<p>Nonlinear optics has extended the reach of integrated photonics to narrow-linewidth lasers, broadband light sources, and highly efficient wavelength converters. However, the dispersion characteristics of these nonlinear photonic devices, which determine the phase-matching conditions of various nonlinear optical processes, remain challenging to modify post-fabrication. Here, we present integrated GeSbS microresonators with dispersion profiles that can be arbitrarily reconfigured. Using intracavity backscattering, we generate standing wave patterns that can imprint or erase Bragg gratings along the microresonator via photosensitivity. The dispersion is synthesized from bidirectional resonances strongly coupled through the Bragg gratings. The tunable dispersion enables the adjustment of parametric oscillator sidebands, the spectral flattening of Kerr microcombs, and the deterministic generation of Brillouin lasers. Our approach allows for arbitrary and dynamic dispersion engineering in integrated photonics, paving the way for scalable and reconfigurable nonlinear devices that can be reprogrammed for multiple applications.</p>