The Operator method for differential equations solving can be applied in nonlinear dynamics for exact solutions finding. Starting from basics of this method and Hankel matrices ranks possibilities of evaluation of DE and special PDE solutions using MAPLE mathematical software will be introduced. Examples of solved ODE and PDE: Huxley, Liouville, KdV equations and their soliton solutions will be presented. It will be shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This result may lead to important findings in a variety of practical applications as nonlinear evolution equations in mathematical physics.

Joint work with Research Group for Mathematical and Numerical Analysis of Dynamical Systems https://nonlinear.fmf.ktu.lt/index.htm. (The talk will be in English)

Not just nonlinear systems, but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequences l2l2 displays chaotic dynamics. We give an outline of the proof starting from Devaney's definition of chaos. Then we construct the corresponding operator on the space of periodic functions and provide its representation involving a principal value integral. We explicitly calculate its eigenfunctions, as well as its periodic points; and also provide examples of chaotic and unbounded trajectories. Joint work with Tamás Kalmár-Nagy. (The talk will be in English.)

Related papers or presentations: paper1, paper2