Institute for Research in Electronics and Applied Physics

Parvez N. Guzdar
Senior Research Scientist

Mailing Address:
Institute for Research in
Electronics and Applied Physics
A.V. Williams Bldg., Rm. 3117
University of Maryland
College Park, Maryland
20742-3511

Phone: (301) 405-1590
Email: guzdar at umd.edu
FAX: (301) 405-1678


Research • Publications


Parvez Guzdar is a Senior Research Scientist in the Institute for Research in Electronics and Applied Physics at the University of Maryland, College Park. He received his doctorate in physics from Gujarat University, India, in 1976. His current research interests are: (1) two-dimensional/three-dimensional nonlinear fluid simulations of fusion, space and magnetospheric plasmas, (2) two-dimensional simulations of Navier-Stokes fluids, (3) two-dimensional modeling of self-focusing of radio waves in high-latitude ionospheric plasmas, and (4) pattern formation in the two-dimensional complex Ginzburg-Landau equations.

Research

b Limit Disruptions in Tokamaks
Kleva, Guzdar

The most serious impediment to the practical utilization of tokamaks as fusion reactors is the limitation on the plasma thermal pressure imposed by disruptions. The ratio b of the thermal pressure to the pressure of the confining magnetic field provides a measure of the efficiency of a magnetic confinement fusion reactor. Operation at high b is very desirable because it yields a large fusion reaction rate relative to the cost of the confining magnetic field. However, experimental attempts to increase the ratio b beyond a critical limit bc have been thwarted by an abrupt, catastrophic loss of confinement. Not only do these disruptions limit b and, therefore, limit the efficiency of a tokamak, but the disruptions themselves can cause serious damage to the reactor. Displayed below are results from 3D MHD simulations performed on the T3E machine at NERSC.

The figure above shows three-dimensional isosurfaces of the pressure as the instability develops along ridges dominantly aligned along the ambient magnetic field.

The figure on the right is a poloidal projection of the pressure that shows the hot plasma fingers as they reach the wall. Our nonlinear simulations of tokamak stability reproduce the salient features of the disruptive loss of confinement. High b toroidal equilibria are linearly unstable to ballooning modes that grow on the pressure gradient on the large R side of the magnetic axis, where R is the major radius of the torus. The convection cells associated with the unstable ballooning modes transport the thermal energy toward the wall at large R in hot plasma ridges whose two-dimensional projection in the poloidal plane resembles fingers. As the hot central plasma is transported out in R in fingers, the average pressure gradient is reduced so that it no longer completely balances the Lorentz force of the confining magnetic field. As a result, there is a small net inward force in R caused by the unbalanced Lorentz force. At lower bthis force generates an axisymmetric flow that opposes the growth of the fingers to the wall at large R, thereby stabilizing the plasma nonlinearly and maintaining confinement. However, as b increases the growth rate of the plasma fingers towards the wall becomes so rapid that there is insufficient time for the self-consistently generated axisymmetric flow to halt their progress before they strike the wall, and confinement is lost.

This research is supported by the U.S. Department of Energy.

High Latitude Ionospheric Modification by Radio Waves
Gondarenko, Guzdar

The artificial generation of large and small-scale ionospheric irregularities using HF-heaters can impact a variety of space and ground-based communications, navigation, and surveillance systems. The understanding of the generation of these irregularities is thus a key objective. One mechanism that can create the irregularities is the thermal self-focusing of the high-powered heater waves. Over the last few years we have developed a 2D nonlinear code which investigates the spatio-temporal development of such irregularities. In the accompanying figure we show the development of the heater wave (left panels), the electron temperature (middle panels) and the plasma density (right panels) at t = 5.6s near the reflection height of a linearly stratified ionosphere. The heater wave generates the irregularities by the process of thermal self-focusing near the critical height and the irregularities, diffuse along the magnetic field lines (vertical direction) producing the field aligned structures. The goal of our studies is to investigate the spectrum of these irregularities for different ionospheric conditions and different polarizations and power density of the heater wave.

This research is supported by the National Science Foundation and the Office of Naval Research.

