Hunter–Saxton equation

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.
Physical background
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 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.
Derivation of the Hunter–Saxton equation
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 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
This assumption reduces the Lagrangian to
and then the Euler–Lagrange equation for the angle φ becomes
There are trivial constant solutions φ=φ0
corresponding to states where the molecules in the liquid crystal are
perfectly aligned.
Linearization around such an equilibrium leads to the linear wave equation
which allows wave propagation in both directions with speed
,
so the nonlinear equation can be expected to behave similarly.
In order to study right-moving waves for large t,
one looks for asymptotic solutions of the form
where
Inserting this into the equation, one finds at the order that
A simple renaming and rescaling of the variables
(assuming that )
transforms this into the Hunter–Saxton equation.
Generalization
The analysis was later generalized by Alì and Hunter, who allowed the director field to point in any direction, but with the spatial dependence still only in the x direction:
Then the Lagrangian is
The corresponding Euler–Lagrange equations are coupled nonlinear wave equations for the angles φ and ψ, with φ corresponding to "splay waves" and ψ to "twist waves". The previous Hunter–Saxton case (pure splay waves) is recovered by taking ψ constant, but one can also consider coupled splay-twist waves where both φ and ψ vary. Asymptotic expansions similar to that above lead to a system of equations, which, after renaming and rescaling the variables, takes the form
where u is related to φ and v to ψ.
This system implies that u satisfies
so (rather remarkably) the Hunter–Saxton equation arises in this context too, but in a different way.
Variational structures and integrability
The integrability of the Hunter–Saxton equation, or, more precisely, that of its x derivative
was shown by Hunter and Zheng, who exploited that this equation is obtained from the Camassa–Holm equation
in the "high frequency limit"
Applying this limiting procedure to a Lagrangian for the Camassa–Holm equation, they obtained a Lagrangian
which produces the Hunter–Saxton equation after elimination of v and w from the Euler–Lagrange equations for u, v, w. Since there is also the more obvious Lagrangian
the Hunter–Saxton has two inequivalent variational structures. Hunter and Zheng also obtained a bihamiltonian formulation and a Lax pair from the corresponding structures for the Camassa–Holm equation in a similar way.
The fact that the Hunter–Saxton equation arises physically in two different ways (as shown above) was used by Alì and Hunter to explain why it has this bivariational (or bihamiltonian) structure.
Geometric Formulation
The periodic Hunter-Saxton equation can be given a geometric interpretation as the geodesic equation on an infinite-dimensional Lie group, endowed with an appropriate Riemannian metric. In more detail, consider the group of diffeomorphisms of the unit circle . Choose some and denote by the subgroup of consisting diffeomorphisms which fix :
The group is an infinite-dimensional Lie group, whose Lie algebra consists of vector fields on which vanish at :
Here is the standard coordinate on . Endow with the homogeneous inner product:
where the subscript denotes differentiation. This inner product defines a right-invariant Riemannian metric on (on the full group this is only a semi-metric, since constant vector fields have norm 0 with respect to . Note that is isomorphic to the right quotient of by the subgroup of translations, which is generated by constant vector fields).
Let
be a time-dependent vector field on such that for all , and let be the flow of , i.e. the solution to:
Then is a periodic solution to the Hunter-Saxton equation if and only if the path is a geodesic on with respect to the right-invariant metric.
In the non-periodic case, one can similarly construct a subgroup of the group of diffeomorphisms of the real line, with a Riemannian metric whose geodesics correspond to non-periodic solutions of the Hunter-Saxton equation with appropriate decay conditions at infinity.
Notes
References
Further reading
Mathematical physics
Solitons
Partial differential equations
Equations of fluid dynamics