We applied the dielectric function method to solve analytically L-NL-L structure problems with negative Kerr nonlinearity. A damped wave in linear and a periodic standing wave in non-linear media had to be matched at boundaries. We gave a formulation of boundary conditions that did not explicitly include a film thickness. The boundary-value of a dielectric function can be expressed through the constant of non-trivial integral of motion. Using it, one generates a family of matched solutions satisfying boundary conditions. Then arbitrary film thickness can be checked against this family of solutions in search of matches. As a result, all fitted solutions are determined straightforwardly.

In the previous work [1], we applied the scattering-type TM standing wave solution to a slab waveguide with negative Kerr nonlinearity. In this work, we analyzed a waveguide problem with positive Kerr nonlinearity. In our formalism, fields are expressed through the value of a dielectric function and a constant of non-flow integral of motion. We formulated the necessary boundary condition for both boundaries with no reference to a film thickness. The condition binds a dielectric function to have two equal or two different values at boundaries leading to the existence of symmetrically and asymmetrically fit wave pieces. Families of such standing wave solutions were constructed. The satisfactory boundary condition is implemented when we start to apply a film thickness to family data. The family of symmetrically fit wave pieces turned out to have a relatively complex structure due to two singularities in a dielectric function and three regions had to be analyzed separately. To look for matches within the data we presented two simple criteria that completely define the length-form and the number of standing wave solutions within a film. Afterward, a calculation of fitted solutions is straightforward.