3D Simulation of the Dynamics of High Latitude Plasma Patches
Gondarenko, Guzdar

The ionospheric plasma at high latitude is known to display varied characteristics. During the phase when the interplanetary magnetic field is southward, large scale plasma patches/blobs, typically hundreds of kilometers, are formed in the cusp region and convect to the polar cap over periods of several hours. These patches are seen to have small scale irregularities, which are typically ten to a hundred times smaller than the size of the patch. We have developed a three-dimensional code for the plasma patch to gain an understanding of this meso-scale structuring. Earlier two-dimensional simulations showed that the structuring would lead to rapid fragmentation of the patch, contrary to observations. Introduction of the third dimension along the earth's field line causes the long wavelength instabilities to be stabilized. Thus this work will hopefully lead to a more complete understanding of the selection of the scalelengths of the structuring in the nonlinear phase and better agreement with observations.

The top figure shows five isosurfaces for the initial patch, with the peak density in red. After about an hour the patch develops irregularities (shown in the lower figure) due to the gradient drift instability and secondary Kelvin-Helmholtz instabilities.
This work is supported by the National Science Foundation.

Novel Algorithms
Guzdar

The study of the nonlinear dynamics of a higher dimensional system represented by one-dimensional and two-dimensional partial differential equations requires fast, efficient algorithms for generating long time-series data sets (for computing suitable thermodynamic averages). The algorithms remove severe time-step restrictions encountered by earlier workers and therefore can be solved on low-end computational platforms, which are readily accessible. One equation that has been solved using such techniques is the Kuramoto-Shivashinsky equation. The accompanying diagram shows the space-time evolution of a scalar field represented by the KS equation. Another equation is the 3D Ginzburg-Landau equation. The stability of 3D scrolls, rings, and spirals has been studied in collaboration with Prof. Edward Ott, Michael Gabbay, and Keeyeol Nam. More recently we have applied these techniques to develop an efficient algorithm for coupled nonlinear Schrodinger equations representing modulation and Raman scattering of laser beams propagating in an optical fiber. This work is in collaboration with Prof. Raj Roy and graduate student Bhaskar Khubchandani.

Publications since 1997
  1. "Large Transport-Induced Operation Limits of Tokamak Plasmas," P. N. Guzdar, R. G. Kleva, P. K. Kaw, R. Singh, B. LaBombard, and M. Greenwald, Phys. Plasmas 14, 020701 (2007).

  2. "Zonal Flow Sawteeth and the Time Period between Edge-Localized Transport Bursts in Tokamaks," Robert G. Kleva and Parvez N. Guzdar, Phys. Plasmas 14, 012303 (2007).

  3. "Two-Dimensional Modeling of Blob Dynamics in Tokamak Edge Plasmas," G. Q. Yu, S. I. Krasheninnikov, and P. N. Guzdar, Phys. Plasmas 13, 0402508 (2006).

  4. "Repetitive Transport Bursts in Simulations of Edge-Localized Modes in Tokamaks," R. G. Kleva and P. N. Guzdar, Phys. Plasmas 13, 072509 (2006).

  5. "Observation of Chaotic Itinerancy in the Light and Carrier Dynamics of a Semiconductor Laser with Optical Feedback," Will Ray, Wing-Shun Lam, Parvez N. Guzdar, and Rajarshi Roy, Phys. Rev. E 73, 026219 (2006).

  6. "Structure and Dynamics of a New Class of Thin Current Sheets," M. I. Sitnov, M. Swisdak, P. N. Guzdar, and A. Runov, J. Geophys. Res. 111, A08204 (2006).

  7. "Simulations of the Scintillation-Producing Irregularities in High-Latitude Plasma Patches," N. A. Gondarenko and P. N. Guzdar, Geophys. Res. Lett. 33, L22107 (2006).

  8. "Nonlinear Three-Dimensional Simulations of Memoscale Structuring by Multiple Drives in High-Latitude Plasma Patches," N. A. Gondarenko and P. N. Guzdar, J. Geophys. Res. 111, A08302 (2006).

  9. "On the Formation of a Plasma Bubble," M. I. Sitnov, P. N. Guzdar, and M. Swisdak, Geophys. Res. Lett. 32, L16103 (2005).

  10. "Intermittency in Two-Dimensional Turbulence with Drag," Y.-K. Tsang, E. Ott, T. Antonsen, and P. Guzdar, Phys. Rev. E 71, 066313 (2005).

  11. "A Theory for the Pressure Pedestal in High (H) Mode Tokamak Discharges," P. N. Guzdar, S. M. Mahajan, and Z. Yoshida, Phys. Plasmas 12, 032502 (2005).

