2024/05/15 by Borluk, Handan, Muslu, Gulcin M., Natali, Fábio · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.09268
In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schrödinger equation with mixed dispersion. The existence of solitary waves can be determined both as minimizers of a constrained complex functional and by using a numerical approach. In addition, for specific values of the frequency associated with the standing wave, one obtains explicit solutions with a hyperbolic secant profile. Despite these explicit solutions being minimizers of the constrained functional, they cannot be seen as a smooth curve of solitary waves, and this fact prevents their determination of stability using classical approaches in the current literature. To overcome this difficulty, we employ a numerical approach to construct a smooth curve of solitary waves. The existence of a smooth curve is useful for showing the existence of a threshold power α0≈ 4.8 of the nonlinear term such that if α∈ (0,α0), the explicit solitary wave is stable, and if α>α0, the wave is unstable. An important feature of our work, caused by the presence of the mixed dispersion term, concerns the fact that the threshold value α0 ≈ 4.8 is not the same as that established for proving the existence of global solutions in the energy space, as is well known for the classical nonlinear Schrödinger equation.