On symmetry of traveling solitary waves for dispersion generalized NLS

We consider dispersion generalized nonlinear Schr"odinger equations (NLS) of the form $i{\mathrm{\partial }}_{t}u=P(D)u-\mathrm{\mid }u{\mathrm{\mid }}^{2\sigma }u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $\sigma \in $\mathbb{N}$$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.