Hunter–Saxton equation

In mathematical physics, the Hunter–Saxton equation In mathematical physics, the Hunter–Saxton equation is an integrable PDE that arises in the theoretical study of nematic liquid crystals. If the molecules in the liquid crystal are initially all aligned, and some of them are then wiggled slightly, this disturbance in orientation will propagate through the crystal, and the Hunter–Saxton equation describes certain aspects of such orientation waves. In the models for liquid crystals considered here, it is assumed that there is no fluid flow, so that only the orientation of the molecules is of interest.Within the elastic continuum theory, the orientation is described by a field of unit vectors n(x,y,z,t). For nematic liquid crystals, there is no difference between orienting a molecule in the n direction or in the −n direction, and the vector field n is then called a director field.The potential energy density of a director field is usually assumed to be given by the Oseen–Frank energy functional where the positive coefficients α {displaystyle alpha } , β {displaystyle eta } , γ {displaystyle gamma } are known as the elastic coefficients of splay, twist, and bend, respectively. The kinetic energy is often neglected because of the high viscosity of liquid crystals. Hunter and Saxton investigated the case when viscous damping is ignored and a kinetic energy term is included in the model. Then the governing equations for the dynamics of the director field are the Euler–Lagrange equations for the Lagrangian where λ {displaystyle lambda } is a Lagrange multiplier corresponding to the constraint |n|=1.They restricted their attention to 'splay waves' where the director field takes the special form