Solutions of the Ginzburg‐Landau Equation of Interest in Shear Flow Transition

Abstract

The Ginzburg-Landau equation may be used to describe the weakly nonlinear 2-dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave-like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier-Stokes equations, describing pulses and fronts of instability in the flow.

Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

“…Stewartson pulse solution is also called a breather since it is additionally periodic in time (also see Holmes [24] or Landman [27]). …”

Slow time-periodic solutions of the Ginzburg-Landau equation

“…The stability of periodic wave solutions (~ = R e itkx+'~t)) is investigated in Stuart and DiPrima [5]; slowly varying waves are studied by Bernhoff [6]. Other types of solutions (bursting solutions, quasiperiodic solutions, homoclinic solutions) are discussed by Hocking and Stewartson [7], Holmes [8], Kramer and Zimmerman [9], Landman [10] and other authors. The equation has been derived by many authors, in various ways.…”

Spatial heteroclinic bifurcation of time periodic solutions to the Ginzburg-Landau equation

“…Thus the number of fixed points is < 4, as expected [18], [26]. Their stability is obtained by studying the sign of the real part of the eigenvalues of J (see, e.g., [26] and [16]).…”

“…reads (see, e.g., [15], [26], [16] The bifurcation is supercritical if R Ra ^3 0 and subcritical if R Rci h < 0 [11], [18], [26]. This property plays an important role in the problem we are considering.…”

“…[22], Van Saarloos and Hohenberg [26] called it a pulse solution. These solutions have been studied by a number of authors (see, e.g., [9], [4], [18], [26] and the references therein). In Sec.…”