Stability of TE0 modes in self-defocusing core nonlinear planar waveguides

Localized standing waves in inhomogeneous Schrödinger equations

“…Even though it is similar to the results reported in [31], there is a significant difference where the asymmetric positive solutions are all stable in their existence region. Moreover, [31] only considers positive solutions. Besides determining the instability of symmetric solutions numerically, we also show it analytically using topological argument techniques as developed in [12,13].…”

“…Figure 2 is similar to Fig. 1 in [31], which is for the case of relatively small η > 0. of (2) becomes the same as Eq. (1) in [31].…”

“…1 in [31], which is for the case of relatively small η > 0. of (2) becomes the same as Eq. (1) in [31]. One important difference is that in our case, asymmetric solutions are all (at least numerically) stable in their existence domain.…”

“…Numerically (not shown here) we observed that P = 1 for the asymmetric positive state. If one could show analytically that P = 1 indeed, one would be able to prove the stability using the theorem presented in [31]. Nonetheless, the solution stability is expected and predicted using the method of [19].…”

“…The method was later rigorously justified by Weinstein in [35], and in [9]. Subsequent studies extend the condition for various situations, including inhomogeneous problems [12,30,31,32,33,19] (see also a recent brief review [25] and references therein). An interesting waveguide system was proposed in [31], consisting of a self-focusing Schrödinger equation and a self-defocusing type inhomogeneity with finite length.…”