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.