Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.     

Delayed fluorescence of porphyrins in different media

Lifetimes of TPPS₄ (meso-tetraphenylporphine tetrasulfonate) triplet states were measured for liquid solutions of different acidity and viscosity and as a function of acceptor concentration for different acceptors (bovine serum albumin, tryptophan and furfuryl alcohol). Triplet lifetimes were estimated by monitoring the decay of TPPS₄ delayed fluorescence of E-type. The lifetime of delayed fluorencence depends on the concentration of O₂, since the latter is an effective quencher of the triplet state. The lifetime is shown to be influenced mainly by degree of aggregation state of TPPS₄ and, therefore, by the pH of the solution, decreasing with pH and for each pH remaining constant over a wide range of acceptor concentrations. The monomeric species is found to have the longest triplet lifetime in aqueous phosphate-buffered saline solution at neutral pH, especially when bound to albumin, despite of the low viscosity and protonated nature of the medium.     

Steep sharp-crested gravity waves on deep water

A new type of steep two-dimensional irrotational symmetric periodic gravity wave with local singular point inside the flow domain is revealed on inviscid incompressible fluid of infinite depth. The speed of fluid particles in the vicinity of the crest of these waves is greater than their phase speed. Corresponding particle trajectories provide insight into how gravity waves overturn and break.     

Uniform Expansions of Periodic Solutions for the Third Superharmonic Resonance

A new method of uniform expansions of periodic solutions to ordinary differential equations has recently been proposed to study quasi-harmonic processes in non-linear dynamical systems, in particular, when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter that appears due to descending the amplitudes of harmonics monotonically with increasing their number (this is the condition that the term quasi-harmonic implies). In this paper, the method is generalized for the third superharmonic resonance (when the first and the third harmonics become of the same magnitude) in a harmonically forced oscillator with arbitrary odd polynomial non-linearity.