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

Abstract This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation in 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 an explicit family of periodic solutions, expressed in terms of Jacobi elliptic functions, which depend smoothly on the wave speed; this is achieved via the inverse function theorem. The spectral stability of this smooth curve of explicit solutions is then rigorously addressed. We perform a detailed spectral analysis of the linearized operator around the explicit wave with 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 nonpositive spectrum of the projected operator in the space of zero mean periodic functions. Finally, by employing Hamiltonian–Krein index theory, we determine the spectral stability and instability of these explicit waves. We also 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.