  12. "Coupled Drift-Wave Zonal Flow Model of Turbulent Transport in the Tokamak Edge," R. Singh. V. Tangri, P. K. Kaw, and P. N. Guzdar, Phys. Plasmas 12, 092307 (2005).

  13. "Edge-Localized Modes and Edge Transport in Spherical Tokamaks," R. G. Kleva and P. N. Guzdar, Phys. Plasmas 12, 052518 (2005).

  14. "Gamma Ray Flashes Due to Plasma Processes in the Atmosphere: Role of whistler Waves," G. M. Milikh, P. N. Guzdar, and A. S. Sharma, J. Geophys. Res. 110, A02308 (2005).

  15. "Perfectly Matched Layers for Radio Wave Propagation in Inhomogeneous Magnetized Plasmas," N. A. Gondarenko, P. N. Guzdar, S. L. Ossakow, and P. A. Bernhardt, J. Comp. Phys. 194 (2), 481-504 (2004). Abstract

  16. "Perfectly Matched Layers for Radio Wave Propagation in Inhomogeneous Magnetized Plasmas," N. A. Gondarenko, P. N. Guzdar, S. L. Ossakow, and P. A. Bernhardt, J. Comp. Phys. 194 (2), 481-504 (2004).

  17. "Spectroscopic Measurements of Plasma Rotation and Ion and Neutral Atom Temperatures in the Maryland Centrifugal Experiment," J. Ghosh, R. Elton, H. Griem, A. Case, R. Ellis, A. Hassam, S. Messer and C. Teodorescu, Phys. Plasmas 11, 3813 (2004).

  18. "Edge Transport and the Low-to-High Transition in Tokamaks with D-Shaped Magnetic Flux Surfaces," R. G. Kleva, P. N. Guzdar, and W. Dorland, Phys. Plasmas 11, 4280 (2004).

  19. "Ballooning Mode Stability of a High Beta Tokamak with a q=1 Magnetic Island," R. G. Kleva and P. N. Guzdar, Phys. Plasmas 11, 4983 (2004).

  20. "Low to High Confinement Transition Theory of Finite-Beta Drift-Wave Driven Shear Flow and Its Comparison with Data from DIII-D," P. N. Guzdar, R. G. Kleva, R. J. Groebner, and P. Gohil, Phys. Plasmas 11, 1109 (2004).

  21. "Density and Electric Field Fluctuations Associated with the Gradient Drift Instability in the High-Latitude Ionosphere," N. A. Gondarenko, N. A. and P. N. Guzdar, Geophys. Res. Lett. 31, L11802 (2004).

  22. "Plasma Patch Structuring by the Nonlinear Evolution of the Gradient Drift Instability in the High-Latitude Ionosphere," N. A. Gondarenko and P. N. Guzdar, J. Geophys. Res. 109, A09301 (2004).

  23. "Zonal Flow and Field Generation by Finite Beta Drift Waves: Finite Ion Temperature Effects," P. N. Guzdar, R. G. Kleva, and N. Chakrabarti, Phys. Plasmas 11, 3324 (2004).

  24. "Current-Driven Instabilities in Forced Current Sheets," M. I. Sitnov, A.T.Y. Liu, P. N. Guzdar, and P. H. Yoon, J. Geophys. Res. 109, A03205 (2004).

  25. "A Model of the Bifurcated Current Sheet 2: Flapping Motions," M. I. Sitnov, M. Swisdak, J. F. Drake, P. N. Guzdar, and B. N. Rogers, Geophys. Res. Lett. 31, L09805 (2004).

  26. "Phase Transition-Like Behavior of Magnetospheric Substorms: Global MHD Simulation Results," X. Shao, M. I. Sitnov, A. S. Sharma, K. Papadopoulos, C. C. Goodrich, P. N. Guzdar, G. M. Milikh, M. J. Wiltberger, and J. G. Lyon, J. Geophys. Res. 108 (A1), 1037, doi: 10.1029/2001JA009237 (2003). Abstract

  27. "Hilbert Phase Analysis of the Dynamics of a Semiconductor Laser with Optical Feedback," Wing-Shun Lam, Parvez N. Guzdar, and Rajarshi Roy, Rapid Communications, Phys. Rev. E 67, 025604(R) (2003). PDF

