Localised nontopological structures: construction of solutions and stability problems

Stability of Spatial Optical Solitons

“…Power dependence P (β) calculated with the help of Eq. (2) for the soliton of the model (1), (5), and (7) is presented in Fig. 2(a), and it is matched with the discrete eigenvalue of the linearized problem (4) shown in Fig.…”

“…(1) and (5). Neglecting the second-order corrections, we find the results for the effective potentials of the linearized eigenvalue problem (4), and…”

“…The study of soliton stability in nonlinear optical materials has a long history, also associated with the study of nonlinear waves in other media such as plasmas and fluids. Stability of one-parameter solitary waves has already been well understood for both fundamental (singlehump and nodeless) solitons [2,3,4,5] and solitons with nodes and multiple humps [6,7]. The pioneering results of N. Vakhitov and A. Kolokolov [2], known these days as the Vakhitov-Kolokolov stability criterion, found their rigorous justification in a general mathematical theory developed by M. Grillakis et al [8].…”

Extraordinary solitons in non-centrally symmetric photorefractive crystals

“…Let φ be the angle between E 0 and b. Then, we find that the hyperbolic case (28) As explained in the preceding section, here the end of the vector n describes a closed curve on a unit sphere. In order to determine the position of this curve, let us find an estimate for its "center."…”

“…For nonintegrable equations, the question of stability is nontrivial and requires separate study in each case [28]. It has been shown [29] that the single component solitons described by Eq.…”