Breather solutions for semilinear wave equations

Abstract

We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations $V(x)u_{tt} - u_{xx} = \Gamma(x) |u|^{p-1} u$ on $\mathbb{R}^2$ for all values of $p\in (1,\infty)$. Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on $V, \Gamma$ beyond the limitations of pure $x$-periodicity. Such an approach requires a detailed analysis of the wave operator acting on time-periodic functions. Hence a generalization of the Floquet-Bloch theory for periodic Sturm-Liouville operators is needed which applies to perturbed periodic operators. For this purpose we develop a suitable functional calculus for the weighted operator $-\frac{1}{V(x)}\frac{\mathrm{d}^2}{\mathrm{d}x^2}$ with an explicit control of its spectral measure. Based on this we prove embedding theorems from the form domain of the wave operator into $L^q$-spaces, which is key to controlling nonlinearities. We complement our existence theory with explicit examples of coefficient functions $V$ and temporal periods $T$ which support breathers.