A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations

We introduce a new family of paraxial asymptotic models that approximate the Vlasov–Maxwell equations in non-relativistic cases. This formulation is nth order accurate in a parameter η, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov–Maxwell equation and then because it allows us to choose the model complexity according to the expected accuracy.