The influence of the power variation on the evolution of an arbitrary beam in strongly nonlocal nonlinear media is investigated on the basis of the mode-decomposition method. The variation of the power changes the longitudinal positions of the beam patterns and induces the longitudinal scaling. Although there is a one-to-one correspondence between the patterns before and after the power variation, the pattern sizes in the two cases differ from each other; therefore the transverse scaling occurs. The effect of the three-dimensional nonuniform scaling effect can be represented with a simple formula, with which the analytical solution for the beam after the power variation can be readily obtained from its counterpart before the power variation.

We investigate quadratic solitons consisting of fundamental waves and oscillatory second harmonics, which can be stable when subject to boundary confinement. A family of quadratic solitons whose fundamental waves can be monopole, dipole, or even multipole are found numerically. The existence of solitons, depending on the sample size and degree of nonlocality, are given for strongly, generally, and weakly nonlocal cases, respectively.

We introduce a type of shape-invariant rotating beam in linear media with harmonic potentials. This type of beam is constructed by superposing a series of properly selected Laguerre-Gaussian beams of different orders. During propagation, the beam is shape invariant, but the pattern rotates and its size varies periodically. The transverse position of the beam at the exit plane of a medium can be steered by tuning the input parameters, even if the transverse position at the entrance plane is fixed.

Under the Dirichlet boundary conditions, a family of bright quadratic solitons exists in the regime where the second harmonic can be regarded as the refractive index of the fundamental wave with an oscillatory nonlocal response. By simplifying the governing equations into the Snyder-Mitchell mode, the approximate analytical solutions are obtained. Taking them as the initial guess and using a numerical code, we found two branches of bright solitons, of which the beam width increases (branch I) and decreases (branch II) with the increase of the sample size, respectively. If the nonlocality is fixed and the sample size is varied, the soliton width varies piecewise and approximately periodically. In each period, solitons only exist in a small range of sample size. Single-hump fundamental wave solitons with the same beam width in narrower samples can be, if the second harmonics are connected smoothly, jointed to be a multihump soliton in a wider sample whose size is the sum of those for the narrower ones. The dynamical simulation shows that the found solitons are unstable.