, J.S. Ross, R.M. Stevenson, P.M. Lushnikov. Mitigation of stimulated Raman scattering in hohlraum plasmas.
      Journal of Physics: Conf. Series, v. 112, p. 022030 (2008). (PDF)
    9. P.M. Lushnikov and H.A.Rose. Collective stimulated Brillouin backscatter. arXiv:0710.0634 (2007).
    10. P.M. Lushnikov and H.A. Rose. Practical Formula for Laser Intensity at Beam Spray Onset.
      Nuclear Weapons Highlights 2007, p. 70, Los Alamos National Laboratory (2007). (PDF).
    11. P.M. Lushnikov and H.A. Rose. Practical Formula for Laser Intensity at Beam Spray Onset.
      LANL report (2006) (arxiv.org/pdf/physics/0609233).
    12. P.M. Lushnikov. Center for Nonlinear Studies. Research Highlights (2001).
    13. P. M. Lushnikov, and M. Saffman. Collapse and generation of ultrashort optical pulses in a nonlinear optical cavity. Quantum Electronics and Laser Science Conference (QELS 2000). Technical Digest. Postconference Edition.TOPS Vol.40, 7-12 May 2000, San Francisco, CA, USA.
    14. P.M. Lushnikov, and A.V. Mamaev. Hexagonal patterns in photorefractive crystals with feedback for scattered light. Proceedings of "Seventh Topical Meeting on Photorefractive Materials, Effects, and Devices". Elsinore, Denmark, 1999.
    15. P.M.Lushnikov. Dynamical criterion of a collapse in the nonlinear Schroedinger equation. Bulletin of the Russian Academy of Sciences. Physics,  v. 61, suppl., no.1, pp.46-51 (1997).

     

    Link to Pavel Lushnikov's web page at the University of New Mexico

    Last modified: 2021-05-05.

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