Diophantine equations in separated variables and polynomial power sums

We consider Diophantine equations of the shape $$ f(x) = g(y) $$ f ( x ) = g ( y ) , where the polynomials f and g are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions (x, y) with a bounded denominator are only possible in trivial cases.

Expanded (G/G2) Expansion Method to Solve Separated Variables for the 2+1-Dimensional NNV Equation

The traditional (G/G2) expansion method is modified to extend the symmetric extension to the negative power term in the solution to the positive power term. The general traveling wave solution is extended to a generalized solution that can separate variables. By using this method, the solution to the detached variables of the symmetric extended form of the 2+1-dimensional NNV equation can be solved, also the soliton structure and fractal structure of Dromion can be studied well.

On the Dynamic Stability of a Reinforced Concrete Plate, Taking into Account the Material Creep

The problem of the dynamic stability of a reinforced concrete plate armoured in two directions parallel to its edges is considered. To describe the viscoelastic properties of concrete, an integral dependence was adopted with an exponential kernel. The use of this dependence led to a linear differential equation of plate vibration. In addition to the creep of concrete, the work of the reinforcement was taken into account. The solution of the differential equation of vibrations of a plate in the form of a series with separated variables is considered, which satisfies the plate fastening conditions. Differential equations are obtained for the time function by the Bubnov-Galerkin method. The task was to find the main areas of dynamic instability. For this, the critical frequency equation was obtained. The influence of the coefficients entering into the equation of critical frequencies on the position of the main regions of dynamic instability is investigated.