# Modulated travelling wave (MTW), $N_g=1$

This is work in progress

A modulated travelling wave with period $T$ satifies $q(x,t+T) = q(x-s T,t)$. Equivalently, using a moving frame to freeze the wave $\xi=x-st$, it holds that $\tilde q(\xi,t+T) = \tilde q(\xi,t)$ where $\tilde q(\xi,t):=q(\xi+st,t)$. Hence, $\tilde q$ is a periodic solution to

$$$\partial_t\tilde q = -sT\cdot\tilde q+F(\tilde q,p).\tag{eqMWP}$$$

Stability of the MTW is determined by the spectrum of the linearized time-$T$ map associated with (eqMWP).

## References

• Beyn

Beyn and Thümmler, Phase Conditions, Symmetries and PDE Continuation.

• Sandstede

Sandstede, Björn. “Stability of Travelling Waves.” In Handbook of Dynamical Systems, 2:983–1055. Elsevier, 2002. https://doi.org/10.1016/S1874-575X(02)80039-X.