Exact and approximate solutions for optical solitary waves in nematic liquid crystals

•Exact solutions for solitary waves in liquid crystals found.•Bistability of solitary waves in liquid crystals found.•Reasons for stability of solitary waves in liquid crystals identified.•Variational approximations for solitary waves found and their accuracy determined.•The existence of minimum power solitary waves investigated.

The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both (1+1)(1+1) and (2+1)(2+1) dimensions. An isolated exact solitary wave solution is found in (1+1)(1+1) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in (2+1)(2+1) dimensions. These exact solutions are used to elucidate what is meant by a nematic liquid crystal to have a nonlocal response and the full role of this nonlocal response in the stability of (2+1)(2+1) dimensional solitary waves. General, approximate solitary wave solutions in (1+1)(1+1) and (2+1)(2+1) dimensions are found using variational methods and they are found to be in excellent agreement with the full numerical solutions. These variational solutions predict that a minimum optical power is required for a solitary wave to exist in (2+1)(2+1) dimensions, as confirmed by a careful examination of the numerical scheme and its solutions. Finally, nematic liquid crystals subjected to two different external electric fields can support the same solitary wave, exhibiting a new type of bistability.