110,179 research outputs found
Non-integrability of a fifth order equation with integrable two-body dynamics
No full textWe consider the fifth order partial differential equation (PDE) u4x,t?5uxxt+4ut+uu5x+2uxu4x?5uu3x?10uxuxx+12uux=0, which is a generalization of the integrable Camassa-Holm equation. The fifth order PDE has exact solutions in terms of an arbitrary number of superposed pulsons, with geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N=2. Numerical simulations show that the pulsons are stable, dominate the initial value problem and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the non-existence of a suitable Lagrangian or bi-Hamiltonian structure, and obtaining negative results from Painlev\'{e} analysis and the Wahlquist-Estabrook method, we assert that the fifth order PDE is not integrable
macounii
No full textGentianopsis virgata subsp. macounii (T. Holm) J.S. PringleGentiana macouniiMacoun's fringed gentian; Macoun's gentian; small fringed gentiangentiane de MacounMargin of Bow River, St.. George's Island, CalgaryOn old sand ba
virgata
No full textGentianopsis virgata subsp. macounii (T. Holm) J.S. PringleMacoun's fringed gentiangentiane de MacounGentiana macouniiScotford Sandhills, 15 miles ne of Fort SaskatchewanOpen, peaty bo
Letter from Holm Bursum to T. B. Wood
No full textBursum describes the business of raising sheep and goats in New Mexico
Stability of stationary solutions for nonintegrable peakon equations
No full textThe Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family.
In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable
A dressing method for soliton solutions of the Camassa-Holm equation
No full textThe soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.18 pages, 2 figure
The r-Camassa-Holm equation: smooth and singular solutions
No full textThis paper introduces the r-Camassa-Holm (r-CH) equation, which describes a
geodesic flow on the manifold of diffeomorphisms acting on the real line
induced by the W1,r metric. The conserved energy is for the problem is given by
the full W1,r norm and the for r = 2, we recover the Camassa-Holm equation. We
compute the Lie symmetries for r-CH and study various symmetry reductions. We
introduce singular weak solutions of the r-CH equation for r >= 2 and
demonstrates their robustness in numerical simulations of their nonlinear
interactions in both overtaking and head-on collisions. Several open questions
are formulated about the unexplored properties of the r-CH weak singular
solutions, including the question of whether they would emerge from smooth
initial conditions.Comment: Revised manuscript after comments from reviewer
<i>P</i> values in Holm-Sidak’s multiple comparison t-test in bounce height.
No full textP values in Holm-Sidak’s multiple comparison t-test in bounce height.</p
Poisson Structures for PDEs Associated with Diffeomorphism Groups
No full textWe study Poisson and Lie-Poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear PDEs. In particular, we provide a precise analytic sense in which the time t map for the Euler equations of an ideal fluid in a region of Rⁿ (or on a smooth compact n-manifold with a boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphisms. The key difficulty in finding a suitable context for that arises from the fact that the integral curves of Euler equations are not differentiable on the Lie algebra of divergence free vector fields of Sobolev class Hs. We overcome this obstacle by utilizing the smoothness that one has in Lagrangian representation and carefully performing a non-smooth Lie-Poisson reduction procedure on the appropriate functional classes.
This technique is generalized to an arbitrary diffeomorphism group possessing a smooth spray. The applications include the Camassa-Holm equation on S¹, the averaged Euler and EPDiff equations on the n-manifold with a boundary. In all cases we prove that time t map is Poisson on the appropriate Lie algebra of Hs vector fields, where s > n/2 + 1 for the Euler equation and s > n/2 + 2 otherwise.</p
Pairwise t-tests using pooled standard deviation and Holm method of adjustment.
No full textPairwise t-tests using pooled standard deviation and Holm method of adjustment.</p
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