# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 1 Introduction  
The solitary wave is a ubiquitous nonlinear dispersive wave form, originally arising in water waves [1; 2; 3] but subsequently found to exist in a wide range of areas, including nonlinear optics [4; 5; 6; 7; 8], plasma physics [9] and biology [10; 11]. A solitary wave is an isolated, most often hump-shaped wavepacket, which generally emerges as the infinite wavelength limit of a (nonlinear) periodic wave solution [2]. A special case of a solitary wave is a soliton, which is a solitary wave solution of a nonlinear dispersive wave equation which is integrable in a Hamiltonian sense through the method of inverse scattering [2; 3]. Solitons exhibit \"clean\" interactions in that \(N\) of them interact with no change of shape or velocity other than a phase shift. Conversely, solitary waves, in general, do not show \"clean\" interactions, with dispersive radiation generated upon collisions. The compact, hump-shaped form of solitary waves allow them to be modelled as particles, especially in their collisions and particularly for the integrable case of solitons for which no radiation is generated on interaction [12]. Solitary waves stem a balance between self-phase modulation (self-focusing) and dispersion (diffraction). As such, they can be generated in nonlinear optical media, such as optical fibres [4; 6] and soft matter [7; 13], for which nonlinear self-effects owing to an intensity-dependent refractive index or self-phase modulation balances diffraction or dispersion. Nematic liquid crystals (NLCs), a family of organic soft matter encompassing optical birefringence and positive uniaxiality in a fluid state with a large degree of orientational order, are an ideal medium in which to excite solitary waves due to the \"huge\" nonlinear response to optical forcing, many orders of magnitude larger than, e.g., in glass fibres [7; 13; 14]. This results in all-optical effects which can be observed at mW powers over millimetre distances [14], rather than the kilometers typical of communication fibres [4]. Since their indisputable demonstration in 2000 [14], there have been extensive studies, both experimental and theoretical/numerical, of nematicons (i.e., solitary waves in NLC) and other optical solitary-type waves in NLC, such as optical vortices--see [7; 8; 15; 16; 17; 18; 19] for reviews of this work. The present paper is a synopsis on the interaction of nematicons andoptical vortices with each other and with regions of refractive index variations in nematic liquid crystal (NLC) samples. The motivation behind such studies is the all-optical control of optical solitary wave trajectories towards applications, such as signal processing and all-optical waveguiding/routing [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Before describing specific cases of solitary wave interactions in planar NLC samples, the equations governing nonlinear optical wavepacket propagation will be summarised to set the examples in context. One of the main difficulties in the theoretical modelling of nematic propagation and dynamics is the lack of any exact general solitary waves solutions of the NLC equations. The only exact solutions which exist are isolated ones for fixed parameter values [31], which are not suitable for modelling the general evolution of light beams in nematic liquid crystals. While the NLC equations can be solved using computational methods, analytical solutions give insight into the dynamics which are not available from numerical solutions. One powerful analytical tool is modulation theory [2], originally based on assuming a slowly varying wavetrain that is an exact solution of the underlying nonlinear dispersive wave equation but with slowly evolving parameters. The basic assumption is that the wavetrain evolves on a slower scale than its wavelength; for slowly varying solitary waves, modulation theory (MT) essentially treats them as particles in a potential [12].