It is pointed out that the higher-order symmetries of the Camassa-Holm (CH) equation are nonlocal and nonlocality poses problems to obtain higher-order conserved densities for this integrable equation (J. Phys. A: Math. Gen. 2005, 38 869-880). This difficulty is circumvented by defining a nolinear hierarchy for the CH equation and an explicit expression is constructed for the nth-order conserve...

In this paper, we study the nonlinear dynamics of an axisymmetric disturbance to the laminar state in non-rotating Poiseuille pipe flows. In particular, we show that the associated Navier–Stokes equations can be reduced to a set of coupled Camassa–Holm type equations. These support inviscid and smooth localized travelling waves, which are numerically computed using the Petviashvili method. In p...

In this paper, we consider a member of an integrable family generalized Camassa–Holm (GCH) equations. We make analysis the point Lie symmetries these equations by using method infinitesimals. derive nonclassical and find new via method, which cannot be obtained symmetry method. employ multiplier to construct conservation laws for GCH Using underlying equation, double reduction is also construct...

and Applied Analysis 3 and in H1 Ω , respectively, observe that H⊥, the orthogonal complement of H in L2 Ω , is {∇p : p ∈ H1 Ω } cf. 11 or 12 . ii We denote P : L2 Ω 3 → H the L2 orthogonal projection, usually referred as Helmholtz-Leray projector, and by A −PΔ the Stokes operator with domain D A H2 Ω 3 ∩ V . Notice that in the case of periodic boundary condition, A −Δ|D A is a selfadjoint posi...

We investigate the problems of exact controllability and asymptotic stabilization of the Camassa-Holm equation in the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up.

We derive the 2-component Camassa–Holm equation and corresponding N = 1 super generalization as geodesic flows with respect to the H metric on the extended Bott-Virasoro and superconformal groups, respectively. Mathematics Subject Classifications (2000): 53A07, 53B50.