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Competition between generic and nongeneric fronts inenvelope equations
Physical Review A (1991)
  • James A. Powell, Utah State University
  • A. C. Newell
  • C. K. R. T. Jones
Arguments are presented for understanding the selection of the speed and the nature of the fronts that join stable and unstable states on the supercritical side of first-order phase transitions. It is suggested that from compact support, nonpositive-definite initial conditions, observable front behavior occurs only when the asymptotic spatial structure of a trajectory in the Galilean ordinary differential equation (ODE) corresponds to the most unstable temporal mode in the governing partial differential equation (PDE). This selection criterion distinguishes between a "nonlinear" front, which has its origin in the first-order nature of the bifurcation, and a "linear" front. The nonlinear front has special properties as a strongly heteroclinic trajectory in the ODE and as an integrable trajectory in the PDE. Many of the characteristics of the linear front are obtained from a steepest-descent linear analysis originally due to Kolmogorov, Petrovsky, and Piscounov [Bull. Univ. Moscow, Ser. Int. , Sec. A 1, 1 (1937)]. Its connection with global stability arguments, and in particular with arguments based on a Lyapunov functional where it exists, is pursued. Finally, the point of view and results are compared and contrasted with those of van Saarloos [Phys. Rev. A 37, 211 (1988);39, 6367 (1989)].
  • Competition,
  • envelope equations,
  • generic front
Publication Date
January 1, 1991
Citation Information
J. Powell, A.C. Newell and C.K.R.T. Jones. “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636–3652, 1991.