Direct and inverse problems for the Kadomtsev–Petviashvili equation in a 2‐D model of a lattice

This study expands the range of previously known physical models for the Korteveg-de Vries (KdV) family of equations. Based on the one-dimensional Fermi-Pasta–Ulam–Tsingou approach, we considered a model of nonlinear oscillations in a thin solid plate with a non-uniform crystal lattice and proved that compression waves propagate in this structure. These waves are described by the Kadomtsev–Petviashvili equation (KP) with variable coefficients, which depend on the lattice structure. We proved that there is a broad class of the KP solutions for which the KP reduces to the KdV equation with variable coefficients. Since the KdV equation in this case is not precisely integrable, we considered the class of piecewise constant coefficients and proved the unique solvability of the Cauchy problem for weak solutions. We proved that the integrals of the solution moments are well-defined. We formulated the inverse problems of determining the coefficients or some its combinations from these integrals and proved that there is no more than one solution to the problems. The results of modeling and numerical solution of the inverse problem confirm all theoretical studies. Computer simulations have shown that using the integrals of the solution moments is the correct method to select input data for the inverse problem. The results obtained demonstrate that developed algorithms restore the complex lattice structures when the initial wave is unknown. The study also solved the problems of detecting small dislocations and structural defects in the crystal lattice. These problems are typical for manufacturing and testing integrated circuits and microchips.