Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift

In this paper, we establish the nonlinear orbital stability of the traveling wave solution of deterministic and stochastic delayed reaction-diffusion equation. Employing the deterministic phase shift and establishing a delayed-integral inequality, we obtain the exponential stability of the traveling wave solution for the deterministic delayed reaction-diffusion equation. Applying a stochastic phase shift and time transformation, we then verify that the traveling wave solution of the deterministic equation retain the nonlinear orbital stability when the noise intensity is small enough and the initial value sufficiently closes the traveling wave.

Automated summary

In this paper, the nonlinear orbital stability of the traveling wave solution for deterministic and stochastic delayed reaction-diffusion equation is established. The exponential stability of the traveling wave solution for the deterministic equation is obtained by employing a deterministic phase shift and establishing a delayed-integral inequality. It is verified that the traveling wave solution of the deterministic equation retains the nonlinear orbital stability when the noise intensity is sufficiently small and the initial value sufficiently closes the traveling wave by applying a stochastic phase shift and time transformation.