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For nonlinear parabolic systems of the form \[ \partial_t w(x,t) =\partial_{x}^2 w(x,t) - \nabla V\bigl(w(x,t)\bigr) \,, \] the followingconclusions are proved to hold generically with respect to the potential $V$:every travelling front invading a minimum point of $V$ is bistable, there is nostanding front, every standing pulse is stable at infinity, the profiles ofthese fronts and pulses approach their limits at $\pm\infty$ tangentially tothe eigenspaces corresponding to the smallest eigenvalues of $D^2V$ at thesepoints, these fronts and pulses are robust with respect to small perturbationsof the potential, and the set of their profiles is discrete. These conclusionsare obtained as consequences of generic transversality results for heteroclinicand homoclinic solutions of the differential systems governing the profiles ofsuch fronts and pulses. Among these results, it is proved that, for a genericHamiltonian system of the form \[ \ddot u=\nabla V(u) \,, \] every asymmetrichomoclinic orbit is transverse and every symmetric homoclinic orbit iselementary.

Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems