On the waveguide with nonlinear insert

Consider a planar metal waveguide with nonlinear insertion in the interval [0,1] along the x -axis. The field u in the waveguide is described by the following problem for the Helmholtz equation with partial radiation conditions. In order to reveal the effects, caused by nonlinearity, we seek approximate solution by the incomplete Galerkin method, restricting ourselves to two modes. Thus, initial problem for partial differentiation equation is reduced to mechanical problem of two bodies with Hamiltonian. Linear conversion of variables allows to separate variables in the Hamiltonian, i.e.bring it to the form.The paper provides calculations, based on derived formulae, and solution plots for given integration constants. Also existence of multivalued solution for some falling amplitudes and existence of solutions corresponding to zero falling amplitudes are proved.