You can use Mathematica software

Some of the recent methods that present exact solutions to the system of coupled PDEs with integral terms can be applied. In the following link , one of my papers that solve these types of equations. you can send me your system and I will try to solve and sending you the solution.

https://www.researchgate.net/publication/236960182_Soliton_solutions_to_a_few_coupled_nonlinear_wave_equations_by_tanh_method?ev=prf_pub

Article Soliton solutions to a few coupled nonlinear wave equations ...

If you are looking for the closed form solutions, you can try to apply the symmetry methods. For the case of equations involving integral terms, the application of these methods is described e.g. in the book by A.V. Bocharov et al., Symmetries and Conservation Laws of Equations of Mathematical Physics (https://www.researchgate.net/publication/233920333_Symmetries_and_Conservation_Laws_for_Differential_Equations_of_Mathematical_Physics), edited by I. S. Krasil'shchik (https://www.researchgate.net/profile/Iosif_Krasilshchik/) and A. M. Vinogradov (https://www.researchgate.net/profile/A_Vinogradov2/), and in the book by S.V. Meleshko (https://www.researchgate.net/profile/Sergey_Meleshko/), Methods for Constructing Exact Solutions of Partial Differential Equations (http://books.google.com/books?id=nc4RrvrHATkC), and the collective monograph Symmetries of Integro-Differential Equations, edited by Yurii N. Grigoriev, Nail H. Ibragimov, Sergey V. Meleshko, Vladimir F. Kovalev (http://books.google.com/books?id=qIzm0yoDG8QC).

Book Symmetries and Conservation Laws for Differential Equations ...