Analysis of the three-dimensional time-dependent Landau−Ginzburg equation and its solutions

The time-dependent Landau-Ginzburg equation of the form eta t+D Delta eta =P( eta ) where Delta is the three-dimensional Laplacian operator and P( eta ) is an odd-order polynomial up to fifth order, has been used to describe the kinetics of non-conservative order parameters eta at and near criticality. The symmetry reduction method has been applied to solve this equation when P( eta ) is given by P( eta )=a+b eta +c eta 3+d eta 5. This can be used to model both first- and second-order phase transitions which take place with (a not=0) or without (a=0) an external field. When either d not=0 or c not=0, two general cases have been found: (i) a=b=0 or a=b=d=0, where the symmetry group involves translations in (3+1)-dimensional spacetime, rotations in three-dimensional space and a dilation; (ii) otherwise, where the symmetry group involves translations in (3+1)-dimensional spacetime and rotations in three-dimensional space. All the reductions to ODE and the corresponding symmetry variables have been derived. The Painleve-type reduced ODE have been solved exactly while the remaining ones can be analysed numerically or approximately. Physical properties of the obtained solutions have been discussed, including their energies.

[1]  R. Jackiw.  Quantum Meaning of Classical Field Theory , 1977 .

[2]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[3]  A. Gordon.  Tricritical phase transitions in ferroelectrics , 1983 .

[4]  P. Ortoleva.  Symmetry Breaking and Far-from-Equilibrium Order , 1980 .

[5]  K. Parlinski,et al.  On the kinetics of the first order phase transitions , 1981 .

[6]  M. Shimizu.  Itinerant electron metamagnetism , 1982 .

[7]  Takao Ohta,et al.  Kinetic Drumhead Model of Interface. I , 1982 .

[8]  L. Keefe.  Integrability and structural stability of solutions to the Ginzburg–Landau equation , 1986 .

[9]  J. Ross,et al.  Stochastic theory of the kinetics of phase transitions , 1976 .

[10]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[11]  J. Tuszynski,et al.  Exact solutions of the multidimensional classical φ6‐field equations obtained by symmetry reduction , 1987 .

[12]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[13]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[14]  James D. Gunton,et al.  Introduction to the Theory of Metastable and Unstable States , 1983 .

[15]  Y. Ishibashi,et al.  The Structure, the Formation and the Activation Energies of a Domain Wall in Crystals Undergoing the First Order Transition , 1984 .

[16]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[17]  D. W. Rand,et al.  ODEPAINLEVE - A MACSYMA package for painlevé analysis of ordinary differential equations , 1986 .

[18]  J. Tuszynski,et al.  Exact results in the three-dimensional Landau-Ginzburg model of magnetic inhomogeneities in uniaxial ferromagnets: I. Continuous transitions , 1988 .

[19]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[20]  W. Khan.  On the magnetic inhomogeneity based on Landau-Ginzburg theory , 1986 .

[21]  R. T. Sharp,et al.  Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups , 1977 .

[22]  P. Banerjee,et al.  Periodic and Aperiodic Solitary Wave Solutions of the Nonlinear Klein-Gordon Equation without Dispersion , 1988 .

[23]  Paul C. Fife,et al.  Asymptotic states for equations of reaction and diffusion , 1978 .

[24]  Victor Martin-Mayor,et al.  Field Theory, the Renormalization Group and Critical Phenomena , 1984 .

[25]  M. Ablowitz,et al.  A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II , 1980 .

[26]  A. Grundland,et al.  A certain class of solutions of the nonlinear wave equation , 1984 .

[27]  Tormod Riste,et al.  Fluctuations, Instabilities, and Phase Transitions , 1975 .

[28]  G. Derrick.  Comments on Nonlinear Wave Equations as Models for Elementary Particles , 1964 .

[29]  D. Sattinger,et al.  Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics , 1986 .

[30]  T. Geballe,et al.  Long range order in solids , 1979 .

[31]  S. Chan.  Steady‐state kinetics of diffusionless first order phase transformations , 1977 .

[32]  M. Slemrod,et al.  DYNAMICS OF FIRST ORDER PHASE TRANSITIONS , 1984 .

[33]  H. Zassenhaus,et al.  Quantum numbers for particles in de Sitter space , 1976 .

[34]  H. Zassenhaus,et al.  The maximal solvable subgroups of the SU(p,q) groups and all subgroups of SU(2,1) , 1974 .

[35]  R. T. Sharp,et al.  Subgroups of the similitude group of three-dimensional Minkowski space , 1976 .

[36]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[37]  J. Tuszynski,et al.  Symmetry reduction for the three-dimensional Landau-Ginzburg equation☆ , 1988 .

[38]  B. Lévy,et al.  Potential energy surface of H2O+: linear approach , 1978 .

[39]  H. Davis.  Introduction to Nonlinear Differential and Integral Equations , 1964 .

[40]  R. T. Sharp,et al.  Subgroups of the Poincaré group and their invariants , 1976 .