We give a description of all polynomial constants of the five-variable Volterra derivation, hence of all polynomial first integrals of its corresponding Volterra system of differential equations. The Volterra system plays a significant role in plasma physics and population biology.
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Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
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This work demonstrates an improved method to simulate long-distance femtosecond pulse propagation in highcontrast nanowaveguides. Different from typical beam propagation methods, the foundational tool here is capable of simulating strong spatiotemporal waveform reshaping and extreme spectral dynamics. Meanwhile, the ability to fully capture effects due to index contrast in the transverse direction is retained, without requiring a decomposition of the electric field in terms of waveguide modes. These simulations can be computationally expensive, however, so cost is reduced in the improved method by considering only the waveguide core. Fields in the cladding are then properly accounted for through a boundary condition suitable for the case of total internal reflection.
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