The existence of solutions to an inhomogeneous higher order differential equation in the Schwartz space

The paper deals with the problem of the existence of solutions to an inhomogeneous linear differential equation of higher even order. The problem arises while studying soliton and soliton-like solutions to partial differential equations of integrable type. The theorem on necessary and sufficient conditions of the existence of solutions to the differential equation in the Schwartz space of rapidly decreasing functions is proved by means of theory of pseudodifferential operators.