On spectral properties of the Sturm–Liouville operator with power nonlinearity

Nonclassical perturbation approach to a nonlinear multiparameter eigenvalue problem arising in electromagnetics

“…The physical problem is interesting in several aspects. In first place, this is a physically realistic setup, which is standard in different wave propagation problems, see linear TE wave propagation case in Adams [1] and Unger [2], nonlinear TE wave propagation case in previous studies [3][4][5], and nonlinear TE-TE case in Smirnov and Valovik [6] and Valovik [7]. The second point is that the linearized problem (i.e., problem of propagation of a sum of two TE waves in a plane layer with constant permittivity) has solutions (eigenvalues and eigenwaves) in an exceptional case only.…”

“…The main problem, which we call problem  = (𝜶), is to determine such triples (𝜆, A 1 , A 2 ) = ( λ, Â1 , Â2 ) for which there exist (nontrivial) real-valued functions u 𝑗 ≡ u 𝑗 (x; λ, 𝜶, A 1 , A 2 ) satisfying (1)- (5), where u 𝑗 are assumed to be twice continuously differentiable with respect to x ∈ Ī.…”

“…Obviously, problem  is a two-parameter problem, where 𝜆 is a spectral parameter and either A 1 or A 2 is an additional parameter. If either A 1 or A 2 is known, then the other one is determined from (5). We assume that A 1 is an unknown parameter and A 2 is determined by the formula…”

“…with the boundary conditions ( 2)-( 4) and additional condition (5). Thus, problem  0 is to determine such triples (𝜆, A 1 , A 2 ) = ( λ, Ã1 , Ã2 ) for which there exist (nontrivial) real-valued functions u 𝑗 ≡ u 𝑗 ( x; λ) satisfying ( 8) and ( 2)- (5), where u 𝑗 are twice continuously differentiable with respect to x ∈ Ī. We note that if system (8) under conditions (2), (4), and ( 5) is solvable, then problem  0 is also solvable.…”

“…If β𝑗+2 = 0, we obtain propagation problem of two TE waves in the nonlinear waveguide Σ: One of the waves depends harmonically on z, and the second one depends harmonically on 𝑦. Also, these waves are linked by condition (5). If β𝑗 = β𝑗+2 = 0 and relation (10) is valid, we obtain the propagation problem of two TE waves in a waveguide filled with a linear medium.…”