We prove that arbitrary (nonpolynomial) scalar evolution equations of order m ≥ 7, that are integrable in the sense of admitting the canonical conserved densities ρ (1) , ρ (2) , and ρ (3) introduced in [1], are polynomial in the derivatives u m−i for i = 0, 1, 2. We also introduce a grading in the algebra of polynomials in u k with k ≥ m − 2 over the ring of functions in x, t, u, . . . , u m−3 and show that integrable equations are scale homogeneous with respect to this grading.

Total Electron Content (TEC) is an important characteristic of the ionosphere relevant to communications. Unpredictable variability of the ionospheric parameters due to various disturbances limits the efficiencies of communications, radar and navigation systems. Therefore forecasting and nowcasting of TEC are important in the planning and operation of Earth-space and satellite-to-satellite communication systems. Near-Earth space processes are complex being highly nonlinear and time varying with random variations in parameters where mathematical modeling is extremely difficult if not impossible. Therefore data driven models such as Neural Network (NN) based models are considered and found promising in modeling such processes. In this paper the NN based METU-NN model is introduced to forecast TEC values for the intervals ranging from 1 to 24 h in advance. Forecast and nowcast of TEC values are also considered based on TEC database. Day-to-day and hourto-hour variability of TEC are also estimated using statistical methods. Another statistical approach based on the clustering technique is developed and a preprocessing approach is demonstrated for the forecast of ionospheric critical frequency foF2.

We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives u k+i = ∂ k+i u/∂x k+i over the ring K (k) of C ∞ functions of u, u 1 , . . . , u k . This grading has the property that the total derivative and the integration by parts with respect to x are filtered algebra maps. In addition, if u satisfies an evolution equation u t = F [u] and F is a level homogeneous differential polynomial, then the total derivative with respect to t, D t , is also a filtered algebra map. Furthermore if ρ is level homogeneous over K (k) , then the top level part of D t ρ depends on u k only. This property allows to determine the dependency of F [u] on u k from the top level part of the conserved density conditions. We apply this structure to the classification of "level homogeneous" scalar evolution equations and we obtain the top level parts of integrable evolution equations of "KdV-type", admitting an unbroken sequence of conserved densities at orders m = 5, 7, 9, 11, 13, 15.

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