Breather solutions for semilinear wave equations

TL;DR

The paper establishes the existence of infinitely many breather solutions for the spatially heterogeneous semilinear wave equation $V(x)u_{tt}-u_{xx}=\Gamma(x)|u|^{p-1}u$ on $\mathbb{R}^2$ by marrying a generalized Floquet-Bloch spectral calculus for the weighted operator $L=-\frac{1}{V(x)}\frac{d^2}{dx^2}$ with a variational scheme on an indefinite energy functional $J$. A key innovation is the development of a functional calculus and explicit spectral measure for $L$, enabling sharp $L^p$ embeddings of the associated Hilbert space $\mathcal{H}$ into nonlinear spaces needed to control the nonlinearity. Under a spectral-gap condition and mild assumptions on $V$ and $\Gamma$ (compact perturbations or asymptotically periodic coefficients), breathers arise as ground states of $J|_{\mathcal{M}}$ on a Nehari-Pankov manifold, with strong $H^2$ regularity and pointwise satisfaction of the PDE. The work extends breather existence beyond strictly periodic media and provides explicit coefficient classes and periods that support breathers, combining spectral analysis, concentration-compactness, and variational methods. This framework advances understanding of time-periodic, spatially localized nonlinear waves in heterogeneous media and offers a toolkit for constructing breathers in nonperiodic settings. $

Abstract

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 $p\in (1,\infty)$. Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on $V, Γ$ 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 $-\frac{1}{V(x)}\frac{\mathrm{d}^2}{\mathrm{d}x^2}$ 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.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Proposition 2.6