Breather solutions for semilinear wave equations

We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations V (x)u$_{tt}$ − u$_{xx}$ = Γ(x)|u|$^{p−1}$u on $\mathbb{R}$$^2$ for all values of p ∈ (1, ∞). Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on V, Γ 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 −$\frac{1}{V (x)}$ $\frac{d^2}{dx^2}$ 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.