Front Propagation into Metastable States

As explained above, the shape and velocity of a front propagating into a metastable state is governed by the nonlinear (interior) part, and the nonlinear term in the reaction function plays an important role. When a front propagates into a metastable state, only one velocity is possible, in contrast to a front propagating into an unstable state. In the latter case, there exists an infinite number of possible velocities, namely all velocities larger than the linear velocity. To obtain the unique velocity of a front propagating into a metastable state, one has to solve the differential equation in the frame co-moving with the front and find the front shape. Since the differential equation is nonlinear, this is not an easy task. Often, one has to resort to some trial parametric solution and substitute it into the differential equation to calculate the parameter values. As a typical example we consider the reaction term F p) = r(p - Pi)(P2 - p) p - P3), where P P2 Ps- This kinetic term has the steady states p = pi, with i = 1,2, 3. The states p and pj, are stable, while P2 is unstable. To obtain the front profile and velocity, we need to solve the differential equation 

Since the front joins the two stable states and propagates into the metastable state, the boundary conditions are given by 

The derivative p must vanish at /O = /O3 and at /O = pi. A possible candidate for the front solution is p = b(p - p ) p - P3), where d is a constant to be determined. Substituting this solution into (4.22), we obtain a cubic polynomial in p. Setting the coefficients to 0 provides a system of four algebraic equations. The equation for the coefficient of p yields b = /rj2D. With this value, the equations for the 

Finally, we calculate tlie front profile by integrating 

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The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

Cubic reaction term. We deal again with the case considered in (4.22) for a front propagating into a metastable state, i.e., connecting pj and pj. To apply (4.70), where it is assumed that the front connects 0 and 1, we rescale (4.22) by defining the new field u = p - P )/(P2, - P )-Equation (4.22) then reads... 

We consider next a front propagating into a metastable state for the Nagumo equation, F p) = 6 p - e)/o(l - p) p - a). The front velocity in the absence of a cutoff is vq = l/ /2 - a /l. Equation (4.86) yields v Exact analytical results for the front solutions can be obtained if the reaction term F(p) with a cutoff is replaced by a piecewise linear approximation. The dependence of the velocity shift on the cutoff displays good agreement with the results by Brunet and Derrida and Kessler et al. [495]. 

Recent studies have applied improved variational principles to deal with the velocity shift due to a cutoff in the reaction term, both for fronts propagating into unstable and metastable states [284, 36]. They confirm the results by Brunet and Derrida and improve the results by Kessler et al. The variational principle given in (4.68) implies that for any admissible trial function a lower bound for the velocity can be found by (4.64). The trial function for which equality in (4.64) holds diverges at p = 0, and it is convenient to consider trial functions that in addition to the requirements g > 0 and < 0 also satisfy g(0) oo. Such trial functions allow us to obtain accurate lower bounds for the front velocity. We perform a change of variables p = p(s), where i — 1/g, and consider s as the independent variable in (4.68). With this change of variables, the variational principle reads... 

W. van Saarloos, M. van Hecke and R. Holyst, Front propagation into unstable and metastable states in smectic-C liquid crystals Linear and nonlinear marginal-stability analysis, Phys. Rev. E, 52, 1773-1777 (1995). 

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