2025/05/19 by Henninger, Julia, Ohrem, Sebastian, Reichel, Wolfgang · 1 citation
#34L05 #49J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35L71 #Secondary: 35B10
paper · doi:10.48550/arxiv.2505.13336
We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations V(x)utt - uxx = Γ(x) |u|p-1 u on ℝ2 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 -(1)/(V(x))(d2)/(dx2) with an explicit control of its spectral measure. Based on this we prove embedding theorems from the form domain of the wave operator into Lq-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.