Instability of Optical Solitons in the Boundary Value Problem for a Medium of Finite Extension

Abstract

Abstract.We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.Mathematics Subject Classification (2010). 37K40 (Soliton theory, asymptotic behavior of solutions), 37C70 (Attractors and repellers, topological structure).

Hamiltonian tools for the analysis of optical polarization control

“…This kind of polarization attraction has been also identified in different nonlinear systems [2] and, in particular, in an optical fiber system pumped by two counterpropagating beams [3]. This latter phenomenon has been the subject of a growing interest these last years, from both the theoretical [4][5][6][7][8][9][10][11][12] and experimental [3,[13][14][15][16] points of view. It finds its origin in pioneering studies of polarization dynamics of optical beams that counterpropagate in optical fibers and whose nonlinear interaction is mediated by the Kerr effect [17][18][19][20][21][22][23][24][25][26][27].…”

“…The equations governing the polarization dynamics of the counterpropagating beams can be written in the following general form [8,9,27]:…”

“…The Stokes vectorsS S x ;S y ;S z andJ J x ;J y ;J z describe, respectively, the polarization states of the forward and backward beams on the Poincaré sphere and "×" denotes the vector product. The matrices I s and I i are diagonal and their coefficients depend on the type of fibers considered [8,9]. The radii of the forward and backward spheres, S 0 and J 0 , are the signal and pump powers, which are conserved quantities of the spatiotemporal dynamics.…”

“…The diagonal matrices read I s diag 0; 0; β and I i αdiag 1; −1; −2 , where α cos 2 φ, β 2 sin 2 φ − cos 2 φ, and φ is the ellipticity of the fiber [8]. The stationary system reads 8 < :…”

“…An efficient polarization attraction has been shown to occur in different types of optical fibers, such as isotropic fibers (IFs) [3,5,6,13], highly birefringent spun fibers (HBSF) [8,11], as well as randomly birefringent fibers (RBFs) used in optical telecommunication systems [9,11,14]. It is important to note that these phenomena of polarization attraction exhibit different properties that depend on the characteristics of the considered optical fiber.…”