Abstract
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration.
Highlights
Let the function Φ( x ) of a real variable x ∈ [0, ∞) be defined by some rather complicated problem.The variable x > 0 can represent, e.g., a coupling constant or concentration of particles
There are a variety of exact solutions: spatial, temporal, dark optical solitons and breathers all follow from the celebrated nonlinear
The method for calculating the critical index α by employing the self-similar root approximants was developed by Gluzman and Yukalov [18]
Summary
Let the function Φ( x ) of a real variable x ∈ [0, ∞) be defined by some rather complicated problem. The key to the success is to introduce the so-called control functions to allow “to sew” the two limit-cases together in the form most natural for each concrete problem [11,12,15,16,17] The example of such an approach is brought up in Appendix B. We think that the idea behind the method of corrected approximants [11,12,16], is the most progressive, since it allows to combine the strength of a few methods together and proceed, in the space of approximations, with piece-wise construction of the approximation sequences, as pointed out recently by Gluzman [16].we present a more expended description of the concept of approximants, applied both to critical and relaxation phenomena, extending the earlier work of Chapter 1 of the book [12]
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