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Collisionless kinetic theory for parametrically pumped magnons

We discuss collisionless kinetic equations describing the non-equilibrium dynamics of magnons in a ferromagnet exposed to an oscillating microwave field. Previously, this problem has been treated within the so-called "S-theory" where the collision integral in the kinetic equation for the magnon distribution is either neglected or taken into account phenomenologically via an effective relaxation time. However, the possibility of magnon condensation has not been included in S-theory. Moreover, the momentum integrations appearing in the magnon self-energies are usually decoupled by retaining only the term where the loop momentum is equal to the external momentum. In this work we critically examine the accuracy of these approximations and develop the proper extensions of S-theory. We show that these extensions can significantly modify the time evolution of the magnon distribution.