  28. "Structuring of High Latitude Plasma Patches with Variable Drive," N. A. Gondarenko, P. N. Guzdar, J. Sojka, M. David, Geophys. Res. Lett. 30 (4), 1165, doi: 10.1029/2002GL016437 (2003). Abstract

  29. "Using Radio-Induced Aurora to Measure the Horizontal Structure of Ion Layers in the Lower Thermosphere," P. A. Bernhardt, N. A. Gondarenko, P. N. Guzdar, F. T. Djuth, C. A. Tepley, M. P. Sulzer, S. L. Ossakow, D. L. Newman, J. Geophys. Res. 108 (A9), 1336, doi: 10.1029/2002JA009712 (2003). Abstract

  30. "Using GRENOUILLE to Characterize Asymmetric Femtosecond Pulses Undergoing Self- and Cross-Phase Modulation in a Polarization-Maintaining Optical Fiber," Bhaskar Khubchandani, A. Christian Silva, Parvez N. Guzdar, and Rajarshi Roy, Optics Express 11 (23), 3063 (2003). PDF

  31. "A Model of the Bifurcated Current Sheet," M. I. Sitnov, P. N. Guzdar, and M. Swisdak, Geophys. Res. Lett. 30 (13), 1712, doi: 10.1029/2003GL017218 (2003). Abstract

  32. "Effect of Spontaneous Emission Noise and Modulation on Semiconductor Lasers near Threshold with Optical Feedback," Wing-Shun Lam, Parvez N. Guzdar, and Rajarshi Roy, Intl. J. Mod. Phys. B 17 (22-24), 4123 (2003). PDF

  33. "Linear Mode Conversion in Inhomogeneous Magnetized Plasmas During Ionospheric Modification by HF Radio Waves," N. A. Gondarenko, P. N. Guzdar, S. L. Ossakow, and P. A. Bernhardt, J. Geophys. Res. 108 (A12), 1470, doi: 10.1029/2003JA009985 (2003). PDF

  34. "Modification of the Electron Density Profile Near the Upper Hybrid Layer During Radio Wave Heating of the Ionosphere," N. A. Gondarenko, P. N. Guzdar, G. Milikh, and S. L. Ossakow, Geophys. Res. Lett. 29 (10), doi: 10.1029/2002GL014934 (2002). Abstract

  35. "Reconnection Onset in the Tail of Earth's Magnetosphere," M. I. Sitnov, A. S. Sharma, P. N. Guzdar, and P. H. Yoon, J. Seophys. Res. 107 (A9), 1256, doi: 10.029/2001JA009148 (2002). Abstract

  36. "Comparison of a Low- to High-Confinement Transition Theory with Experimental Data from DIII-D," P. N. Guzdar, R. G. Kleva, R. J. Groebner, and P. Gohil, Phys. Rev. Lett. 89 (26), 265004 (2002). PDF

  37. "Influence of Stochasticity on Multiple Four-Wave-Mixing Processes in an Optical Fiber," B. Khubchandani, P. N. Guzdar, R. Roy, Phys. Rev. E 66, 066609 (2002). PDF

  38. "Stabilization of Sawteeth in Tokamaks with Toroidal Flows," Robert G. Kleva and Parvez N. Guzdar, Phys. Plasmas 9 (7), 3013 (2002). PDF

  39. "Toroidal Flow Stabilization of Disruptive High Beta Tokamaks," Robert G. Kleva and Parvez N. Guzdar, Phys. Plasmas 9 (6), 2655 (2002). PDF

  40. "Shear Flow Generation by Drift Waves Revisited," P. N. Guzdar, R. G. Kleva, and Liu Chen, Phys. Plasmas 8, 459 (2001).PDF

  41. "Modeling Ionospheric Absorption Modified by Anomalous Heating during Substorms," G. M. Milikh, X. Shao, E. Burns, P. N. Guzdar, M. Wiltberger, C. C. Goodrich, K. Papadopoulos, A. S. Sharma, J. Lyon, and J. Fedder, Geophys. Res. Lett. 28, 487 (2001).Abstract

  42. "Three-Dimensional MHD Simulations of the Steady-State Magnetosphere with Northward IMF," P. N. Guzdar, X. Shao, M. Wiltberger, C. C. Goodrich, K. Papadopoulos, and J. Lyon, J. Geophys. Res. 106, 271 (2001).Abstract

