Multistability Fronts advancing on metastable states

3 Multistability Fronts advancing on meta-stable states 

Essential to the discussion of the previous Section was the existence of a linearly unstable state. The growth of perturbations on top of it pull the rest of the front structure. We now consider front solutions connecting linearly stable states. This happens in situations of multistability, as the ones mentioned in Sect. 3.1.5. If the initial condition is such that initially different parts of the system approach locally one of the possible stable states, competition between them makes the interfaces between the states to move. In this case any front motion should be pushed by the nonlinearities since the linear approximation predicts that any perturbation is damped and does not propagate. The situation is illustrated in Fig. 4.5. 

Counting arguments dealing with the manifolds of the fixed points of 

These ideas can be made more concrete for the simple case (4.22), for which explicit exact solutions are available. For instance, one can check by substitution that an exact solution of (4.23) joining jf( = — oo) = C2 with ( = 00) = C is  

The simple structure of (4.22) allows additional analytic solutions (Sanati and Saxena, 1998) which may be representative of other bistable systems. For example, the potential analogy in Fig. 4.6 shows that, if dissipation is absent (v = 0, which means that we are looking for a steady solution) there is generically a trajectory which starts at the metastable C, rolls down passing Cu in the direction of C2 but, before arriving, returns back and stops again precisely at C. Such solution, homoclinic to C, has the shape of a pulse as shown 

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