Shape-Invariant Solitons in Nematic Liquid Crystals: The Influence of Noise
In a numerical study we investigate the influence of noise on the shape-invariant solitons in nematic liquid crystals. We use the modified Petviashvili’s method for finding eigenvalues and eigenfunctions of the evolution partial differential equations, to
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stract In a numerical study we investigate the influence of noise on the shapeinvariant solitons in nematic liquid crystals. We use the modified Petviashvili’s method for finding eigenvalues and eigenfunctions of the evolution partial differential equations, to determine the shape-invariant solitons in a realistic scalar three-dimensional model that includes the highly nonlocal nature of uniaxial nematic liquid crystals. We check the stability of such solutions by propagating them for long distances, without or within the presence of white noise. Without noise, we find them stable. In the presence of noise (added to the medium), we find them breathing, which renders shape-invariant solitons difficult to observe. After prolonged propagation, the noise leads to the dissipation of solitons.
1 Introduction 1.1 Fundamental Solitons Fundamental optical spatial solitons are laser beams that propagate in nonlinear media without changing their transverse profiles [1]. Such shape-invariant solutions commonly appear in (1 C 1)-dimensional [(1 C 1)D] nonlinear systems, especially the ones based on the nonlinear Schrödinger equation (NLSE). In that and other nonlinear evolution partial differential equations (PDEs), the inverse scattering M.R. Beli´c () Texas A&M University at Qatar, P. O. Box 23874, Doha, Qatar e-mail: [email protected] M.S. Petrovi´c Institute of Physics, P. O. Box 57, 11001, Belgrade, Serbia A.I. Strini´c N.B. Aleksi´c Institute of Physics, University of Belgrade, P. O. Box 68, 11001, Belgrade, Serbia R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__20, © Springer International Publishing Switzerland 2014
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theory, formulated to treat such equations, guaranties their existence [2]. However, the situation is more complex in the multidimensional and multicomponent systems. No complete inverse scattering theory is formulated in more than one dimension and even when the localized solutions are found, no rigorous procedure for guaranteeing their stability is established. In fact, wave instability and collapse of solutions are overriding concerns in multidimensional nonlinear systems [3]. Additional compounding difficulties arise in the multicomponent vector models or in the scalar nonlocal models, in which the medium response is driven by the optical field itself. Such are the models describing the generation of solitary waves—nematicons— in nematic liquid crystals (NLCs).
1.2 Nonlocality Nonlocality is an important feature of many nonlinear media. Generically, it refers to a situation where the medium response at a point depends not only on the excitation at that point, but also on the excitation in the region around the point. A highly nonlocal situation arises in a nonlocal nonlinear medium in which the characteristic size of the medium response is much wider than the size of the excitation itself [4,5]. In nematic liquid crystals, both experiments [6,7] and the
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