Stability of a Traveling Wave on a Saddle-Node Trajectory

For semilinear partial differential equations, we consider the solution in the form of a plane wave traveling with a constant velocity. This solution is determined from an ordinary differential equation. A wave that stabilizes at infinity to equilibria corresponds to a phase trajectory connecting fixed points. The fundamental problem of the possibility of using such solutions in applications is their stability in the linear approximation. The stability problem is solved for a wave that corresponds to a trajectory from a saddle to a node. It is known that the velocity is determined ambiguously in this case. In this paper, a method is indicated for finding the limit of the velocity of stable waves for parabolic and hyperbolic equations, which can easily be implemented numerically.