  43. "Zonal Flow and Zonal Magnetic Field Generation by Finite Beta Drift Waves: A Theory for L-H Transitions in Tokamaks," P. N. Guzdar, R. G. Kleva, A. Das, and P. K. Kaw, Phys. Rev. Lett. 87, 15001 (2001).PDF

  44. "Zonal Flow and Zonal Magnetic Field Generation by Finite Beta Drift Waves and Kinetic Shear Alfven Waves," P. N. Guzdar, R. G. Kleva, A. Das, and P. K. Kaw, Phys. Plasmas 8, 3907 (2001).PDF

  45. "Three-Dimensional MHD Simulations of the Earth's Magnetosphere on Feb. 9-10, 1995 for Northward Interplanetary Magnetic Field and Comparison of the Lobe Field with Geotail Observations," X. Shao, P. N. Guzdar, K. Papadopoulos, C. C. Goodrich, A. S. Sharma, G. M. Milikh, M. J. Wiltberger, and J. G. Lyon, Geophys. Res. Lett. 28, 3835 (2001).Abstract

  46. "Three-dimensional Structuring Characteristics of High Latitude Plasma Patches," N. A. Gondarenko and P. N. Guzdar, J. Geophys. Res. 106, 24,611 (2001).

  47. "Comparing Ground Magnetic Field Perturbations from Global MHD Simulations with Magnetometer Data for the January 10, 1997 Magnetic Storm Event," X. Shao, P. N. Guzdar, G. M. Milikh, K. Papadopoulos, C. C. Goodrich, A. S. Sharma, E. Burns, M. J. Wiltberger, and J. G. Lyon, to appear in Geophys. Res. (2001).

  48. "Gamma Ray Flashes by Plasma Effects in the Middle Atmosphere," P. K. Kaw, G. M. Milikh, A. S. Sharma, P. N. Guzdar, and K. Papadopoulos, Phys. Plasmas 8, 4954 (2001). PDF

  49. "Nonlinear Stability Limit in High b Tokamaks," Robert G. Kleva and Parvez N. Guzdar, Phys. Plasmas 7, 1163 (2000).PDF

  50. "Diffraction Model of Ionospheric Irregularity-Induced Heater-Wave Pattern Detected on the WIND satellite," P. N. Guzdar, N. A. Gondarenko, K. Papadopoulos, G. M. Milikh, A. S. Sharma, P. Rodriguez, Yu. V. Tokarev, Yu. I. Belov, and S. L. Ossakow, Geophys. Res. Lett. 27, 317-320 (2000).Abstract

  51. "Lagrangian Chaos and the Effect of Drag on the Enstrophy Cascade in Two-Dimensional Turbulence," Keeyeol Nam, Edward Ott, Thomas M. Antonsen, Jr., and Parvez N. Guzdar, Phys. Rev. Lett. 84, 5134 (2000).PDF

  52. "Power Spectrum of Passive Scalars in Two-Dimensional Chaotic Flows," G.-C. Yuan, K. Nam, T. M. Antonsen, Jr., E. Ott, and P. N. Guzdar, Chaos 10, 39 (2000).PDF

  53. "The Effects of Finite Lifetime of Passive Scalars and Vorticity on Their Power Spectra," Keeyeol Nam, Thomas Antonsen, Jr., Parvez N. Guzdar, and Edward Ott, Physica A 288, 265-279 (2000).Abstract

  54. "Target Waves in the Complex Ginzburg-Landau Equation," Matthew Hendrey, Keeyeol Nam, Parvez Guzdar, and Edward Ott, Phys. Rev. E 62, 7627 (2000).PDF

  55. "Riometer," G. M. Milikh, X. Shao, E. Burns, P. N. Guzdar, M. Wiltberger, C. C. Goodrich, K. Papadopoulos, A. S. Sharma, J. Lyon, and J. Fedder, Geophys. Res. Lett (2000).
  56. "Collisionless Nonideal Ballooning Modes," Robert G. Kleva and Parvez N. Guzdar, Phys. Plasmas 6, 116-121 (1999). PDF

  57. "Reply to Comment on b Limit Disruptions in Tokamaks," R. G. Kleva and P. N. Guzdar, Phys. Rev. Lett. 82, 5414 (1999).PDF

