One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation

We apply the Lie theory to determine the infinitesimal generators of the one-parameter point transformations which leave invariant the 3 + 1 Kudryashov–Sinelshchikov equation. We solve the classification problem of the one-dimensional optimal system, while we derive all the possible independent Lie invariants; that is, we determine all the independent similarity transformations which lead to different reductions. For an application, the results are applied to prove the existence of travel-wave solutions. Furthermore, the method of singularity analysis is applied where we show that the 3 + 1 Kudryashov–Sinelshchikov equation possess the Painlevé property and its solution can be written by using a Laurent expansion.

Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures

We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectra, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola–Kanai model as a particular case. Explicit solutions are given in both the classical and quantum pictures.

A (Needle) Case in Point: Transformations in the Carpathian Basin During the Early Middle Ages (Late Avar Period, 8th–9th century ad)

In the late Avar period (eighth to ninth century ad), vast quantities of utilitarian artefacts were produced in series in the Carpathian Basin, a phenomenon not seen since the end of the Roman period. The distribution of these articles reflects not only the region's settlement pattern, but also how these artefacts were disseminated. The communication network in the Carpathian Basin underwent a significant transformation between the early and late Avar period: its major nodes, equated with population centres but not necessarily with elite centres, contributed to moulding a social and cultural milieu that included specialized craftsmen. An early single hub in southern Transdanubia was replaced by multiple centres by the late Avar period. Around ad 700, a bipolar settlement pattern emerges in the southern part of the Carpathian Basin. It seems that the Great Hungarian Plain began to play an equal, if not dominant, role in the communication network of the Carpathian Basin at this time.

Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd

Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L1) of the sequence $\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g0. The general result is applied to a family of C1+α-smooth Markov maps in ℝd.

Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.