Considerable work has been dedicated to hyperspectral single image super-resolution to improve the spatial resolution of hyperspectral images and fully exploit their potential. However, most of these methods are supervised and require some data with ground truth for training, which is often non-available. To overcome this problem, we propose a new unsupervised training strategy for the super-resolution of hyperspectral remote sensing images, based on the use of synthetic abundance data. Its first step decomposes the hyperspectral image into abundances and endmembers by unmixing. Then, an abundance super-resolution neural network is trained using synthetic abundances, which are generated using the dead leaves model in such a way as to faithfully mimic real abundance statistics. Next, the spatial resolution of the considered hyperspectral image abundances is increased using this trained network, and the high resolution hyperspectral image is finally obtained by recombination with the endmembers. Experimental results show the training potential of the synthetic images, and demonstrate the method effectiveness.