  58. "k Spectrum of Finite Lifetime Passive Scalars in Lagrangian Chaotic Fluid Flows," Keeyeol Nam, Thomas M. Antonsen, Jr., and Parvez N. Guzdar, Phys. Rev. Lett. 83, 3426 (1999).PDF

  59. "Cross-Field Transport Due to Low-Frequency Oscillations in the Auroral Region: A Three-Dimensional Simulations," Supriya B. Ganguli, Parvez N. Guzdar, Valeriy V. Gavrishchaka, Warren A. Krueger, and Paul E. Blanchard, J. Geophys. Res. 104, 4297 (1999).Abstract

  60. "Spatio-Temporal Development of Filaments Due to Thermal Self-Focusing Instability near the Critical Surface in the Ionosphere," N. A. Gondarenko, P. N. Guzdar, G. M. Milikh, A. S. Sharma, K. Papadopoulos, and S. L. Ossakow, Izv. VUZ Radiof. 42 (7), 670-681 (1999).Abstract

  61. "Excitation of Short-Scale Density Structures by Drift Waves during Ionospheric Heating," S. N. Antani and P. N. Guzdar, Geophys. Res. Lett. 26, 3285-3288 (1999).Abstract

  62. "Gradient Drift Instability in High Latitude Plasma Patches: Ion Inertial Effects," N. A. Gondarenko and P. N. Guzdar, Geophys. Res. Lett. 26, 3345-3348 (1999).Abstract

  63. "Three-Dimensional Simulations of the Ionospheric Plasma Transport in the Presence of the Structured Field-Aligned Flows," Valeriy V. Gavrishchaka, Supriya B. Ganguli, and Parvez N. Guzdar, J. Geophys. Res. 104, 22,511-22,524 (1999).Abstract

  64. "Three-Dimensional Nonlinear Simulations of the Gradient Drift Instability in a High-Latitude Ionosphere," P. N. Guzdar, N. A. Gondarenko, and P. K. Chaturvedi, Radio Sci. 33, 1901-1913 (1998). Abstract

  65. "The Thermal Self-Focusing Instability near the Critical Surface in the High-Latitude Ionosphere," P. N. Guzdar, P. K. Chaturvedi, K. Papadopoulos, and S. L. Ossakow, J. Geophys. Res. 103, 2231-2237 (1998). Abstract

  66. "b Limit Disruptions in Tokamaks," R. G. Kleva and P. N. Guzdar, Phys. Rev. Lett. 80, 3081 (1998). PDF

  67. "The Derivation of Equations for Fluctuations and Transport in Flux-Tube Geometries," J. J. Martinell, P. N. Guzdar, and A. B. Hassam, Phys. Plasmas 5, 1273 (1998). PDF

  68. "Route to Chaos for a Two-Dimensional Externally Driven Flow," R. Braun, F. Feudel, and P. Guzdar, Phys. Rev. E 58, 1927 (1998).PDF

  69. "The Dynamics of Scroll Wave Filaments in the Complex Ginzburg-Landau Equation," Michael Gabbay, Edward Ott, and Parvez N. Guzdar, Physica D 118, 371 (1998).Abstract

  70. "Reconnection of Vortex Filaments in the Complex Ginzburg-Landau Equation," Michael Gabbay, Edward Ott, and Parvez N. Guzdar, Phys. Rev. E 58, 2576 (1998).PDF

  71. "Stability of Spiral Wave Vortex Filaments with Phase Twists," Keeyeol Nam, Edward Ott, Parvez N. Guzdar, and Michael Gabbay, Phys. Rev. E 58, 2580 (1998).PDF

  72. "Turbulence and the Formation of Transport Barriers in Finite Plasma," B. N. Rogers, J. F. Drake, Y. T. Lau, P. N. Guzdar, A. B. Hassam, S. V. Novakovskii, and A. Zeiler, Plasma Phys. Controlled Nucl. Fusion Res., IAEA, Vienna, 1997.

  73. "Charging of Substrates Irradiated by Particle Beams," P. N. Guzdar, A. S. Sharma, and S. K. Guharay, Appl. Phys. Lett. 71, 3302 (1997).PDF

  74. "Motion of Scroll Wave Filaments in the Complex Ginzburg-Landau Equation," Michael Gabbay, Edward Ott, and Parvez N. Guzdar, Phys. Rev. Lett. 78, 2012 (1997).PDF
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