Traveling Waves and Functional Differential Equations of Pointwise Type. What Is Common?
For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problem, an important class of solutions are traveling wave solutions (soliton solutions). In turn, soliton solutions for finite-difference analogs of the equations of mathematical physics are in one-to-one correspondence with solutions of induced functional differential equations of pointwise type (FDEPT). The presence of a wide range of numerical methods for constructing FDEPT solutions, as well as the existence of appropriate existence and uniqueness theorems for the solution, a continuous dependence on the initial and boundary conditions, the "rudeness" of such equations, allows us to construct soliton solutions for the initial equations of mathematical physics. Within the framework of the presented work, on the example of a problem from the theory of plastic deformation the mentioned correspondence between solutions of the traveling wave type and the solutions of the induced functional differential equation will be demonstrated.