In this paper, the differential-q-difference 2D Toda lattice is studied. Hirota’s bilinear technique is applied to it. Soliton solutions are constructed through the resulting bilinear form for the differential-q-difference 2D Toda equation.

<abstract><p>In this paper, the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) equations are studied using the sine-cosine method, the tanh-coth method, and the Kudryashov method. As a result, analytical solutions in the form of dark solitons, bright solitons, and periodic wave solutions are obtained. Finally, the dynamic behavior of the solutions is illustrated by choosing the appropriate parameters using 2D and 3D plots. The obtained results show that the proposed methods are straightforward and powerful and can provide more forms of traveling wave solutions, which are expected to be useful for the study of the theory of traveling waves in physics.</p></abstract>

In this work, (3+1)-dimensional Kudryashov-Sinelshchikov equation is investigated by using the sine-cosine method and modification of the truncated expansion method. A variety of exact solutions are obtained.

The object of the study is coal sludge and coal fines of the Shubarkol deposit and the Kuznetsk coal basin (Republic of Kazakhstan) for the production of coal-water fuel, which allows replacing liquid and gaseous expensive products. The resulting fuel (after treatment of coal seams and burial) from industrial waste should not harm the environment, which requires certain economic investments. For crushing coal and coal sludge in the crushing and grinding unit, an electrohydroimpulse device for fine grinding of materials was used, consisting of a control unit with a protection system, a pulse capacitor and a high-voltage generator (capacitor bank capacity 0.75 µF, pulse discharge voltage 15–30 kV, length of the interelectrode distance 7–10 mm). After grinding, fine coal particles rise to the surface of the water, and impurities settle at the bottom of the device, which allows enriching the product (flotation). Surface structures and coal fraction sizes were obtained using a Tescan Mira 3 scanning electron microscope. The main characteristics of coal-water fuel during vortex combustion were: the diameter of the fraction 0–250 microns – 63–74 %, process water – 36–24 %, special additive – 1–2 %. Coal-water fuel is similar to liquid fuel, and when transferring heat-generating plants to combustion of suspension, no significant changes in the design of boilers (units) are required. This makes it easy to mechanize and automate the processes of receiving, feeding and burning fuel, and the vortex combustion technology at a temperature of 950–1,050 °C guarantees fuel efficiency of more than 97 %. The given optimal parameters of electrohydroimpulse technology when introduced into production will allow not only grinding, but also enriching the coal product

The Toda lattice is a non-linear evolution equation describing an infinite system of masses on a line that interacts through an exponential force. The paper analyzes the construction of soliton solution for the q-Toda lattice in the two-dimensional case. For this purpose, the equation of motion is taken and the transformation of the dependent variable is used to convert the nonlinear equation into a bilinear form, which is written as the Hirota polynomial. As one of the most effective methods for constructing multisoliton solutions of integrable nonlinear evolution equations, Hirota method is applicable to a wide class of equations, including nonlinear differential, nonlinear differential-difference equations. Using the Hirota method, the bilinear form was obtained for the two-dimensional q-Toda lattice on the basis of which the q-soliton solution was found. The dynamics of the q-soliton solution for two-dimensional q-Toda lattice is presented. Note that the soliton is conserved due to the equilibrium between the action of the nonlinear environment with dispersion. In addition, the soliton behaves like a particle: does not collapse when interacting with each other or other disturbances, while maintaining the structure and continues to move. This quality has the ability to use when transferring data or information over long distances with virtually no interference. In addition, the study of the Toda lattice and the application to it of different methods in different dimensions allows one to proceed to the understanding of such complex terms as matrix models that can be used to describe different physical systems.