Stability properties of multidimensional finite-amplitude solitions

Abstract

The existence and stability of three-dimensional envelope solitons in nonlinear media are investigated. In the first part, the scalar exponential nonlinear schr\"odinger equation is considered to demonstrate the main method for a simple example. In the second part, the investigation is based on the vectorial exponential nonlinear Schr\"odinger equation, describing envelope solitons in plasma physics and nonlinear optics. When considering the stability of the stationary spherically symmetric solutions, one finds that (i) for large ${\ensuremath{\eta}}^{2}$ (nonlinear frequency shift) three-dimensional solitons are completely stable; (ii) a stability threshold in ${\ensuremath{\eta}}^{2}$ exists; and (iii) below this threshold, instability occurs. Physically, these results mean that the amplitudes for stable solitons are bounded from below by ${\ensuremath{\eta}}^{2}=0.10125$. The proof is based on a Liapunov functional for stability. The exponential nonlinear Schr\"odinger equation is a quite simple model for plasma cavitons. Therefore in the third part of the paper, the investigation is generalized to the vectorial nonlinear Schr\"odinger equation for the high-frequency electric field and the hydrodynamic equations for the ions. The low-frequency part of the electron density is calculated by a Boltzmann distribution including the ponderomotive and ambipolar potential. Again, this model can be handled analytically. It is physically reasonable and covers the main processes apart from kinetic effects. A Liapunov functional is constructed from the constants of motion; it shows the same definiteness properties as in the previous more simple case. Thus the existence of stable spherically symmetric envelope solitons is proved. In the final part of the paper the existence of stable nonenvelope solitons, described by Korteweg---deVries---type equations, is briefly reviewed.