Elliptically polarised cnoidal waves in a medium with spatial dispersion of cubic nonlinearity

Approximate solutions to a nonintegrable problem of propagation of elliptically polarised waves in an isotropic gyrotropic nonlinear medium, and periodic analogues of multisoliton complexes

“…At the same time among particular solutions found in [5], there are no analogues of both the anti-phase solution (12), in which r ± (t) oscillate at a frequency W + , and of the solutions of a general character in the form of beats r ± (t) at frequencies W ± . Figure 2 shows the dependence of the normalised frequencies W ± '{r - u', r + u'}.…”

“…In the vicinity of this minimum, the degenerate particular solution to (1) exists if Dk + = Dk -= Dk (d) [5]:…”

“…In the general case, the approximate solutions (12) are linear asymptotics of new periodic (at commensurable normal frequencies) solutions, undergoing a transition, by increasing their period, to the analogues of multisoliton complexes [16 - 19], what drastically differs them from the particular solutions found in [5].…”

“…Without going into detail about the approximate solution (3) in the vicinity of the equilibrium point (13), we note only that for the corresponding particular solutions obtained in [5], the amplitudes of the components A ± are proportional to the elliptic Jacobi functions cn( gt, m) and dn( gt, m) [13]. In the linear limit ( m ® 0) their scale factor g and eigenvalues are given by [5] (13) are equal to g and 2g, respectively.…”

“…Thus, Makarov and Petrov [4], on the assumption of a linear relationship between A + and A -, found particular solutions (1) in the form of soliton pairs. Under the condition of the formation of waveguides of the same profile for circularly polarised components of the field, the authors of papers [5,10] obtained the solutions, in which |A ± | are proportional to the Jacobi elliptic functions [13], and arg{A ± } are linearly dependent on z. In [11,12] under the same assumptions, the authors found exact and approximate solutions, where arg{A ± } depend nonlinearly on t and linearly on z.…”