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

The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both ( 1 + 1 ) and ( 2 + 1 ) dimensions. An isolated exact solitary wave solution is found in ( 1 + 1 ) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in ( 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 ) dimensional solitary waves. General, approximate solitary wave solutions in ( 1 + 1 ) and ( 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 ) 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.