Integration of the soliton hierarchy with mixed sources

We propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the soliton hierarchy with mixed sources which consist of both the sum of square of eigenfunctions and integral over the square of eigenfunctions. Then we determine the evolution equations of scattering data corresponding to the soliton hierarchy with mixed sources and solve the equation in the soliton hierarchy with mixed sources by inverse scattering transformation. The new mKdV hierarchy with mixed sources is used as an example to illustrate the method we proposed, and the nonlinear Schrödinger equation hierarchy with mixed sources, the KdV hierarchy with mixed sources are integrated. This approach can be applied to all other (1+1)-dimensional soliton hierarchies with mixed sources.