The Hermite-Gaussian (HG) beam has potentials for many advanced applications due to its distinctive modes and intensity distributions. For example, in optical communications, electron acceleration, nonlinear optics and bio-optical disease detection. These researches are significant in the development of optics, medicine and quantum technology. However, the control of the evolution of HG beams with quadratic phase modulation (QPM) in fractional systems under variable coefficients and potentials has rarely been reported. In this paper, the propagation dynamics of the HG beam with QPM are investigated based on the fractional Schrödinger equation (FSE) under different variable coefficients and potentials by using a split-step Fourier algorithm. In the free space, the focusing position of the beam gets larger as the positive QPM coefficient increases or the Lévy index decreases. The QPM coefficient has little effect on the focusing amplitude as the Lévy index is 2. When the QPM coefficient is negative, the focusing of the beam disappears. Under the combined effect of cosine modulations and QPM, the transmission of the beam oscillates not by the cosine law, it shows one large and one small breathing structure. The positive and negative coefficients of QPM only alter the breathing sequence. The evolution period and width of the beam decrease as the modulation frequency increases. The trajectory of split beams becomes into a parabolic shape under the linear modulation. In the combined influence of linear modulations and QPM, the HG beam exhibits focusing or not focusing. Furthermore, the focusing position and focal plane of the beam decrease as the Lévy index increases. When the Lévy index is small, the beam keeps a straight-line transmission without distortion at a longer distance under the combined effect of the power function modulation and a positive QPM. The transmission of the beam also stabilizes and the beam width gets larger with a negative QPM. Under a linear potential, the splitting of the HG beam disappears with the increase of the linear coefficient and shows a periodic evolution. The propagation trajectory of the beam shows a serrated pattern. The beam is significantly amplified with the addition of a QPM. Additionally, the evolution period of the beam is inversely proportional to the linear coefficient, and the transverse amplitude gets larger as the Lévy index increases. The interference among beams is strong, but it also exhibits an autofocus-defocusing effect under the combined action of a parabolic potential and QPM. In addition, the positive and negative coefficients of QPM only affect the focusing time of the beam. The frequency of focusing increases as the Lévy index and parabolic coefficient rises. These features are significant for applications of optical manipulations and optical focusing.
