In this paper, we investigate the Cauchy problem for modified Korteweg de Vries-sine-Gordon equation with a Riesz fractional-order derivative containing time-dependent variable coefficients(vcRfmKdV-sG). We show that the inverse scattering transform(IST) method can also be used to obtain a soliton solutions of the vcRfmKdV-sG equation. We have determined the time evolution of the scattering data for the Ablowitz-Kaup-Newell-Segur system associated with the solution of the vcRfmKdV-sG equation. Then, using the solution of the inverse scattering problem with respect to the time-dependent scattering data, we recover the desired solution of the vcRfmKdV-sG equation. For the one- and two-soliton cases, explicit formulas for the solutions of the problem under consideration are derived as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. The results indicate that both the variable coefficients and the fractional order significantly influence the velocity of soliton propagation, while no energy dissipation occurs throughout the entire motion process. This study contributes valuable insights into nonlinear dynamics and soliton behavior, improving the understanding of multifaceted wave interactions in nonlinear fractional equations.
Babajanov et al. (Mon,) studied this question.