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A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local 2+1 dimensional bi-Hamiltonian formulations and consequent 2+1 dimensional hereditary recursion operators.

Integrable couplings of soliton equations by perturbations I: A general theory and application to the KDV hierarchy