 Solution

Substitution gives $({\dot{X}}^2-c^2)\varphi \prime \prime -(\ddot{X}+\gamma \dot{X})\varphi \prime +{\omega }^2\varphi =0$. Multiply by $\varphi \prime$ and integrate over $\mathbb{R}$; decay kills the first and last integrals, leaving $(\ddot{X}+\gamma \dot{X})\int (\varphi \prime {)}^2=0$. Thus a nonconstant profile requires $X=X_{\infty }+C{e}^{-\gamma t}$ and approaches a fixed position.