On the Orbital Stability of Solitary Waves for the Fourth Order Nonlinear Schrödinger Equation

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 explicitly and by using a numerical approach. These explicit solutions 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 ≈ α4.8 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.