Paper detail

Qualitative Aspects of Periodic Traveling Waves for the Sinh-Gordon equation

This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation within a periodic setting. Our investigation proceeds in three main stages. First we establish the existence of periodic solutions for a fixed wave speed and varying periods by applying the mountain pass theorem. Subsequently, for a fixed period, we construct a family of periodic solutions that depend smoothly on the wave speed; this is achieved via the implicit function theorem. The spectral stability of these waves is then rigorously addressed. We perform a detailed spectral analysis of the linearized operator around the wave of fixed period. A key element in this analysis is the monotonicity of the period map, which, when combined with Morse index theory, enables us to fully characterize the non-positive spectrum of the projected operator in the space of zero-mean periodic functions. Finally, by employing the Hamiltonian-Krein index analysis, we determine the spectral stability and instability of the constructed waves. We additionally discuss qualitative aspects of the Cauchy problem associated with the sinh-Gordon equation, including local well-posedness and blow-up phenomena. The former supports a new linearization of the problem, while the latter predicts the spectral instability of the wave.