  $\beta$-\ce{Ga2O3} currently stands for the most attractive isomorph of
gallium oxide polymorphs, in view of the increasing number of potential
applications especially in optoelectronics and photonics for solar technology,
as for instance in blind ultraviolet photodetectors solar cells (SBUV). For its
use in these applications, but also in the prospect of predicting novel
possible features, knowledge of its fundamental properties is paramount. Thus
optical, thermal and electronic properties of $\beta$-\ce{Ga2O3} have been
studied expriementally giving evidence of a wide-band, inorganic and
transparent semiconductor with a Kerr nonlinearity. Thermo-optical properties
of the material, probed in photo-thermal SBUV sensing experiments, reveal a
sizable heat diffusion characterized by a temperature gradient along the path
of optical beams, quadratic in beam position and promoting a refractive-index
change with temperature. The experimentally observed Kerr nonlinearity together
with the thermally induced birefringence, point unambiguously to a possible
formation of soliton molecules during propagation of high-intensity fields in
$\beta$-\ce{Ga2O3}. To put this conjecture on a firm ground we propose a
theoretical analysis, based on the cubic nonlinear Schr\"odinger equation in
1+1 spatial dimension, in which thermal lensing creates an effective potential
quadratic in beam position coordinate. Using the non-isospectral
inverse-scattering transform method, the exact one-soliton solution to the
propagation equation is obtained. This solution features a bound state of
entangled pulses forming a soliton molecule, in which pulses are more or less
entangled depending on characteristic parameters of the system.
