We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation ut + uux = 14 ( R x −∞ u 2 x dx− R∞ x ux dx) with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ edD(u0, v0).

Zero curvature formulations, pseudo-potentials, modified versions, “Miura transformations”, and nonlocal symmetries of the Korteweg–de Vries, Camassa–Holm and Hunter–Saxton equations are investigated from an unified point of view: these three equations belong to a two–parameters family of equations “describing pseudo-spherical surfaces”, and therefore their basic integrability properties can be...

A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric. AMS Subject Classification (2000): 37K10, 17A70.

Abstract We consider integrability structures of the generalized Hunter–Saxton equation. obtain Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra higher symmetries, prove existence infinite number cosymmetries order. Further, we give examples employing order to constructing exact...

We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data. 1 Introduction In this paper we investigate the Cauchy problem

Abstract In the article a convergent numerical method for conservative solutions of Hunter–Saxton equation is derived. The based on piecewise linear projections, followed by evolution along characteristics where time step chosen in order to prevent wave breaking. Convergence obtained when proportional square root spatial size, which milder restriction than common CFL condition conservation laws.

We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misio lek as Euler equations on the Virasoro algebra for H 1 α,β-metrics. Our result demonstrates a unive...

A variety of approaches to wavefront sensing and reconstruction are surveyed as they are used in adaptive optics and related applications. These include the Gerchberg-Saxton algorithm; shearing interferometry; and Shack-Hartmann, curvature, and pyramid wavefront sensing. Emphasis is placed on the relevant optics and mathematics, which are developed in some detail for Shack-Hartmann and curvatur...