The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R simeq mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on
the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $mathbb{D}_h$ symmetries are presented.
Several consecutive experiments with specifically built set-up are described. Performing of the consecutive experimental tasks enables possibility to determine Boltzmanns constant $k_mathrm{_B}$. The fluctuations of the voltage $U(t)$ of series of ca
pacitors connected in parallel with a constant resistance are measured. The voltage is amplified 1~million times $Y=10^6$. The amplified voltage $YU(t)$ is applied to a device, which give the voltage mean quared in time $U_2=left<(Y U(t))^2right>/U_0$. This voltage $U_2$ is measured with a multimeter. A series of measurements gives the possibility to determine the Boltzmanns constant from the equipartition theorem $Cleft<U^2 right>=k_mathrm{_B}T$. In order to determine the set-up constant $U_0$ a series of problems connected with Ohms law are given that are addressed to the senior students. For the junior high school students, the basic problem is to analyse the analog mean squaring. The students works are graded in four age groups S, M, L, XL. The last age group contains problems that are for university students (XL category) and include theoretical research of the set-up as an engineering device. This problem is given at the Fifth Experimental Physics Olympiad Day of the Electron, on December 2017 in Sofia, organized by the Sofia Branch of the Union of Physicists in Bulgaria with the cooperation of the Physics Faculty of Sofia University and the Society of Physicists of the Republic of Macedonia, Strumica.
Several consecutive experiments are described with a printed circuit board PCB set-up, especially designed for these experiments. Doing the consecutive experimental tasks opens up possibility to determine the value of electron charge $q_e.$ The fluct
uations of the voltage $U(t)$ should be measured for different illuminations of a photodiode. The voltage is amplified 1 million times $Y=10^6$. The amplified voltage $YU(t)$ is applied to the device, which gives the result of the value of the time averaged square of the voltage $U_mathrm{S}=left<(Y U(t))^2right>/U_0$. This voltage $U_mathrm{S}$ is measured with a multimeter. The series of measurements gives the possibility to determine the $q_e$ using the well known Schottky formula for the spectral density of the current noise $(I^2)_f=2q_eleft<Iright>.$ For the junior high school students, the basic problem is to analyze the analog squaring. Students work is separated and graded in four categories S, M, L, XL divided by age of students. For the last XL categories, the tasks contain problems oriented to physics university education program and include theoretical research of the PCB set-up as an engineering device. This is the problem of EPO6, December 2018 ``Day of the Charge considered. EPO6 is organized by Sofia branch of Union of physicists in Bulgaria in cooperation with Faculty of physics of Sofia University and Society of Physicists of Republic of Macedonia.
We have derived a new system of mKdV-type equations which can be related to the affine Lie algebra $A_{5}^{(2)}$. This system of partial differential equations is integrable via the inverse scattering method. It admits a Hamiltonian formulation and t
he corresponding Hamiltonian is also given. The Riemann-Hilbert problem for the Lax operator is formulated and its spectral properties are discussed.
We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D^(1)_4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered
. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
We have derived a family of equations related to the untwisted affine Lie algebras $A^{(1)}_{r}$ using a Coxeter $mathbb{Z}_{r+1}$ reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. We also give s
ome particular examples and impose additional reductions.
