In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space Diff1(R) equipped with the homogenous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric. Here Diff1(R) denotes the extension of the group of all either comp...

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).

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.

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 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...

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.