On the connection of solutions of the Malyuzhinets equations and the high-order functional difference equation with meromorphic coefficients in the problem of localized waves propagating along the angular junction of thin elastic membranes

The paper studies the connection of the Kontorovich–Lebedev and Sommerfeld integral representations for solving the problem of localized acoustic waves propagating along the contact line of the angular junction of thin elastic membranes. The construction of solutions in the form of the Kontorovich–Lebedev integral is reduced to solving a sixth-order functional difference equation with a meromorphic potential of a special kind. On the other hand, explicit formulas (i.e. in quadratures) are obtained using Sommerfeld integrals and constructing meromorphic solutions of the Malyuzhinets equations. In this work we establish a connection between solutions of the sixth-order functional difference equation and solutions of the Malyuzhinets equation.